CMSA/Tsinghua Math-Science Literature Lecture: Large cardinals and small sets: The AD+ Duality Program
2022-11-09 09:30 - 11:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture

Prof. Hugh Woodin will present a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Wednesday, November 9, 2022
Time: 9:30 – 11:00 am ET
Location: Via Zoom Webinar and Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

Title: Large cardinals and small sets: The AD+ Duality Program
Abstract: The determinacy axiom, AD, was introduced by Mycielski and Steinhaus over 60 years ago as an alternative to the Axiom of Choice for the study of arbitrary sets of real numbers. The modern view is that determinacy axioms concern generalizations of the borel sets, and deep connections with large cardinal axioms have emerged.
The study of determinacy axioms has led to a specific technical refinement of AD, this is the axiom AD+. The further connections with large axioms have in turn implicitly led to a duality program, this is the AD+ Duality Program.
The main open problems here are intertwined with those of the Inner Model Program, which is the central program in the study of large cardinal axioms.
This has now all been distilled into a series of specific conjectures.

Talk chair: Horng-Tzer Yau (Harvard Mathematics & CMSA)
Moderator: Alejandro Poveda Ruzafa (Harvard CMSA)

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

Representation Theory, Calabi–Yau Manifolds, and Mirror Symmetry
2022-11-28 09:00 - 2022-12-01 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Videos are available on the CMSA Youtube Playlist.
On November 28 – Dec 1, 2022, the CMSA hosted a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry.
Organizers: An Huang (Brandeis University) | Siu-Cheong Lau (Boston University) | Tsung-Ju Lee (CMSA, Harvard) | Andrew Linshaw (University of Denver)
Scientific Advisor: Shing-Tung Yau (Harvard, Tsinghua)
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

The conference was held in hybrid format, both in-person and online.
The workshop was partially supported by Simons and NSF Grant DMS-2227199.

Speakers:

Tomoyuki Arakawa (Kyoto)
Thomas Creutzig (Edmonton)
Jonathan Mboyo Esole (Northeastern)
Fei Han (National University of Singapore)
Shinobu Hosono (Gakushuin University)
Flor Orosz Hunziker (Colorado)
Cuipo Jiang (Shanghai)
Shashank Kanade (Denver)
Matt Kerr (Washington University in St. Louis)
Carl Lian (Humboldt-Universität zu Berlin)
Nai-Chung Conan Leung (CUHK)
Ivan Loseu (Yale)
Robert McRae (Tsinghua University)
Anne Moreau (Université Paris-Saclay, Orsay)
Tony Pantev (University of Pennsylvania)
Mauricio Romo (Tsinghua University)
Bailin Song (USTC)
Cumrun Vafa (Harvard University)
Chin-Lung Wang (National Taiwan University)
Weiqiang Wang (Virginia)
Yaping Yang (University of Melbourne)
Shing-Tung Yau (Tsinghua University)
Chenglong Yu (Tsinghua University)
Gufang Zhao (University of Melbourne)

Schedule (Eastern Time)
Schedule (pdf)
11/28 (Monday)

08:30am – 08:55am
Refreshments

08:55am – 09:00am
Opening remarks by Horng-Tzer Yau

09:00am – 09:45am
Shing-Tung Yau*
Title: The Hull-Strominger system through conifold transitions
Abstract: In this talk I discuss the geometry of C-Y manifolds outside of the Kähler regime and especially describe the Hull-Strominger system through the conifold transitions.

10:00am – 10:45am
Chenglong Yu*
Title: Commensurabilities among Lattices in PU(1,n)
Abstract: In joint work with Zhiwei Zheng, we study commensurabilities among certain subgroups in PU(1,n). Those groups arise from the monodromy of hypergeometric functions. Their discreteness and arithmeticity are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of the lattices among them up to conjugation and finite index (commensurability) is not completed. When n=1, it is the commensurabilities of hyperbolic triangles. The cases of n=2 are almost resolved by Deligne-Mostow and Sauter’s commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some higher dimensional Calabi-Yau type varieties instead of complex reflection groups. We obtain some relations and commensurability indices for higher n and also give new proofs for existing pairs in n=2.

11:00am – 11:45am
Thomas Creutzig*
Title: Shifted equivariant W-algebras
Abstract: The CDO of a compact Lie group is a family of VOAs whose top level is the space of functions on the Lie group. Similar structures appear at the intersections of boundary conditions in 4-dimensional gauge theories, I will call these new families of VOAs shifted equivariant W-algebras. I will introduce these algebras, construct them and explain how they can be used to quickly prove the GKO-coset realization of principal W-algebras.

11:45am – 1:30 pm
Lunch

01:30pm – 02:15pm
Cumrun Vafa
Title: Reflections on Mirror Symmetry
Abstract: In this talk I review some of the motivations leading to the search and discovery of mirror symmetry as well as some of the applications it has had.

02:30pm – 03:15pm
Jonathan Mboyo Esole
Title: Algebraic topology and matter representations in F-theory
Abstract: Recently, it was observed that representations appearing in geometric engineering in F-theory all satisfy a unique property: they correspond to characteristic representations of embedding of Dynkin index one between Lie algebras. However, the reason why that is the case is still being understood. In this talk, I will present new insights, giving a geometric explanation for this fact using K-theory and the topology of Lie groups and their classifying spaces. In physics, this will be interpreted as conditions on the charge of instantons and the classifications of Wess-Zumino-Witten terms.

03:15pm – 03:45 pm
Break

03:45pm – 04:30pm
Weiqiang Wang
Title: A Drinfeld presentation of affine i-quantum groups
Abstract: A quantum symmetric pair of affine type (U, U^i) consists of a Drinfeld-Jimbo affine quantum group (a quantum deformation of a loop algebra) U and its coideal subalgebra U^i (called i-quantum group). A loop presentation for U was formulated by Drinfeld and proved by Beck. In this talk, we explain how i-quantum groups can be viewed as a generalization of quantum groups, and then we give a Drinfeld type presentation for the affine quasi-split i-quantum group U^i. This is based on joint work with Ming Lu (Sichuan) and Weinan Zhang (Virginia).

04:45pm – 05:30pm
Tony Pantev
Title: Decomposition, anomalies, and quantum symmetries
Abstract: Decomposition is a phenomenon in quantum physics which converts quantum field theories with non-effectively acting gauge symmetries into equivalent more tractable theories in which the fields live on a disconnected space. I will explain the mathematical content of decomposition which turns out to be a higher categorical version of Pontryagin duality. I will examine how this duality interacts with quantum anomalies and secondary quantum symmetries and will show how the anomalies can be canceled by homotopy coherent actions of diagrams of groups. I will discuss in detail the case of 2-groupoids which plays a central role in anomaly cancellation, and will describe a new duality operation that yields decomposition in the presence of anomalies. The talk is based on joint works with Robbins, Sharpe, and Vandermeulen.

11/29 (Tuesday)

Refreshments

09:00am – 09:45am
Robert MacRae*
Title: Rationality for a large class of affine W-algebras
Abstract: One of the most important results in vertex operator algebras is Huang’s theorem that the representation category of a “strongly rational” vertex operator algebra is a semisimple modular tensor category. Conversely, it has been conjectured that every (unitary) modular tensor category is the representation category of a strongly rational (unitary) vertex operator algebra. In this talk, I will describe my results on strong rationality for a large class of affine W-algebras at admissible levels. This yields a large family of modular tensor categories which generalize those associated to affine Lie algebras at positive integer levels, as well as those associated to the Virasoro algebra.

10:00am – 10:45am
Bailin Song*
Title: The global sections of chiral de Rham complexes on compact Calabi-Yau manifolds
Abstract: Chiral de Rham complex is a sheaf of vertex algebras on a complex manifold. We will describe the space of global sections of the chiral de Rham complexes on compact Calabi-Yau manifolds.

11:00am – 11:45am
Carl Lian*
Title: Curve-counting with fixed domain
Abstract: The fixed-domain curve-counting problem asks for the number of pointed curves of fixed (general) complex structure in a target variety X subject to incidence conditions at the marked points. The question comes in two flavors: one can ask for a virtual count coming from Gromov-Witten theory, in which case the answer can be computed (in principle) from the quantum cohomology of X, or one can ask for the “honest” geometric count, which tends to be more subtle. The answers are conjectured to agree in the presence of sufficient positivity, but do not always. I will give an overview of some recent results and open directions. Some of this work is joint with Alessio Cela, Gavril Farkas, and Rahul Pandharipande.

11:45am – 01:30pm
Lunch

01:30pm – 02:15pm
Chin-Lung Wang
Title: A blowup formula in quantum cohomology
Abstract: We study analytic continuations of quantum cohomology $QH(Y)$ under a blowup $phi: Y to X$ of complex projective manifolds along the extremal ray variable $q^{ell}$. Under $H(Y) = phi^* H(X) oplus K$ where $K = ker phi_*$, we show that (i) the restriction of quantum product along the $phi^*H(X)$ direction, denoted by $QH(Y)_X$, is meromorphic in $x := 1/q^ell$, (ii) $K$ deforms uniquely to a quantum ideal $widetilde K$ in $QH(Y)_X$, (iii) the quotient ring $QH(Y)_X/widetilde K$ is regular over $x$, and its restriction to $x = 0$ is isomorphic to $QH(X)$. This is a joint work (in progress) with Y.-P. Lee and H.-W. Lin.

02:30pm – 03:15pm
Ivan Loseu
Title: Quantizations of nilpotent orbits and their Lagrangian subvarieties
Abstract: I’ll report on some recent progress on classifying quantizations of the algebras of regular functions of nilpotent orbits (and their covers) in semisimple Lie algebras, as well as the classification of quantizations of certain Lagrangian subvarieties. An ultimate goal here is to understand the classification of unitary representations of real semisimple Lie groups.

03:15pm – 03:45pm
Break

03:45pm – 04:30pm
Matt Kerr*
Title: $K_2$ and quantum curves
Abstract: The basic objects for this talk are motives consisting of a curve together with a $K_2$ class, and their mixed Hodge-theoretic invariants.
My main objective will be to explain a connection (recently proved in joint work with C. Doran and S. Sinha Babu) between (i) Hodge-theoretically distinguished points in the moduli of such motives and (ii) eigenvalues of operators on L^2(R) obtained by quantizing the equations of the curves.
By local mirror symmetry, this gives evidence for a conjecture in topological string theory (due to M. Marino, A. Grassi, and others) relating enumerative invariants of toric CY 3-folds to spectra of quantum curves.

04:45pm – 05:30pm
Flor Orosz Hunziker
Title: Tensor structures associated to the N=1 super Virasoro algebra
Abstract: We have recently shown that there is a natural category of representations associated to the N=1 super Virasoro vertex operator algebras that have braided tensor structure. We will describe this category and discuss the problem of establishing its rigidity at particular central charges. This talk is based on joint work in progress with Thomas Creutzig, Robert McRae and Jinwei Yang.

11/30 (Wednesday)

08:30am – 09:00am
Refreshments

09:00am – 09:45am
Tomoyuki Arakawa
Title: 4D/2D duality and representation theory
Abstract: This talk is about the 4D/2D duality discovered by Beem et al. rather recently in physics. It associates a vertex operator algebra (VOA) to any 4-dimensional superconformal field theory, which is expected to be a complete invariant of thl theory. The VOAs appearing in this manner may be regarded as chiralization of various symplectic singularities and their representations are expected to be closely related with the Coulomb branch of the 4D theory. I will talk about this remarkable 4D/2D duality from a representation theoretic perspective.

10:00am – 10:45am
Shashank Kanade
Title: Combinatorics of principal W-algebras of type A
Abstract: The combinatorics of principal W_r(p,p’) algebras of type A is controlled by cylindric partitions. However, very little seems to be known in general about fermionic expressions for the corresponding characters. Welsh’s work explains the case of Virasoro minimal models W_2(p,p’). Andrews, Schilling and Warnaar invented and used an A_2 version of the usual (A_1) Bailey machinery to give fermionic characters (up to a factor of (q)_infty) of some, but not all, W_3(3,p’) modules. In a recent joint work with Russell, we have given a complete set of conjectures encompassing all of the remaining modules for W_3(3,p’), and proved our conjectures for small values of p’. In another direction, characters of W_r(p,p’) algebras also arise as appropriate limits of certain sl_r coloured Jones invariants of torus knots T(p,p’), and we expect this to provide further insights on the underlying combinatorics.

11:00am – 11:45am
Gufang Zhao
Title: Quasimaps to quivers with potentials
Abstract: This talk concerns non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential. The construction borrows ideas from the theory of gauged linear sigma models as well as recent development in shifted symplectic geometry and Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials are discussed. This is based on work in progress, in collaboration with Yalong Cao.

11:45am – 01:30pm
Group Photo, Lunch

01:30pm – 02:15pm
Yaping Yang
Title: Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
Abstract: Let X be a smooth local toric Calabi-Yau 3-fold. On the cohomology of the moduli spaces of certain sheaves on X, there is an action of the cohomological Hall algebra (COHA) of Kontsevich and Soibelman via “raising operators”. I will discuss the “double” of the COHA that acts on the cohomology of the moduli space by adding the “lowering operators”. We associate a root system to X. The double COHA is expected to be the shifted Yangian of this root system. We also give a prediction for the shift in terms of an intersection pairing. We provide evidence of the aforementioned expectation in various examples. This is based on my joint work with M. Rapcak, Y. Soibelman, and G. Zhao

02:30pm – 03:15pm
Fei Han
Title: Graded T-duality with H-flux for 2d sigma models
Abstract: T-duality in string theory can be realised as a transformation acting on the worldsheet fields in the two-dimensional nonlinear sigma model. Bouwknegt-Evslin-Mathai established the T-duality in a background flux for the first time upon compactifying spacetime in one direction to a principal circle by constructing the T-dual maps transforming the twisted cohomology of the dual spacetimes. In this talk, we will describe our recent work on how to promote the T-duality maps of Bouwknegt-Evslin-Mathai in two aspects. More precisely, we will introduce (1) graded T-duality, concerning the graded T-duality maps of all levels of twistings; (2) the 2-dimensional sigma model picture, concerning the double loop space of spacetimes. This represents our joint work with Mathai.

03:15pm – 3:45pm
Break

03:45pm – 04:30pm
Mauricio Romo
Title: Networks and BPS Counting: A-branes view point
Abstract: I will review the countings of BPS invariants via exponential/spectral networks and present an interpretation of this counting as a count of certain points in the moduli space of A-branes corresponding to degenerate Lagrangians.

04:45pm – 05:30pm
Shinobu Hosono
Title: Mirror symmetry of abelian fibered Calabi-Yau manifolds with ρ = 2
Abstract: I will describe mirror symmetry of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces, which have Picard number two. Finding a mirror family over a toric variety explicitly, I observe that mirror symmetry of all related Calabi-Yau manifods arises from the corresponding boundary points, which are not necessarily toric boundary points. Calculating Gromov-Witten invariants up to genus 2, I find that the generating functions are expressed elliptic (quasi-)modular forms, which reminds us the modular anomaly equation found for elliptic surfaces. This talk is based on a published work with Hiromichi Takaki (arXiv:2103.08150).

06:00pm
Banquet @ Royal East Restaurant, 782 Main St, Cambridge, MA 02139

12/1 (Thursday)

08:30am – 09:00am
Refreshments

09:00am – 09:45am
Conan Nai Chung Leung*
Title: Quantization of Kahler manifolds
Abstract: I will explain my recent work on relationships among geometric quantization, deformation quantization, Berezin-Toeplitz quantization and brane quantization.

10:00am – 10:45am
Cuipo Jiang*
Title: Cohomological varieties associated to vertex operator algebras
Abstract: We define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties. As examples, we construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational $C_2$-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. This talk is based on joint work with Antoine Caradot and Zongzhu Lin.

11:00am – 11:45am
Anne Moreau*
Title: Action of the automorphism group on the Jacobian of Klein’s quartic curve
Abstract: In a joint work with Dimitri Markouchevitch, we prove that the quotient variety of the 3-dimensional Jacobian of the plane Klein quartic curve by its full automorphism group of order 336 is isomorphic to the 3-dimensional weighted projective space with weights 1,2,4,7.
The latter isomorphism is a particular case of the general conjecture of Bernstein and Schwarzman suggesting that a quotient of the n-dimensional complex space by the action of an irreducible complex crystallographic group generated by reflections is a weighted projective space.
In this talk, I will explain this conjecture and the proof of our result. An important ingredient is the computation of the Hilbert function of the algebra of invariant theta-functions on the Jacobian.

11:45am – 11:50am
Closing remarks

11:50am
Free discussions and departure

* = Online speaker
CMSA COVID-19 Policies

Third Annual Yip Lecture
2023-02-02 19:00 - 20:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Andrew Strominger will give the Third Annual Yip Lecture on February 2, 2023.
Time: 7:00-8:00 pm ET
Location: Harvard Science Center Hall A

Title: Black Holes: The Most Mysterious Objects in the Universe
Abstract: In the last decade black holes have come to center stage in both theoretical and observational science. Theoretically, they were shown a half-century ago by Stephen Hawking and others to obey a precise but still-mysterious set of laws which imply they are paradoxically both the simplest and most complex objects in the universe. Compelling progress on this paradox has occurred recently. Observationally, they have finally and dramatically been seen in the sky, including at LIGO and the Event Horizon Telescope. Future prospects for progress on both fronts hinge on emergent symmetries occurring near the black holes. An elementary presentation of aspects of these topics and their interplay will be given.
Andrew Strominger is the Gwill E. York Professor of Physics and a senior faculty member at the Black Hole Initiative at Harvard University.
Introduction: Peter Galison (Harvard Physics & Black Hole Initiative)
Moderator: Daniel Kapec (Harvard CMSA)
The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

The previous Yip Lecture featured Avi Loeb (Harvard), who spoke on Extraterrestrial Life.

Special Lectures on Machine Learning and Protein Folding
2023-02-09 15:30 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The CMSA hosted a series of three 90-minute lectures on the subject of machine learning for protein folding.
Thursday Feb. 9, Thursday Feb. 16, & Thursday March 9, 2023, 3:30-5:00 pm ET
Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138 & via Zoom

Speaker: Nazim Bouatta, Harvard Medical School
Abstract: AlphaFold2, a neural network-based model which predicts protein structures from amino acid sequences, is revolutionizing the field of structural biology. This lecture series, given by a leader of the OpenFold project which created an open-source version of AlphaFold2, will explain the protein structure problem and the detailed workings of these models, along with many new results and directions for future research.
Thursday, Feb. 9, 2023

Thursday, Feb. 9, 2023
3:30–5:00 pm ET
Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm space
A brief intro to protein biology. AlphaFold2 impacts on experimental structural biology. Co-evolutionary approaches. Space of ‘algorithms’ for protein structure prediction. Proteins as images (CNNs for protein structure prediction). End-to-end differentiable approaches. Attention and long-range dependencies. AlphaFold2 in a nutshell.

Thursday, Feb. 16, 2023
3:30–5:00 pm ET
Lecture 2: Machine learning for protein structure prediction, Part 2: AlphaFold2 architecture
Turning the co-evolutionary principle into an algorithm: EvoFormer. Structure module and symmetry principles (equivariance and invariance). OpenFold: retraining AlphaFold2 and insights into its learning mechanisms and capacity for generalization. Applications of variants of AlphaFold2 beyond protein structure prediction: AlphaFold Multimer for protein complexes, RNA structure prediction.

Thursday, March 9, 2023
3:30–5:00 pm ET
Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFold
Limitations of AlphaFold2 and evolutionary ML pipelines. OpenFold: retraining AlphaFold2 yields new insights into its capacity for generalization.

Biography: Nazim Bouatta received his doctoral training in high-energy theoretical physics, and transitioned to systems biology at Harvard Medical School, where he received training in cellular and molecular biology in the group of Prof. Judy Lieberman. He is currently a Senior Research Fellow in the Laboratory of Systems Pharmacology led by Prof. Peter Sorger at Harvard Medical School, and an affiliate of the Department of Systems Biology at Columbia, in the group of Prof. Mohammed AlQuraishi. He is interested in applying machine learning, physics, and mathematics to biology at multiple scales. He recently co-supervised the OpenFold project, an optimized, trainable, and completely open-source version of AlphaFold2. OpenFold has paved the way for many breakthroughs in biology, including the release of the ESM Metagenomic Atlas containing over 600 million predicted protein structures.

Chair: Michael Douglas (Harvard CMSA)
Moderators: Farzan Vafa & Sergiy Verstyuk (Harvard CMSA)

Lecture 1: Machine learning for protein structure prediction, Part 1: Algorithm space

Lecture 2: Machine learning for protein structure prediction, Part 2: AlphaFold2 architecture

Lecture 3: Machine learning for protein structure prediction, Part 3: AlphaFold2 limitations and insights learned from OpenFold

Conference on Geometry and Statistics
2023-02-27 09:00 - 2023-03-01 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On Feb 27-March 1, 2023 the CMSA will host a Conference on Geometry and Statistics.
Location: G10, CMSA, 20 Garden Street, Cambridge MA 02138
Organizing Committee:
Stephan Huckemann (Georg-August-Universität Göttingen)
Ezra Miller (Duke University)
Zhigang Yao (Harvard CMSA and Committee Chair)
Scientific Advisors:
Horng-Tzer Yau (Harvard CMSA)
Shing-Tung Yau (Harvard CMSA)
Speakers:

Tamara Broderick (MIT)
David Donoho (Stanford)
Ian Dryden (Florida International University in Miami)
David Dunson (Duke)
Charles Fefferman (Princeton)
Stefanie Jegelka (MIT)
Sebastian Kurtek (OSU)
Lizhen Lin (Notre Dame)
Steve Marron (U North Carolina)
Ezra Miller (Duke)
Hans-Georg Mueller (UC Davis)
Nicolai Reshetikhin (UC Berkeley)
Wolfgang Polonik (UC Davis)
Amit Singer (Princeton)
Zhigang Yao (Harvard CMSA)
Bin Yu (Berkeley)

Moderator: Michael Simkin (Harvard CMSA)

SCHEDULE
Monday, Feb. 27, 2023 (Eastern Time)

8:30 am
Breakfast

8:45–8:55 am
Zhigang Yao
Welcome Remarks

8:55–9:00 am
Shing-Tung Yau*
Remarks

Morning Session Chair: Zhigang Yao

9:00–10:00 am
David Donoho
Title: ScreeNOT: Exact MSE-Optimal Singular Value Thresholding in Correlated Noise
Abstract: Truncation of the singular value decomposition is a true scientific workhorse. But where to Truncate?
For 55 years the answer, for many scientists, has been to eyeball the scree plot, an approach which still generates hundreds of papers per year.
I will describe ScreeNOT, a mathematically solid alternative deriving from the many advances in Random Matrix Theory over those 55 years. Assuming a model of low-rank signal plus possibly correlated noise, and adopting an asymptotic viewpoint with number of rows proportional to the number of columns, we show that ScreeNOT has a surprising oracle property.
It typically achieves exactly, in large finite samples, the lowest possible MSE for matrix recovery, on each given problem instance – i.e. the specific threshold it selects gives exactly the smallest achievable MSE loss among all possible threshold choices for that noisy dataset and that unknown underlying true low rank model. The method is computationally efficient and robust against perturbations of the underlying covariance structure.
The talk is based on joint work with Matan Gavish and Elad Romanov, Hebrew University.

10:00–10:10 am
Break

10:10–11:10 am
Steve Marron
Title: Modes of Variation in Non-Euclidean Spaces
Abstract: Modes of Variation provide an intuitive means of understanding variation in populations, especially in the case of data objects that naturally lie in non-Euclidean spaces. A variety of useful approaches to finding useful modes of variation are considered in several non-Euclidean contexts, including shapes as data objects, vectors of directional data, amplitude and phase variation and compositional data.

11:10–11:20 am
Break

11:20 am–12:20 pm
Zhigang Yao
Title: Manifold fitting: an invitation to statistics
Abstract: While classical statistics has dealt with observations which are real numbers or elements of a real vector space, nowadays many statistical problems of high interest in the sciences deal with the analysis of data which consist of more complex objects, taking values in spaces which are naturally not (Euclidean) vector spaces but which still feature some geometric structure. This manifold fitting problem can go back to H. Whitney’s work in the early 1930s (Whitney (1992)), and finally has been answered in recent years by C. Fefferman’s works (Fefferman, 2006, 2005). The solution to the Whitney extension problem leads to new insights for data interpolation and inspires the formulation of the Geometric Whitney Problems (Fefferman et al. (2020, 2021a)): Assume that we are given a set $Y subset mathbb{R}^D$. When can we construct a smooth $d$-dimensional submanifold $widehat{M} subset mathbb{R}^D$ to approximate $Y$, and how well can $widehat{M}$ estimate $Y$ in terms of distance and smoothness? To address these problems, various mathematical approaches have been proposed (see Fefferman et al. (2016, 2018, 2021b)). However, many of these methods rely on restrictive assumptions, making extending them to efficient and workable algorithms challenging. As the manifold hypothesis (non-Euclidean structure exploration) continues to be a foundational element in statistics, the manifold fitting Problem, merits further exploration and discussion within the modern statistical community. The talk will be partially based on a recent work Yao and Xia (2019) along with some on-going progress. Relevant reference:https://arxiv.org/abs/1909.10228

12:20–1:50 pm
12:20 pm Group Photo
followed by Lunch

Afternoon Session Chair: Stephan Huckemann

1:50–2:50 pm
Bin Yu*
Title: Interpreting Deep Neural Networks towards Trustworthiness
Abstract: Recent deep learning models have achieved impressive predictive performance by learning complex functions of many variables, often at the cost of interpretability. This lecture first defines interpretable machine learning in general and introduces the agglomerative contextual decomposition (ACD) method to interpret neural networks. Extending ACD to the scientifically meaningful frequency domain, an adaptive wavelet distillation (AWD) interpretation method is developed. AWD is shown to be both outperforming deep neural networks and interpretable in two prediction problems from cosmology and cell biology. Finally, a quality-controlled data science life cycle is advocated for building any model for trustworthy interpretation and introduce a Predictability Computability Stability (PCS) framework for such a data science life cycle.

2:50–3:00 pm
Break

3:00-4:00 pm
Hans-Georg Mueller
Title: Exploration of Random Objects with Depth Profiles and Fréchet Regression
Abstract: Random objects, i.e., random variables that take values in a separable metric space, pose many challenges for statistical analysis, as vector operations are not available in general metric spaces. Examples include random variables that take values in the space of distributions, covariance matrices or surfaces, graph Laplacians to represent networks, trees and in other spaces. The increasing prevalence of samples of random objects has stimulated the development of metric statistics, an emerging collection of statistical tools to characterize, infer and relate samples of random objects. Recent developments include depth profiles, which are useful for the exploration of random objects. The depth profile for any given object is the distribution of distances to all other objects (with P. Dubey, Y. Chen 2022).
These distributions can then be subjected to statistical analysis. Their mutual transports lead to notions of transport ranks, quantiles and centrality. Another useful tool is global or local Fréchet regression (with A. Petersen 2019) where random objects are responses and scalars or vectors are predictors and one aims at modeling conditional Fréchet means. Recent theoretical advances for local Fréchet regression provide a basis for object time warping (with Y. Chen 2022). These approaches are illustrated with distributional and other data.

4:00-4:10 pm
Break

4:10-5:10 pm
Stefanie Jegelka
Title: Some benefits of machine learning with invariances
Abstract: In many applications, especially in the sciences, data and tasks have known invariances. Encoding such invariances directly into a machine learning model can improve learning outcomes, while it also poses challenges on efficient model design. In the first part of the talk, we will focus on the invariances relevant to eigenvectors and eigenspaces being inputs to a neural network. Such inputs are important, for instance, for graph representation learning. We will discuss targeted architectures that can universally express functions with the relevant invariances – sign flips and changes of basis – and their theoretical and empirical benefits.
Second, we will take a broader, theoretical perspective. Empirically, it is known that encoding invariances into the machine learning model can reduce sample complexity. For the simplified setting of kernel ridge regression or random features, we will discuss new bounds that illustrate two ways in which invariances can reduce sample complexity. Our results hold for learning on manifolds and for invariances to (almost) any group action, and use tools from differential geometry.
This is joint work with Derek Lim, Joshua Robinson, Behrooz Tahmasebi, Lingxiao Zhao, Tess Smidt, Suvrit Sra, and Haggai Maron.

Tuesday, Feb. 28, 2023 (Eastern Time)

8:30-9:00 am
Breakfast

Morning Session Chair: Zhigang Yao

9:00-10:00 am
Charles Fefferman*
Title: Lipschitz Selection on Metric Spaces
Abstract: The talk concerns the problem of finding a Lipschitz map F from a given metric space X into R^D, subject to the constraint that F(x) must lie in a given compact convex “target” K(x) for each point x in X. Joint work with Pavel Shvartsman and with Bernat Guillen Pegueroles.

10:00-10:10 am
Break

10:10-11:10 am
David Dunson
Title: Inferring manifolds from noisy data using Gaussian processes
Abstract: In analyzing complex datasets, it is often of interest to infer lower dimensional structure underlying the higher dimensional observations. As a flexible class of nonlinear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the original data with lower dimensional coordinates without providing an estimate of the manifold in the observation space or using the manifold to denoise the original data. This article proposes a new methodology for addressing these problems, allowing interpolation of the estimated manifold between fitted data points. The proposed approach is motivated by novel theoretical properties of local covariance matrices constructed from noisy samples on a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing application of Gaussian processes for probabilistic manifold reconstruction. In addition to theory justifying the algorithm, we provide simulated and real data examples to illustrate the performance. Joint work with Nan Wu – see https://arxiv.org/abs/2110.07478

11:10-11:20 am
Break

11:20 am-12:20 pm
Wolfgang Polonik
Title: Inference in topological data analysis
Abstract: Topological data analysis has seen a huge increase in popularity finding applications in numerous scientific fields. This motivates the importance of developing a deeper understanding of benefits and limitations of such methods. Using this angle, we will present and discuss some recent results on large sample inference in topological data analysis, including bootstrap for Betti numbers and the Euler characteristics process.

12:20–1:50 pm
Lunch

Afternoon Session Chair: Stephan Huckemann

1:50-2:50 pm
Ezra Miller
Title: Geometric central limit theorems on non-smooth spaces
Abstract: The central limit theorem (CLT) is commonly thought of as occurring on the real line, or in multivariate form on a real vector space. Motivated by statistical applications involving nonlinear data, such as angles or phylogenetic trees, the past twenty years have seen CLTs proved for Fréchet means on manifolds and on certain examples of singular spaces built from flat pieces glued together in combinatorial ways. These CLTs reduce to the linear case by tangent space approximation or by gluing. What should a CLT look like on general non-smooth spaces, where tangent spaces are not linear and no combinatorial gluing or flat pieces are available? Answering this question involves figuring out appropriate classes of spaces and measures, correct analogues of Gaussian random variables, and how the geometry of the space (think “curvature”) is reflected in the limiting distribution. This talk provides an overview of these answers, starting with a review of the usual linear CLT and its generalization to smooth manifolds, viewed through a lens that casts the singular CLT as a natural outgrowth, and concluding with how this investigation opens gateways to further advances in geometric probability, topology, and statistics. Joint work with Jonathan Mattingly and Do Tran.

2:50-3:00 pm
Break

3:00-4:00 pm
Lizhen Lin
Title: Statistical foundations of deep generative models
Abstract: Deep generative models are probabilistic generative models where the generator is parameterized by a deep neural network. They are popular models for modeling high-dimensional data such as texts, images and speeches, and have achieved impressive empirical success. Despite demonstrated success in empirical performance, theoretical understanding of such models is largely lacking. We investigate statistical properties of deep generative models from a nonparametric distribution estimation viewpoint. In the considered model, data are assumed to be observed in some high-dimensional ambient space but concentrate around some low-dimensional structure such as a lower-dimensional manifold structure. Estimating the distribution supported on this low-dimensional structure is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. We obtain convergence rates with respect to the Wasserstein metric of distribution estimators based on two methods: a sieve MLE based on the perturbed data and a GAN type estimator. Such an analysis provides insights into i) how deep generative models can avoid the curse of dimensionality and outperform classical nonparametric estimates, and ii) how likelihood approaches work for singular distribution estimation, especially in adapting to the intrinsic geometry of the data.

4:00-4:10 pm
Break

4:10-5:10 pm
Conversation session

Wednesday, March 1, 2023 (Eastern Time)

8:30-9:00 am
Breakfast

Morning Session Chair: Ezra Miller

9:00-10:00 am
Amit Singer*
Title: Heterogeneity analysis in cryo-EM by covariance estimation and manifold learning
Abstract: In cryo-EM, the 3-D molecular structure needs to be determined from many noisy 2-D tomographic projection images of randomly oriented and positioned molecules. A key assumption in classical reconstruction procedures for cryo-EM is that the sample consists of identical molecules. However, many molecules of interest exist in more than one conformational state. These structural variations are of great interest to biologists, as they provide insight into the functioning of the molecule. Determining the structural variability from a set of cryo-EM images is known as the heterogeneity problem, widely recognized as one of the most challenging and important computational problem in the field. Due to high level of noise in cryo-EM images, heterogeneity studies typically involve hundreds of thousands of images, sometimes even a few millions. Covariance estimation is one of the earliest methods proposed for heterogeneity analysis in cryo-EM. It relies on computing the covariance of the conformations directly from projection images and extracting the optimal linear subspace of conformations through an eigendecomposition. Unfortunately, the standard formulation is plagued by the exorbitant cost of computing the N^3 x N^3 covariance matrix. In the first part of the talk, we present a new low-rank estimation method that requires computing only a small subset of the columns of the covariance while still providing an approximation for the entire matrix. This scheme allows us to estimate tens of principal components of real datasets in a few minutes at medium resolutions and under 30 minutes at high resolutions. In the second part of the talk, we discuss a manifold learning approach based on the graph Laplacian and the diffusion maps framework for learning the manifold of conformations. If time permits, we will also discuss the potential application of optimal transportation to heterogeneity analysis. Based on joint works with Joakim Andén, Marc Gilles, Amit Halevi, Eugene Katsevich, Joe Kileel, Amit Moscovich, and Nathan Zelesko.

10:00-10:10 am
Break

10:10-11:10 am
Ian Dryden
Title: Statistical shape analysis of molecule data
Abstract: Molecular shape data arise in many applications, for example high dimension low sample size cryo-electron microscopy (cryo-EM) data and large temporal sequences of peptides from molecular dynamics simulations. In both applications it is of interest to summarize the shape evolution of the molecules in a succinct, low-dimensional representation. However, Euclidean techniques such as principal components analysis (PCA) can be problematic as the data may lie far from in a flat manifold. Principal nested spheres gives a fundamentally different decomposition of data from the usual Euclidean subspace based PCA. Subspaces of successively lower dimension are fitted to the data in a backwards manner with the aim of retaining signal and dispensing with noise at each stage. We adapt the methodology to 3D sub-shape spaces and provide some practical fitting algorithms. The methodology is applied to cryo-EM data of a large sliding clamp multi-protein complex and to cluster analysis of peptides, where different states of the molecules can be identified. Further molecular modeling tasks include resolution matching, where coarse resolution models are back-mapped into high resolution (atomistic) structures. This is joint work with Kwang-Rae Kim, Charles Laughton and Huiling Le.

11:10-11:20 am
Break

11:20 am-12:20 pm
Tamara Broderick
Title: An Automatic Finite-Sample Robustness Metric: Can Dropping a Little Data Change Conclusions?
Abstract: One hopes that data analyses will be used to make beneficial decisions regarding people’s health, finances, and well-being. But the data fed to an analysis may systematically differ from the data where these decisions are ultimately applied. For instance, suppose we analyze data in one country and conclude that microcredit is effective at alleviating poverty; based on this analysis, we decide to distribute microcredit in other locations and in future years. We might then ask: can we trust our conclusion to apply under new conditions? If we found that a very small percentage of the original data was instrumental in determining the original conclusion, we might not be confident in the stability of the conclusion under new conditions. So we propose a method to assess the sensitivity of data analyses to the removal of a very small fraction of the data set. Analyzing all possible data subsets of a certain size is computationally prohibitive, so we provide an approximation. We call our resulting method the Approximate Maximum Influence Perturbation. Our approximation is automatically computable, theoretically supported, and works for common estimators. We show that any non-robustness our method finds is conclusive. Empirics demonstrate that while some applications are robust, in others the sign of a treatment effect can be changed by dropping less than 0.1% of the data — even in simple models and even when standard errors are small.

12:20-1:50 pm
Lunch

Afternoon Session Chair: Ezra Miller

1:50-2:50 pm
Nicolai Reshetikhin*
Title: Random surfaces in exactly solvable models in statistical mechanics.
Abstract: In the first part of the talk I will be an overview of a few models in statistical mechanics where a random variable is a geometric object such as a random surface or a random curve. The second part will be focused on the behavior of such random surfaces in the thermodynamic limit and on the formation of the so-called “limit shapes”.

2:50-3:00 pm
Break

3:00-4:00 pm
Sebastian Kurtek
Title: Robust Persistent Homology Using Elastic Functional Data Analysis
Abstract: Persistence landscapes are functional summaries of persistence diagrams designed to enable analysis of the diagrams using tools from functional data analysis. They comprise a collection of scalar functions such that birth and death times of topological features in persistence diagrams map to extrema of functions and intervals where they are non-zero. As a consequence, variation in persistence diagrams is encoded in both amplitude and phase components of persistence landscapes. Through functional data analysis of persistence landscapes, under an elastic Riemannian metric, we show how meaningful statistical summaries of persistence landscapes (e.g., mean, dominant directions of variation) can be obtained by decoupling their amplitude and phase variations. This decoupling is achieved via optimal alignment, with respect to the elastic metric, of the persistence landscapes. The estimated phase functions are tied to the resolution parameter that determines the filtration of simplicial complexes used to construct persistence diagrams. For a dataset obtained under geometric, scale and sampling variabilities, the phase function prescribes an optimal rate of increase of the resolution parameter for enhancing the topological signal in a persistence diagram. The proposed approach adds to the statistical analysis of data objects with rich structure compared to past studies. In particular, we focus on two sets of data that have been analyzed in the past, brain artery trees and images of prostate cancer cells, and show that separation of amplitude and phase of persistence landscapes is beneficial in both settings. This is joint work with Dr. James Matuk (Duke University) and Dr. Karthik Bharath (University of Nottingham).

4:00-4:10 pm
Break

4:10-5:10 pm
Conversation session

5:10-5:20 pm
Stephan Huckemann, Ezra Miller, Zhigang Yao
Closing Remarks

* Virtual Presentation

2023 Ding Shum Lecture
2023-03-21 17:00 - 18:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
On March 21, 2023, the CMSA hosted the fourth annual Ding Shum Lecture, given by Cynthia Dwork (Harvard SEAS and Microsoft Research).

Time: 5:00-6:00 pm ET
Location: Harvard University Science Center Hall D
This event was be held in person and via Zoom webinar.

Title: Measuring Our Chances: Risk Prediction in This World and its Betters
Abstract: Prediction algorithms score individuals, assigning a number between zero and one that is often interpreted as an individual probability: a 0.7 “chance” that this child is in danger in the home; an 80% “probability” that this woman will succeed if hired; a 1/3 “likelihood” that they will graduate within 4 years of admission. But what do words like “chance,” “probability,” and “likelihood” actually mean for a non-repeatable activity like going to college? This is a deep and unresolved problem in the philosophy of probability. Without a compelling mathematical definition we cannot specify what an (imagined) perfect risk prediction algorithm should produce, nor even how an existing algorithm should be evaluated. Undaunted, AI and machine learned algorithms churn these numbers out in droves, sometimes with life-altering consequences.
An explosion of recent research deploys insights from the theory of pseudo-random numbers – sequences of 0’s and 1’s that “look random” but in fact have structure – to yield a tantalizing answer to the evaluation problem, together with a supporting algorithmic framework with roots in the theory of algorithmic fairness.
We can aim even higher. Both (1) our qualifications, health, and skills, which form the inputs to a prediction algorithm, and (2) our chances of future success, which are the desired outputs from the ideal risk prediction algorithm, are products of our interactions with the real world. But the real world is systematically inequitable. How, and when, can we hope to approximate probabilities not in this world, but in a better world, one for which, unfortunately, we have no data at all? Surprisingly, this novel question is inextricably bound with the very existence of nondeterminism.

Professor Cynthia Dwork is Gordon McKay Professor of Computer Science at the Harvard University John A. Paulson School of Engineering and Applied Sciences, Affiliated Faculty at Harvard Law School, and Distinguished Scientist at Microsoft. She uses theoretical computer science to place societal problems on a firm mathematical foundation.
Her recent awards and honors include the 2020 ACM SIGACT and IEEE TCMF Knuth Prize, the 2020 IEEE Hamming Medal, and the 2017 Gödel Prize.

Talk Chair: Horng-Tzer Yau (Harvard Mathematics & CMSA)

Moderator: Faidra Monachou (Harvard CMSA)

The 2020-2022 Ding Shum lectures were postponed due to Covid-19.

The 2019 Ding Shum Lecture featured Ronald Rivest on “Election Security.”

This event is made possible by the generous funding of Ding Lei and Harry Shum.

Watch the Lecture on Youtube:

CMSA/MATH Bi-Annual Gathering
2023-03-24 16:30 - 18:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On Friday, March 24th, 4:30PM – 6PM, the CMSA will host the CMSA/MATH Bi-Annual Gathering for Harvard CMSA and Math affiliates in the Common Room at 20 Garden Street, Cambridge MA 02138.

Current Developments in Mathematics Conference 2023
2023-04-07 14:00 - 2023-04-08 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Current Developments in Mathematics 2023
Harvard University Science Center, Lecture Hall C
April 7-8, 2023
Speakers:
Amol Aggarwal – Columbia University
Bhargav Bhatt – Institute for Advanced Study, Princeton University, & University of Michigan
Paul Bourgade – New York University, Courant Institute
Vesselin Dimitrov – Institute for Advanced Study & Georgia Institute of Technology
Greta Panova – University of Southern California

For more information, and to register, please visit:
Current Developments in Mathematics 2023

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau
Sponsored by the National Science Foundation, Harvard University Mathematics, Harvard University Center of Mathematical Sciences and Applications, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

Workshop on Global Categorical Symmetries
2023-05-07 09:00 - 2023-05-12 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The CMSA will be hosting a Workshop on Global Categorical Symmetries from May 7 – 12, 2023
Participation in the workshop is by invitation.
Public Lectures
There will be three lectures on Thursday, May 11, 2023, which are open to the public.
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Note: The public lectures will be held in-person only.
2:00 – 2:50 pm
Speaker: Kantaro Ohmori (U Tokyo )
Title: Fusion Surface Models: 2+1d Lattice Models from Higher Categories
Abstract: Generalized symmetry in general dimensions is expected to be described by higher categories. Conversely, one might expect that, given a higher category with appropriate structures, there exist models that admit the category as its symmetry. In this talk I will explain a construction of such 2+1d lattice models for fusion 2-categories defined by Douglas and Reutter, generalizing the work of Aasen, Fendley and Mong on anyon chains. The construction is by decorating a boundary of a topological Freed-Teleman-Moore sandwich into a non-topological boundary. In particular we can construct a family of candidate lattice systems for chiral topological orders.

3:00 – 3:50 pm
Speaker: David Jordan (Edinburgh)
Title: Langlands duality for 3-manifolds
Abstract: Originating in number theory, and permeating representation theory, algebraic geometry, and quantum field theory, Langlands duality is a pattern of predictions relating pairs of mathematical objects which have no clear a priori mathematical relation. In this talk I’ll explain a new conjectural appearance of Langlands duality in the setting of 3-manifold topology, I’ll give some evidence in the form of special cases, and I’ll survey how the conjecture relates to both the arithmetic and geometric Langlands duality conjectures.
3:50 – 4:30 pm
Tea/Snack Break
4:30 – 5:30 pm
Speaker: Ken Intriligator (UCSD)
Colloquium
Title: QFT Aspects of Symmetry
Abstract: Everything in the Universe, including the photons that we see and the quarks and electrons in our bodies, are actually ripples of quantum fields. Quantum field theory (QFT) is the underlying mathematical framework of Nature, and in the case of electrons and photons it is the most precisely tested theory in science. Strongly coupled aspects, e.g. the confinement of quarks and gluons at long distances, remain challenging. QFT also describes condensed matter systems, connects to string theory and quantum gravity, and describes cosmology. Symmetry has deep and powerful realizations and implications throughout physics, and this is especially so for the study of QFT. Symmetries play a helpful role in characterizing the phases of theories and their behavior under renormalization group flows (zooming out). Quantum field theory has also been an idea generating machine for mathematics, and there has been increasingly fruitful synergy in both directions. We are currently exploring the symmetry-based interconnections between QFT and mathematics in our Simons Collaboration on Global Categorical Symmetry, which is meeting here this week. I will try to provide an accessible, colloquium-level introduction to aspects of symmetries and QFT, both old and new.

GRAMSIA: Graphical Models, Statistical Inference, and Algorithms
2023-05-16 09:00 - 2023-05-19 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On May 16 – May 19, 2023 the CMSA hosted a four-day workshop on GRAMSIA: Graphical Models, Statistical Inference, and Algorithms. The workshop was held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. This workshop was organized by David Gamarnik (MIT), Kavita Ramanan (Brown), and Prasad Tetali (Carnegie Mellon).
The purpose of this workshop is to highlight various mathematical questions and issues associated with graphical models and message-passing algorithms, and to bring together a group of researchers for discussion of the latest progress and challenges ahead. In addition to the substantial impact of graphical models on applied areas, they are also connected to various branches of the mathematical sciences. Rather than focusing on the applications, the primary goal is to highlight and deepen these mathematical connections.
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

Speakers:

Jake Abernethy (Georgia Tech)
Guy Bresler (MIT)
Florent Krzakala (Ecole Polytechnique Federale de Lausanne)
Lester Mackey (Microsoft Research New England)
Theo McKenzie (Harvard)
Andrea Montanari (Stanford)
Elchanan Mossel (MIT)
Yury Polyanskiy (MIT)
Patrick Rebeschini (Oxford)
Subhabrata Sen (Harvard)
Devavrat Shah (MIT)
Pragya Sur (Harvard)
Alex Wein (UC Davis)
Yihong Wu (Yale)
Sarath Yasodharan (Brown)
Horng-Tzer Yau (Harvard)
Christina Lee Yu (Cornell)
Ilias Zadik (MIT)

Schedule:
Tuesday, May 16, 2023

9:00 am
Breakfast

9:15 – 9:30 am
Introductory remarks by organizers

9:30 – 10:20 am
Subhabrata Sen (Harvard)
Title: Mean-field approximations for high-dimensional Bayesian regression
Abstract: Variational approximations provide an attractive computational alternative to MCMC-based strategies for approximating the posterior distribution in Bayesian inference. The Naive Mean-Field (NMF) approximation is the simplest version of this strategy—this approach approximates the posterior in KL divergence by a product distribution. There has been considerable progress recently in understanding the accuracy of NMF under structural constraints such as sparsity, but not much is known in the absence of such constraints. Moreover, in some high-dimensional settings, the NMF is expected to be grossly inaccurate, and advanced mean-field techniques (e.g. Bethe approximation) are expected to provide accurate approximations. We will present some recent work in understanding this duality in the context of high-dimensional regression. This is based on joint work with Sumit Mukherjee (Columbia) and Jiaze Qiu (Harvard University).

10:30 – 11:00 am
Coffee break

11:00 – 11:50 am
Elchanan Mossel (MIT)
Title: Some modern perspectives on the Kesten-Stigum bound for reconstruction on trees.
Abstract: The Kesten-Stigum bound is a fundamental spectral bound for reconstruction on trees. I will discuss some conjectures and recent progress on understanding when it is tight as well as some conjectures and recent progress on what it signifies even in cases where it is not tight.

12:00 – 2:00 pm
Lunch

2:00 – 2:50 pm
Christina Lee Yu (Cornell)
Title: Exploiting Neighborhood Interference with Low Order Interactions under Unit Randomized Design
Abstract: Network interference, where the outcome of an individual is affected by the treatment assignment of those in their social network, is pervasive in many real-world settings. However, it poses a challenge to estimating causal effects. We consider the task of estimating the total treatment effect (TTE), or the difference between the average outcomes of the population when everyone is treated versus when no one is, under network interference. Under a Bernoulli randomized design, we utilize knowledge of the network structure to provide an unbiased estimator for the TTE when network interference effects are constrained to low order interactions among neighbors of an individual. We make no assumptions on the graph other than bounded degree, allowing for well-connected networks that may not be easily clustered. Central to our contribution is a new framework for balancing between model flexibility and statistical complexity as captured by this low order interactions structure.

3:00 – 3:30 pm
Coffee break

3:30 – 4:20 pm
Theo McKenzie (Harvard)
Title: Spectral statistics for sparse random graphs
Abstract: Understanding the eigenvectors and eigenvalues of the adjacency matrix of random graphs is fundamental to many algorithmic questions; moreover, it is related to longstanding questions in quantum physics. In this talk we focus on random models of sparse graphs, giving some properties that are unique to these sparse graphs, as well as some specific obstacles. Based on this, we show some new results on spectral statistics of sparse random graphs, as well as some conjectures.

4:40 – 6:30 pm
Lightning talk session + welcome reception

Wednesday, May 17, 2023

9:00 am
Breakfast

9:30 – 10:20
Ilias Zadik (MIT)
Title: Revisiting Jerrum’s Metropolis Process for the Planted Clique Problem
Abstract: Jerrum in 1992 (co-)introduced the planted clique model by proving the (worst-case initialization) failure of the Metropolis process to recover any o(sqrt(n))-sized clique planted in the Erdos-Renyi graph G(n,1/2). This result is classically cited in the literature of the problem, as the “first evidence” the o(sqrt(n))-sized planted clique recovery task is “algorithmically hard”.
In this work, we show that the Metropolis process actually fails to work (under worst-case initialization) for any o(n)-sized planted clique, that is the failure applies well beyond the sqrt(n) “conjectured algorithmic threshold”. Moreover we also prove, for a large number of temperature values, that the Metropolis process fails also under “natural initialization”, resolving an open question posed by Jerrum in 1992.

10:30 – 11:00
Coffee break

11:00 – 11:50
Florent Krzakala (Ecole Polytechnique Federale de Lausanne)
Title: Are Gaussian data all you need for machine learning theory?
Abstract: Clearly, no! Nevertheless, the Gaussian assumption remains prevalent among theoreticians, particularly in high-dimensional statistics and physics, less so in traditional statistical learning circles. To what extent are Gaussian features merely a convenient choice for certain theoreticians, or genuinely an effective model for learning? In this talk, I will review recent progress on these questions, achieved using rigorous probabilistic approaches in high-dimension and techniques from mathematical statistical physics. I will demonstrate that, despite its apparent limitations, the Gaussian approach is sometimes much closer to reality than one might expect. In particular, I will discuss key findings from a series of recent papers that showcase the Gaussian equivalence of generative models, the universality of Gaussian mixtures, and the conditions under which a single Gaussian can characterize the error in high-dimensional estimation. These results illuminate the strengths and weaknesses of the Gaussian assumption, shedding light on its applicability and limitations in the realm of theoretical statistical learning.

12:00 – 2:00 pm
Lunch

2:00 – 2:50 pm
Andrea Montanari (Stanford)
Title: Dimension free ridge regression
Abstract: Random matrix theory has become a widely useful tool in high-dimensional statistics and theoretical machine learning. However, random matrix theory is largely focused on the proportional asymptotics in which the number of columns grows proportionally to the number of rows of the data matrix. This is not always the most natural setting in statistics where columns correspond to covariates and rows to samples. With the objective to move beyond the proportional asymptotics, we revisit ridge regression. We allow the feature vector to be high-dimensional, or even infinite-dimensional, in which case it belongs to a separable Hilbert space and assume it to satisfy a certain convex concentration property. Within this setting, we establish non-asymptotic bounds that approximate the bias and variance of ridge regression in terms of the bias and variance of an ‘equivalent’ sequence model (a regression model with diagonal design matrix). Previously, such an approximation result was known only in the proportional regime and only up to additive errors: in particular, it did not allow to characterize the behavior of the excess risk when this converges to 0. Our general theory recovers earlier results in the proportional regime (with better error rates). As a new application, we obtain a completely explicit and sharp characterization of ridge regression for Hilbert covariates with regularly varying spectrum. Finally, we analyze the overparametrized near-interpolation setting and obtain sharp ‘benign overfitting’ guarantees.
[Based on joint work with Chen Cheng]

3:00 – 3:50 pm
Yury Polyanskiy (MIT)
Title: Recent results on broadcasting on trees and stochastic block model
Abstract: I will survey recent results and open questions regarding the q-ary stochastic block model and its local version (broadcasting on trees, or BOT). For example, establishing uniqueness of non-trivial solution to distribution recursions (BP fixed point) implies a characterization for the limiting mutual information between the graph and community labels. For q=2 uniqueness holds in all regimes. For q>2 uniqueness is currently only proved above a certain threshold that is asymptotically (for large q) is close to Kesten-Stigum (KS) threshold. At the same time between the BOT reconstruction and KS we show that uniqueness does not hold, at least in the presence of (arbitrary small) vertex-level side information. I will also discuss extension of the robust reconstruction result of Janson-Mossel’2004.
Based on joint works with Qian Yu (Princeton) and Yuzhou Gu (MIT).

4:00 – 4:30 pm
Coffee break

4:30 – 5:20 pm
Alex Wein (UC Davis)
Title: Is Planted Coloring Easier than Planted Clique?
Abstract: The task of finding a planted clique in the random graph G(n,1/2) is perhaps the canonical example of a statistical-computational gap: for some clique sizes, the task is statistically possible but believed to be computationally hard. Really, there are multiple well-studied tasks related to the planted clique model: detection, recovery, and refutation. While these are equally difficult in the case of planted clique, this need not be true in general. In the related planted coloring model, I will discuss the computational complexity of these three tasks and the interplay among them. Our computational hardness results are based on the low-degree polynomial model of computation.By taking the complement of the graph, the planted coloring model is analogous to the planted clique model but with many planted cliques. Here our conclusion is that adding more cliques makes the detection problem easier but not the recovery problem.

Thursday, May 18, 2023

9:00
Breakfast

9:30 – 10:20
Guy Bresler (MIT)
Title: Algorithmic Decorrelation and Planted Clique in Dependent Random Graphs
Abstract: There is a growing collection of average-case reductions starting from Planted Clique (or Planted Dense Subgraph) and mapping to a variety of statistics problems, sharply characterizing their computational phase transitions. These reductions transform an instance of Planted Clique, a highly structured problem with its simple clique signal and independent noise, to problems with richer structure. In this talk we aim to make progress in the other direction: to what extent can these problems, which often have complicated dependent noise, be transformed back to Planted Clique? Such a bidirectional reduction between Planted Clique and another problem shows a strong computational equivalence between the two problems. We develop a new general framework for reasoning about the validity of average-case reductions based on low sensitivity to perturbations. As a concrete instance of our general result, we consider the planted clique (or dense subgraph) problem in an ambient graph that has dependent edges induced by randomly adding triangles to the Erdos-Renyi graph G(n,p), and show how to successfully eliminate dependence by carefully removing the triangles while approximately preserving the clique (or dense subgraph). Joint work with Chenghao Guo and Yury Polyanskiy.

10:30 – 11:00
Coffee break

11:00 – 11:50
Sarath Yasodharan (Brown)
Title: A Sanov-type theorem for unimodular marked random graphs and its applications
Abstract: We prove a Sanov-type large deviation principle for the component empirical measures of certain sequences of unimodular random graphs (including Erdos-Renyi and random regular graphs) whose vertices are marked with i.i.d. random variables. Specifically, we show that the rate function can be expressed in a fairly tractable form involving suitable relative entropy functionals. As a corollary, we establish a variational formula for the annealed pressure (or limiting log partition function) for various statistical physics models on sparse random graphs. This is joint work with I-Hsun Chen and Kavita Ramanan.

12:00 – 12:15 pm
12:15 – 2:00 pm
Group Photo
Lunch

2:00 – 2:50 pm
Patrick Rebeschini (Oxford)
Title: Implicit regularization via uniform convergence
Abstract: Uniform convergence is one of the main tools to analyze the complexity of learning algorithms based on explicit regularization, but it has shown limited applicability in the context of implicit regularization. In this talk, we investigate the statistical guarantees on the excess risk achieved by early-stopped mirror descent run on the unregularized empirical risk with the squared loss for linear models and kernel methods. We establish a direct link between the potential-based analysis of mirror descent from optimization theory and uniform learning. This link allows characterizing the statistical performance of the path traced by mirror descent directly in terms of localized Rademacher complexities of function classes depending on the choice of the mirror map, initialization point, step size, and the number of iterations. We will discuss other results along the way.

3:00 – 3:50 pm
Pragya Sur (Harvard)
Title: A New Central Limit Theorem for the Augmented IPW estimator in high dimensions
Abstract: Estimating the average treatment effect (ATE) is a central problem in causal inference. Modern advances in the field studied estimation and inference for the ATE in high dimensions through a variety of approaches. Doubly robust estimators such as the augmented inverse probability weighting (AIPW) form a popular approach in this context. However, the high-dimensional literature surrounding these estimators relies on sparsity conditions, either on the outcome regression (OR) or the propensity score (PS) model. This talk will introduce a new central limit theorem for the classical AIPW estimator, that applies agnostic to such sparsity-type assumptions. Specifically, we will study properties of the cross-fit version of the estimator under well-specified OR and PS models, and the proportional asymptotics regime where the number of confounders and sample size diverge proportional to each other. Under assumptions on the covariate distribution, our CLT will uncover two crucial phenomena among others: (i) the cross-fit AIPW exhibits a substantial variance inflation that can be quantified in terms of the signal-to-noise ratio and other problem parameters, (ii) the asymptotic covariance between the estimators used while cross-fitting is non-negligible even on the root-n scale. These findings are strikingly different from their classical counterparts, and open a vista of possibilities for studying similar other high-dimensional effects. On the technical front, our work utilizes a novel interplay between three distinct tools—approximate message passing theory, the theory of deterministic equivalents, and the leave-one-out approach.

4:00 – 4:30 pm
Coffee break

4:30 – 5:20 pm
Yihong Wu (Yale)
Title: Random graph matching at Otter’s threshold via counting chandeliers

Abstract: We propose an efficient algorithm for graph matching based on similarity scores constructed from counting a certain family of weighted trees rooted at each vertex. For two Erdős–Rényi graphs G(n,q) whose edges are correlated through a latent vertex correspondence, we show that this algorithm correctly matches all but a vanishing fraction of the vertices with high probability, provided that nqtoinfty and the edge correlation coefficient ρ satisfies ρ^2>α≈0.338, where α is Otter’s tree-counting constant. Moreover, this almost exact matching can be made exact under an extra condition that is information-theoretically necessary. This is the first polynomial-time graph matching algorithm that succeeds at an explicit constant correlation and applies to both sparse and dense graphs. In comparison, previous methods either require ρ=1−o(1) or are restricted to sparse graphs. The crux of the algorithm is a carefully curated family of rooted trees called chandeliers, which allows effective extraction of the graph correlation from the counts of the same tree while suppressing the undesirable correlation between those of different trees. This is joint work with Cheng Mao, Jiaming Xu, and Sophie Yu, available at https://arxiv.org/abs/2209.12313

Friday, May 19, 2023

9:00
Breakfast

9:30 – 10:20
Jake Abernethy (Georgia Tech)
Title: Optimization, Learning, and Margin-maximization via Playing Games
Abstract: A very popular trick for solving certain types of optimization problems is this: write your objective as the solution of a two-player zero-sum game, endow both players with an appropriate learning algorithm, watch how the opponents compete, and extract an (approximate) solution from the actions/decisions taken by the players throughout the process. This approach is very generic and provides a natural template to produce new and interesting algorithms. I will describe this framework and show how it applies in several scenarios, including optimization, learning, and margin-maximiation problems. Along the way we will encounter a number of novel tools and rediscover some classical ones as well.

10:30 – 11:00
Coffee break

11:00 – 11:50
Devavrat Shah (MIT)
Title: On counterfactual inference with unobserved confounding via exponential family
Abstract: We are interested in the problem of unit-level counterfactual inference with unobserved confounders owing to the increasing importance of personalized decision-making in many domains: consider a recommender system interacting with a user over time where each user is provided recommendations based on observed demographics, prior engagement levels as well as certain unobserved factors. The system adapts its recommendations sequentially and differently for each user. Ideally, at each point in time, the system wants to infer each user’s unknown engagement if it were exposed to a different sequence of recommendations while everything else remained unchanged. This task is challenging since: (a) the unobserved factors could give rise to spurious associations, (b) the users could be heterogeneous, and (c) only a single trajectory per user is available.
We model the underlying joint distribution through an exponential family. This reduces the task of unit-level counterfactual inference to simultaneously learning a collection of distributions of a given exponential family with different unknown parameters with single observation per distribution. We discuss a computationally efficient method for learning all of these parameters with estimation error scaling linearly with the metric entropy of the space of unknown parameters – if the parameters are an s-sparse linear combination of k known vectors in p dimension, the error scales as O(s log k/p). En route, we derive sufficient conditions for compactly supported distributions to satisfy the logarithmic Sobolev inequality.
Based on a joint work with Raaz Dwivedi, Abhin Shah and Greg Wornell (all at MIT) with manuscript available here: https://arxiv.org/abs/2211.08209

12:00 – 2:00 pm
Lunch

2:00 – 2:50 pm
Lester Mackey (Microsoft Research New England)
Title: Advances in Distribution Compression
Abstract: This talk will introduce two new tools for summarizing a probability distribution more effectively than independent sampling or standard Markov chain Monte Carlo thinning:
1. Given an initial n-point summary (for example, from independent sampling or a Markov chain), kernel thinning finds a subset of only square-root n-points with comparable worst-case integration error across a reproducing kernel Hilbert space.
2. If the initial summary suffers from biases due to off-target sampling, tempering, or burn-in, Stein thinning simultaneously compresses the summary and improves the accuracy by correcting for these biases.
These tools are especially well-suited for tasks that incur substantial downstream computation costs per summary point like organ and tissue modeling in which each simulation consumes 1000s of CPU hours.
Based on joint work with Raaz Dwivedi, Marina Riabiz, Wilson Ye Chen, Jon Cockayne, Pawel Swietach, Steven A. Niederer, Chris. J. Oates, Abhishek Shetty, and Carles Domingo-Enrich.

3:00 – 3:30 pm
Coffee break

3:30 – 4:20 pm
Horng-Tzer Yau (Harvard)
Title: On the spectral gap of mean-field spin glass models.
Abstract: We will discuss recent progress regarding spectral gaps for the Glauber dynamics of spin glasses at high temperature. In addition, we will also report on estimating the operator norm of the covariance matrix for the SK model.

Moderators: Benjamin McKenna, Harvard CMSA & Changji Xu, Harvard CMSA

CMSA COVID-19 Policies

Big Data Conference 2023
2023-08-31 09:00 - 2023-09-01 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
On August 31-Sep 1, 2023 the CMSA hosted the ninth annual Conference on Big Data. The Big Data Conference features speakers from the Harvard community as well as scholars from across the globe, with talks focusing on computer science, statistics, math and physics, and economics.
Speakers:

Jacob Andreas, MIT
Morgane Austern, Harvard
Albert-László Barabási, Northeastern
Rachel Cummings, Columbia
Melissa Dell, Harvard
Jianqing Fan, Princeton
Tommi Jaakkola, MIT
Ankur Moitra, MIT
Mark Sellke, Harvard
Marinka Zitnik, Harvard Medical School

Organizers:

Michael Douglas, CMSA, Harvard University
Yannai Gonczarowski, Economics and Computer Science, Harvard University
Lucas Janson, Statistics and Computer Science, Harvard University
Tracy Ke, Statistics, Harvard University
Horng-Tzer Yau, Mathematics and CMSA, Harvard University
Yue Lu, Electrical Engineering and Applied Mathematics, Harvard University

Schedule
(PDF download)
Thursday, August 31, 2023

9:00 AM
Breakfast

9:30 AM
Introductions

9:45–10:45 AM
Albert-László Barabási (Northeastern, Harvard)
Title: From Network Medicine to the Foodome: The Dark Matter of Nutrition
Abstract: A disease is rarely a consequence of an abnormality in a single gene but reflects perturbations to the complex intracellular network. Network medicine offer a platform to explore systematically not only the molecular complexity of a particular disease, leading to the identification of disease modules and pathways, but also the molecular relationships between apparently distinct (patho) phenotypes. As an application, I will explore how we use network medicine to uncover the role individual food molecules in our health. Indeed, our current understanding of how diet affects our health is limited to the role of 150 key nutritional components systematically tracked by the USDA and other national databases in all foods. Yet, these nutritional components represent only a tiny fraction of the over 135,000 distinct, definable biochemicals present in our food. While many of these biochemicals have documented effects on health, they remain unquantified in any systematic fashion across different individual foods. Their invisibility to experimental, clinical, and epidemiological studies defines them as the ‘Dark Matter of Nutrition.’ I will speak about our efforts to develop a high-resolution library of this nutritional dark matter, and efforts to understand the role of these molecules on health, opening novel avenues by which to understand, avoid, and control disease.
https://youtu.be/UmgzUwi6K3E

10:45–11:00 AM
Break

11:00 AM–12:00 PM
Rachel Cummings (Columbia)
Title: Differentially Private Algorithms for Statistical Estimation Problems
Abstract: Differential privacy (DP) is widely regarded as a gold standard for privacy-preserving computation over users’ data. It is a parameterized notion of database privacy that gives a rigorous worst-case bound on the information that can be learned about any one individual from the result of a data analysis task. Algorithmically it is achieved by injecting carefully calibrated randomness into the analysis to balance privacy protections with accuracy of the results.
In this talk, we will survey recent developments in the development of DP algorithms for three important statistical problems, namely online learning with bandit feedback, causal interference, and learning from imbalanced data. For the first problem, we will show that Thompson sampling — a standard bandit algorithm developed in the 1930s — already satisfies DP due to the inherent randomness of the algorithm. For the second problem of causal inference and counterfactual estimation, we develop the first DP algorithms for synthetic control, which has been used non-privately for this task for decades. Finally, for the problem of imbalanced learning, where one class is severely underrepresented in the training data, we show that combining existing techniques such as minority oversampling perform very poorly when applied as pre-processing before a DP learning algorithm; instead we propose novel approaches for privately generating synthetic minority points.
Based on joint works with Marco Avella Medina, Vishal Misra, Yuliia Lut, Tingting Ou, Saeyoung Rho, and Ethan Turok.
https://youtu.be/0cPE6rb1Roo

12:00–1:30 PM
Lunch

1:30–2:30 PM
Morgane Austern (Harvard)
Title: To split or not to split that is the question: From cross validation to debiased machine learning
Abstract: Data splitting is a ubiquitous method in statistics with examples ranging from cross-validation to cross-fitting. However, despite its prevalence, theoretical guidance regarding its use is still lacking. In this talk, we will explore two examples and establish an asymptotic theory for it. In the first part of this talk, we study the cross-validation method, a ubiquitous method for risk estimation, and establish its asymptotic properties for a large class of models and with an arbitrary number of folds. Under stability conditions, we establish a central limit theorem and Berry-Esseen bounds for the cross-validated risk, which enable us to compute asymptotically accurate confidence intervals. Using our results, we study the statistical speed-up offered by cross-validation compared to a train-test split procedure. We reveal some surprising behavior of the cross-validated risk and establish the statistically optimal choice for the number of folds. In the second part of this talk, we study the role of cross-fitting in the generalized method of moments with moments that also depend on some auxiliary functions. Recent lines of work show how one can use generic machine learning estimators for these auxiliary problems, while maintaining asymptotic normality and root-n consistency of the target parameter of interest. The literature typically requires that these auxiliary problems are fitted on a separate sample or in a cross-fitting manner. We show that when these auxiliary estimation algorithms satisfy natural leave-one-out stability properties, then sample splitting is not required. This allows for sample reuse, which can be beneficial in moderately sized sample regimes.
https://youtu.be/L_pHxgoQSgU

2:30–2:45 PM
Break

2:45–3:45 PM
Ankur Moitra (MIT)
Title: Learning from Dynamics
Abstract: Linear dynamical systems are the canonical model for time series data. They have wide-ranging applications and there is a vast literature on learning their parameters from input-output sequences. Moreover they have received renewed interest because of their connections to recurrent neural networks.
But there are wide gaps in our understanding. Existing works have only asymptotic guarantees or else make restrictive assumptions, e.g. that preclude having any long-range correlations. In this work, we give a new algorithm based on the method of moments that is computationally efficient and works under essentially minimal assumptions. Our work points to several missed connections, whereby tools from theoretical machine learning including tensor methods, can be used in non-stationary settings.
https://youtu.be/UmgzUwi6K3E

3:45–4:00 PM
Break

4:00–5:00 PM
Mark Sellke (Harvard)
Title: Algorithmic Thresholds for Spherical Spin Glasses
Abstract: High-dimensional optimization plays a crucial role in modern statistics and machine learning. I will present recent progress on non-convex optimization problems with random objectives, focusing on the spherical p-spin glass. This model is related to spiked tensor estimation and has been studied in probability and physics for decades. We will see that a natural class of “stable” optimization algorithms gets stuck at an algorithmic threshold related to geometric properties of the landscape. The algorithmic threshold value is efficiently attained via Langevin dynamics or by a second-order ascent method of Subag. Much of this picture extends to other models, such as random constraint satisfaction problems at high clause density.
https://youtu.be/JoghiwiIbT8

6:00 – 8:00 PM
Banquet for organizers and speakers

Friday, September 1, 2023

9:00 AM
Breakfast

9:30 AM
Introductions

9:45–10:45 AM
Jacob Andreas (MIT)
Title: What Learning Algorithm is In-Context Learning?
Abstract: Neural sequence models, especially transformers, exhibit a remarkable capacity for “in-context” learning. They can construct new predictors from sequences of labeled examples (x,f(x)) presented in the input without further parameter updates. I’ll present recent findings suggesting that transformer-based in-context learners implement standard learning algorithms implicitly, by encoding smaller models in their activations, and updating these implicit models as new examples appear in the context, using in-context linear regression as a model problem. First, I’ll show by construction that transformers can implement learning algorithms for linear models based on gradient descent and closed-form ridge regression. Second, I’ll show that trained in-context learners closely match the predictors computed by gradient descent, ridge regression, and exact least-squares regression, transitioning between different predictors as transformer depth and dataset noise vary, and converging to Bayesian estimators for large widths and depths. Finally, we present preliminary evidence that in-context learners share algorithmic features with these predictors: learners’ late layers non-linearly encode weight vectors and moment matrices. These results suggest that in-context learning is understandable in algorithmic terms, and that (at least in the linear case) learners may rediscover standard estimation algorithms. This work is joint with Ekin Akyürek at MIT, and Dale Schuurmans, Tengyu Ma and Denny Zhou at Stanford.
https://youtu.be/UNVl64G3BzA

10:45–11:00 AM
Break

11:00 AM–12:00 PM
Tommi Jaakkola (MIT)
Title: Generative modeling and physical processes
Abstract: Rapidly advancing deep distributional modeling techniques offer a number of opportunities for complex generative tasks, from natural sciences such as molecules and materials to engineering. I will discuss generative approaches inspired from physical processes including diffusion models and more recent electrostatic models (Poisson flow), and how they relate to each other in terms of embedding dimension. From the point of view of applications, I will highlight our recent work on SE(3) invariant distributional modeling over backbone 3D structures with ability to generate designable monomers without relying on pre-trained protein structure prediction methods as well as state of the art image generation capabilities (Poisson flow). Time permitting, I will also discuss recent analysis of efficiency of sample generation in such models.
https://youtu.be/GLEwQAWQ85E

12:00–1:30 PM
Lunch

1:30–2:30 PM
Marinka Zitnik (Harvard Medical School)
Title: Multimodal Learning on Graphs
Abstract: Understanding biological and natural systems requires modeling data with underlying geometric relationships across scales and modalities such as biological sequences, chemical constraints, and graphs of 3D spatial or biological interactions. I will discuss unique challenges for learning from multimodal datasets that are due to varying inductive biases across modalities and the potential absence of explicit graphs in the input. I will describe a framework for structure-inducing pretraining that allows for a comprehensive study of how relational structure can be induced in pretrained language models. We use the framework to explore new graph pretraining objectives that impose relational structure in the induced latent spaces—i.e., pretraining objectives that explicitly impose structural constraints on the distance or geometry of pretrained models. Applications in genomic medicine and therapeutic science will be discussed. These include TxGNN, an AI model enabling zero-shot prediction of therapeutic use across over 17,000 diseases, and PINNACLE, a contextual graph AI model dynamically adjusting its outputs to contexts in which it operates. PINNACLE enhances 3D protein structure representations and predicts the effects of drugs at single-cell resolution.
https://youtu.be/hjt4nsN_8iM

2:30–2:45 PM
Break

2:45–3:45 PM
Jianqing Fan (Princeton)
Title: UTOPIA: Universally Trainable Optimal Prediction Intervals Aggregation
Abstract: Uncertainty quantification for prediction is an intriguing problem with significant applications in various fields, such as biomedical science, economic studies, and weather forecasts. Numerous methods are available for constructing prediction intervals, such as quantile regression and conformal predictions, among others. Nevertheless, model misspecification (especially in high-dimension) or sub-optimal constructions can frequently result in biased or unnecessarily-wide prediction intervals. In this work, we propose a novel and widely applicable technique for aggregating multiple prediction intervals to minimize the average width of the prediction band along with coverage guarantee, called Universally Trainable Optimal Predictive Intervals Aggregation (UTOPIA). The method also allows us to directly construct predictive bands based on elementary basis functions. Our approach is based on linear or convex programming which is easy to implement. All of our proposed methodologies are supported by theoretical guarantees on the coverage probability and optimal average length, which are detailed in this paper. The effectiveness of our approach is convincingly demonstrated by applying it to synthetic data and two real datasets on finance and macroeconomics. (Joint work Jiawei Ge and Debarghya Mukherjee).
https://youtu.be/WY6dr1oEOrk

3:45–4:00 PM
Break

4:00–5:00 PM
Melissa Dell (Harvard)
Title: Efficient OCR for Building a Diverse Digital History
Abstract: Many users consult digital archives daily, but the information they can access is unrepresentative of the diversity of documentary history. The sequence-to-sequence architecture typically used for optical character recognition (OCR) – which jointly learns a vision and language model – is poorly extensible to low-resource document collections, as learning a language-vision model requires extensive labeled sequences and compute. This study models OCR as a character-level image retrieval problem, using a contrastively trained vision encoder. Because the model only learns characters’ visual features, it is more sample-efficient and extensible than existing architectures, enabling accurate OCR in settings where existing solutions fail. Crucially, it opens new avenues for community engagement in making digital history more representative of documentary history.
https://youtu.be/u0JY9vURUAs

Information about the 2022 Big Data Conference can be found here.

CMSA/Math Fall Gathering
2023-09-22 16:00 - 18:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Friday, Sep 22, 2023

4:00 pm

All CMSA and Math affiliates are invited.

Mathematics in Science: Perspectives and Prospects
2023-10-27 - 2023-10-28 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Mathematics in Science: Perspectives and Prospects
A showcase of mathematics in interaction with physics, computer science, biology, and beyond.
October 27–28, 2023
Location: Harvard University Science Center Hall D & via Zoom.
Directions and Recommended Lodging
Mathematics in Science: Perspectives and Prospects Youtube Playlist

Speakers

Nima Arkani-Hamed (IAS)
Constantinos Daskalakis (MIT)
Alison Etheridge (Oxford)
Mike Freedman (Harvard CMSA)
Greg Moore (Rutgers)
Bernd Sturmfels (MPI Leipzig)

Organizers

Michael R. Douglas (Harvard CMSA)
Dan Freed (Harvard Math & CMSA)
Mike Hopkins (Harvard Math)
Cumrun Vafa (Harvard Physics)
Horng-Tzer Yau (Harvard Math)

Schedule
Friday, October 27, 2023

2:00–3:15 pm

Greg Moore (Rutgers)
Title: Remarks on Physical Mathematics
Abstract: I will describe some examples of the vigorous modern dialogue between mathematics and theoretical physics (especially high energy and condensed matter physics). I will begin by recalling Stokes’ phenomenon and explain how it is related to some notable developments in quantum field theory from the past 30 years. Time permitting, I might also say something about the dialogue between mathematicians working on the differential topology of four-manifolds and physicists working on supersymmetric quantum field theories. But I haven’t finished writing the talk yet, so I don’t know how it will end any more than you do.
Slides (PDF)

3:15–3:45 pm
Break

3:45–5:00 pm

Bernd Sturmfels (MPI Leipzig)
Title: Algebraic Varieties in Quantum Chemistry
Abstract: We discuss the algebraic geometry behind coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schroedinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Pluecker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations. This is joint work with Fabian Faulstich and Svala Sverrisdóttir.
Slides (PDF)

Saturday, October 28, 2023

9:00 am
Breakfast

9:30–10:45 am

Mike Freedman (Harvard CMSA)
Title: ML, QML, and Dynamics: What mathematics can help us understand and advance machine learning?
Abstract: Vannila deep neural nets DNN repeatedly stretch and fold. They are reminiscent of the logistic map and the Smale horseshoe. What kind of dynamics is responsible for their expressivity and trainability. Is chaos playing a role? Is the Kolmogorov Arnold representation theorem relevant? Large language models are full of linear maps. Might we look for emergent tensor structures in these highly trained maps in analogy with emergent tensor structures at local minima of certain loss functions in high-energy physics.
Slides (PDF)

10:45–11:15 am
Break

11:15 am–12:30 pmvia Zoom

Nima Arkani-Hamed (IAS)
Title: All-Loop Scattering as A Counting Problem
Abstract: I will describe a new understanding of scattering amplitudes based on fundamentally combinatorial ideas in the kinematic space of the scattering data. I first discuss a toy model, the simplest theory of colored scalar particles with cubic interactions, at all loop orders and to all orders in the topological ‘t Hooft expansion. I will present a novel formula for loop-integrated amplitudes, with no trace of the conventional sum over Feynman diagrams, but instead determined by a beautifully simple counting problem attached to any order of the topological expansion. A surprisingly simple shift of kinematic variables converts this apparent toy model into the realistic physics of pions and Yang-Mills theory. These results represent a significant step forward in the decade-long quest to formulate the fundamental physics of the real world in a new language, where the rules of spacetime and quantum mechanics, as reflected in the principles of locality and unitarity, are seen to emerge from deeper mathematical structures.

12:30–2:00 pm
Lunch break

2:00–3:15 pm

Constantinos Daskalakis (MIT)
Title: How to train deep neural nets to think strategically
Abstract: Many outstanding challenges in Deep Learning lie at its interface with Game Theory: from playing difficult games like Go to robustifying classifiers against adversarial attacks, training deep generative models, and training DNN-based models to interact with each other and with humans. In these applications, the utilities that the agents aim to optimize are non-concave in the parameters of the underlying DNNs; as a result, Nash equilibria fail to exist, and standard equilibrium analysis is inapplicable. So how can one train DNNs to be strategic? What is even the goal of the training? We shed light on these challenges through a combination of learning-theoretic, complexity-theoretic, game-theoretic and topological techniques, presenting obstacles and opportunities for Deep Learning and Game Theory going forward.
Slides (PDF)

3:15–3:45 pm
Break

3:45–5:00 pm

Alison Etheridge (Oxford)
Title: Modelling hybrid zones
Abstract: Mathematical models play a fundamental role in theoretical population genetics and, in turn, population genetics provides a wealth of mathematical challenges. In this lecture we investigate the interplay between a particular (ubiquitous) form of natural selection, spatial structure, and, if time permits, so-called genetic drift. A simple mathematical caricature will uncover the importance of the shape of the domain inhabited by a species for the effectiveness of natural selection.
Slides (PDF)

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to mathsci2023@cmsa.fas.harvard.edu no later than October 9, 2023.
Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

CMSA/Tsinghua Math-Science Literature Lecture: Scott Kominers
2023-11-20 09:00 - 10:30 | Virtual
CMSA/Tsinghua Math-Science Literature Lecture

Prof. Scott Kominers will present a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Monday, November 20, 2023
Time: 9:00 – 10:30 am ET
Location: Via Zoom Webinar
Title: 60 Years of Matching: From Gale and Shapley to Trading Networks
Abstract: Gale and Shapley’s 1962 American Mathematical Monthly paper, “College Admissions and the Stability of Marriage,” is by now one of the most cited articles in the journal’s history, having served as the foundation for an entire branch of the field of market design. This success owes in large part to the beautiful, applicable, and surprisingly general theory of matching mechanisms uncovered in Gale and Shapley’s work. This talk traces the history and evolution of matching theory from that paper forward to the present day, along the way touching on real-world applications to everything from medical residency matching to electricity markets.
Moderator: Sergiy Verstyuk

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

CMSA/MATH Bi-Annual Gathering
2024-01-26 16:00 - 18:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
On Friday, Jan. 26, 2024 the CMSA will host the CMSA/MATH Bi-Annual Gathering for Harvard CMSA and Math affiliates in the CMSA Common Room at 20 Garden Street, Cambridge MA 02138.

Arithmetic Quantum Field Theory Program
2024-02-05 09:00 - 2024-03-29 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Program
Dates: Feb. 5–Mar. 29, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge MA 02138
Arithmetic Quantum Field Theory Program Youtube Playlist
Organizers:

David Ben-Zvi (University of Texas Austin)
Solomon Friedberg (Boston College)
Natalie Paquette (University of Washington Seattle)
Brian Williams (Boston University)

This program features a weekly seminar series, workshops, and a conference.
The object of the program is to develop and disseminate exciting new connections emerging between quantum field theory and algebraic number theory, and in particular between the fundamental invariants of each: partition functions and L-functions.
On one hand, there has been tremendous progress in the past decade in our understanding of the algebraic structures underlying quantum field theory as expressed in terms of the geometry and topology of low-dimensional manifolds, both on the level of states (via the Atiyah-Segal / Baez-Dolan / Lurie formalism of extended, functorial field theory) and on the level of observables (via the Beilinson–Drinfeld / Costello–Gwilliam formalism of factorization algebras). On the other hand, Weil’s Rosetta Stone and the Mazur–Morishita–Kapranov–Reznikov arithmetic topology (the “knots and primes” dictionary) provide a sturdy bridge between the topology of 2- and 3-manifolds and the arithmetic of number fields. Thus, one can now port over quantum field theoretic ideas to number theory, as first proposed by Minhyong Kim with his arithmetic counterpart of Chern-Simons theory. Most recently, the work of Ben-Zvi–Sakellaridis–Venkatesh applies an understanding of the Langlands program as an arithmetic avatar of electric-magnetic duality in four-dimensional gauge theory to reveal a hidden quantum mechanical nature of the theory of $L$-functions.
The program will bring together a wide range of mathematicians and physicists working on adjacent areas to explore the emerging notion of arithmetic quantum field theory as a tool to bring quantum physics to bear on questions of interest for the theory of automorphic forms, harmonic analysis and L-functions. Conversely, we will explore potential geometric and physical consequences of arithmetic ideas, for example, the Langlands correspondence theory of L-functions for 3-manifolds.

Schedule
The first week of the program will feature several lecture series aimed at a broad local community of mathematicians and physicists, aiming to introduce the main ideas underlying our program and help establish a common reference point.
The program will host a weekly seminar series on Fridays.
The speakers will be selected with the aim of covering a wide panorama of the subjects over the course of the program.
The program will conclude with a week-long Conference on Arithmetic Quantum Field Theory March 25–29, 2024.

AQFT Youtube Playlist
Lecture series
All lectures take place in Room G10, Harvard CMSA, 20 Garden Street Cambridge.
Week 1: Feb. 5–9, 2024
Abstract: In this lecture series we will introduce some of the themes underlying the CMSA program on Arithmetic Quantum Field Theory taking place this winter and the upcoming conference March 25-29, 2024.
Some of the themes we plan to discuss include:
Structures in QFT (like factorization for observables and functorial QFT for states and their relation to geometric / deformation quantization) that are sufficiently algebraic and formal to allow for arithmetic analogs.
The setup of arithmetic topology as a bridge between the background of QFT to that of arithmetic (both “global” and “local”), including the “middle realm” of positive characteristic function fields.
Questions and structures in arithmetic that have been / might be amenable to inspiration from QFT, in particular the theory of L-functions and the Langlands program.
Schedule

Monday, Feb. 5, 2024

11:00 am – 12:00 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 1 pdf)

1:30 – 2:30 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

2:30 – 3:30 pm
David Ben-Zvi
The Langlands program via arithmetic QFT
Video

Wednesday, Feb. 7, 2024

11:00 am – 12:00 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 2 pdf)

2:30 – 3:30 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

Thursday, Feb.8, 2024

2:30 – 3:30 pm
Minhyong Kim
Arithmetic topology and field theory
Video
(Slides part 3 pdf)

4:00 – 5:00 pm
David Ben-Zvi
The Langlands program via arithmetic QFT
Video

Friday, Feb. 9, 2024

1:00 – 2:00 pm
Brian Williams
Algebraic quantum field theory
Video
(Lecture Notes)

2:00 – 3:00 pm
David Ben-Zvi
The Langlands program via arithmetic QFT 1
Video

3:30 – 4:30 pm
David Ben-Zvi
The Langlands program via arithmetic QFT 2
Video

Monday, Feb. 26, 2024

1:00 – 2:00 pm
Omer Offen (Brandeis)
Period integrals of automorphic forms and the residue method
Video

Tuesday, Feb. 27, 2024

2:00 – 3:00 pm
Wei Zhang (MIT)
Shtuka special cycles and their generating series
Video

Friday, March 1, 2024

11:00 am – 12:00 pm
Chen Wan (Rutgers Newark)
Some examples of the relative Langlands duality
Video

2:00 – 3:00 pm
Peng Shan (Tsinghua)
Skein algebras and quantized Coulomb branches
Video

Thursday, March 7, 2024

1:30 – 2:30 pm
An Huang (Brandeis)
Tate’s thesis and p-adic strings
Video

3:00 – 4:00 pm
John Francis (Northwestern)
Integrating braided categories over 3-manifolds
Video

Friday, March 8, 2024

1:00 – 2:00 pm
Dihua Jiang (U Minnesota)
Shalika Periods: Functoriality and Arithmetic
Video

Friday, March 15, 2024

11:45 – 1:00 pm
Baiying Liu (Purdue)
Recent progress on certain problems related to local Arthur packets of classical groups
Video

2:15 – 3:30 pm
Tasho Kaletha (Michigan)
Covers of reductive groups and functoriality
Video

Monday, March 18, 2024

1:00 – 3:00 pm
Xinwen Zhu (Stanford)
The tame categorical local Langlands correspondence
Video

4:30 – 5:30 pm
Natalie Paquette (U Washington)
Koszul duality & twisted holography for asymptotically flat spacetimes

Wednesday, March 20, 2024

11:00 – 12:15 pm
Stephen D. Miller (Rutgers)
What 4-graviton scattering amplitudes had to say about the unitary dual

Friday, March 22, 2024

1:45 – 3:00 pm
Jayce Getz (Duke)
The Poisson summation conjecture and the fiber bundle method
Video

Program Visitors

Mina Aganagic, University of California, Berkeley
Anne-Marie Aubert, Institut de Mathématiques de Jussieu-Paris Rive Gauche, March 15-29
Clark Barwick, University of Edinburgh, February 19-March 15
Alexander Braverman, Perimeter Institute
Alejandra Castro, Cambridge University, March 25-29
YoungJu Choie, Pohang University of Science and Technology, February 12-16; March 22-28
John Francis, Northwestern University, March 1-14
David Gaiotto, Perimeter Institute, March 25-29
Jayce Getz, Duke University, March 18-22
Ezra Getzler, Northwestern University, March 11-22
Sam Gunningham, Montana State University, February 9-12
Sarah Harrison, Northeastern University
Dihua Jiang, University of Minnesota, February 29-March 9
Tasho Kaletha, University of Michigan, March 12-20
Minhyong Kim, University of Edinburgh, February 1-29
Axel Kleinschmidt, Max Planck Institute for Gravitational Physics, Potsdam, March 18-28
Kim Klinger-Logan, Kansas State University, March 25-29
Kobi Kremnitzer, Oxford University, March 25-29

Baiying Liu, Purdue University, March 13-16

Steven Miller, Rutgers University

Greg Moore, Rutgers University, February 5-9
David Nadler, University of California, Berkeley, March 17-30
Bảo Châu Ngô, University of Chicago, March 25-29
George Pappas, Michigan State University, March 25-29
Daniel Persson, Chalmers Institute of Technology, March 25-29
Sam Raskin, Yale University, March 26-29
Yiannis Sakellaridis, Johns Hopkins University, March 18-22
Peng Shan, Tsinghua University, February 12-April 14
Akshay Venkatesh, Institute for Advanced Study
Roberto Volpato, University of Padova, February 4-10
Chen Wan, Rutgers University, February 29-March 9
Fei Yan, Brookhaven National Laboratory, March 18-29
Xinwen Zhu, Stanford University

CMSA/Tsinghua Math-Science Literature Lecture: Amie Wilkinson
2024-02-07 09:00 - 10:30 | Virtual
CMSA/Tsinghua Math-Science Literature Lecture
Prof. Amie Wilkinson gave a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Wednesday, February 7, 2024
Time: 9:00–10:30 am ET
Title: Stretching and shrinking: 85 years of the Hopf argument for ergodicity
Abstract: The early 20th century witnessed an explosion of activity, much of it centered at Harvard, on rigorizing the property of ergodicity first proposed by Boltzmann in his 1898 Ergodic Hypothesis for ideal gases. Earlier, in the 1880’s, Henri Poincaré and Felix Klein had also initiated a study of discrete groups of hyperbolic isometries. The geodesics in hyperbolic manifolds were discovered to carry a rich structure, first investigated from a topological perspective by Emil Artin and Marston Morse. The time was ripe to investigate geodesics in hyperbolic manifolds from an ergodic theoretic (i.e., statistical) perspective, and indeed Gustav Hedlund proved in 1934 that the geodesic flow for closed hyperbolic surfaces is ergodic.

In 1939, Eberhard Hopf published a proof of the ergodicity of geodesic flows for negatively curved surfaces containing a novel method, now known as the Hopf argument. The Hopf argument, a “soft” argument for ergodicity of systems with some hyperbolicity (the “stretching and shrinking” in the title) has since seen wide application in geometry, representation theory and dynamics. I will discuss three results relying on the Hopf argument:

Theorem (E. Hopf, 1939, D. Anosov, 1967): In a closed manifold of negative sectional curvatures, almost every geodesic is directionally equidistributed.

Theorem (G. Mostow, 1968) Let M and N be closed hyperbolic manifolds of dimension at least 3, and let f:M->N be a homotopy equivalence. Then f is homotopic to a unique isometry.

Theorem (R. Mañé, 1983, A. Avila- S. Crovisier- A.W., 2022) The C^1 generic symplectomorphism of a closed symplectic manifold with positive entropy is ergodic.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

Math Science Lectures in Honor of Raoul Bott: Maggie Miller: Fibered ribbon knots vs. major 4D conjectures
2024-02-20 16:00 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Fibered ribbon knots vs. major 4D conjectures
Location: Harvard University Science Center Hall A & via Zoom webinar
Dates: Feb 20 & 22, 2024
Time: 4:00-5:30 pm
Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow.
This is the fourth annual Math Science Lecture Series held in Honor of Raoul Bott.
Talk topic: Fibered ribbon knots vs. major 4D conjectures

Feb. 20, 2024
Title: Fibered ribbon knots and the Poincaré conjecture
Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of Casson-Gordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for non-fibered examples.

Feb. 22, 2024
Title: Fibered knots and the slice-ribbon conjecture
Abstract: The slice-ribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of Casson-Gordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a non-ribbon disk. In 2022, Dai-Kong-Mallick-Park-Stoffregen showed that unfortunately( ?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)-cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz.

Talk Chair: Cliff Taubes (Harvard Mathematics)
Moderator: Freid Tong (Harvard CMSA)

Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem.

Fourth Annual Yip Lecture | Josh Tenenbaum | How to grow a mind from a brain: From guessing and betting to thinking and talking
2024-02-29 16:00 - 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Josh Tenenbaum gave the Fourth Annual Yip Lecture on February 29, 2024.
Title: How to grow a mind from a brain: From guessing and betting to thinking and talking
Time: 4:00-5:00 pm ET
Location: Harvard Science Center
The Yip Lecture takes place thanks to the support of Dr. Shing-Yiu Yip.

The previous Yip Lecture featured Andrew Strominger (Harvard), who spoke on Black Holes.

CMSA/Tsinghua Math-Science Literature Lecture: Cameron Gordon
2024-03-20 09:00 - 10:30 | Hybrid
CMSA/Tsinghua Math-Science Literature Lecture
Prof. Cameron Gordon presented a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.

Date: Wednesday, March 20, 2024
Time: 9:00–10:30 am ET
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA and via Zoom Webinar
Title: The Unknotting Number of a Knot
Abstract: One of the oldest and most natural knot invariants is the unknotting number, which is the minimum number of times a knot must be allowed to pass through itself in order to unknot it. Although this invariant was discussed by Tait almost 150 years ago, it is still poorly understood. For instance it is not known if it is algorithmically computable, and indeed there is an 8-crossing knot whose unknotting number is unknown. Nevertheless, the many developments in knot theory since Tait have led to some understanding of unknotting number, for example through its connection with 4-dimensional topology. We will give a historical account of this progress, and discuss some of the questions that are still open.

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

Arithmetic Quantum Field Theory Conference
2024-03-25 09:00 - 2024-03-29 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Arithmetic Quantum Field Theory Conference
Dates: March 25-29, 2024
Location: Room G10, Harvard CMSA, 20 Garden Street, Cambridge MA 02138
Arithmetic Quantum Field Theory Conference Youtube Playlist
Organizers:

David Ben-Zvi (University of Texas Austin)
Solomon Friedberg (Boston College)
Natalie Paquette (University of Washington Seattle)
Brian Williams (Boston University)

Scientific Goals: On one hand, there has been tremendous progress in the past decade in our understanding of the algebraic structures underlying quantum field theory as expressed in terms of the geometry and topology of low-dimensional manifolds, both on the level of states (via the formalism of extended, functorial field theory) and on the level of observables (via the formalism of factorization algebras). On the other hand, the arithmetic topology (or “knots and primes”) dictionary provides a sturdy bridge between the topology of 2- and 3-manifolds and the arithmetic of number fields. Thus, one can now port over quantum field theoretic ideas to number theory, as first proposed by Minhyong Kim with his arithmetic counterpart of Chern-Simons theory. Moreover, automorphic objects appear in string theory where they play a role in the study of graviton scattering. Most recently, the work of Kapustin-Witten has been extended towards an understanding of the Langlands program as an arithmetic avatar of electric-magnetic duality in four-dimensional gauge theory to reveal a hidden quantum mechanical nature of the theory of L-functions.
The conference will bring together a wide range of mathematicians and physicists working on adjacent areas to explore the emerging notion of arithmetic quantum field theory as a tool to bring quantum physics to bear on questions of interest for the theory of automorphic forms, representation theory, harmonic analysis and L-functions. Conversely, we will explore potential geometric and physical consequences of arithmetic ideas. Our program will also build on the significant interactions between number theorists and physicists arising from the frequent appearance of modular and automorphic forms in partition functions, scattering amplitudes, and other quantities of interest in quantum field theory and quantum gravity.
Monday, March 25: Connections for Women in Mathematics and Physics
Speakers

Charlotte Chan (U Michigan)
Kim Klinger-Logan (Kansas State)
Sarah Harrison (Northeastern)
Melanie Matchett Wood (Harvard)
Fei Yan (Brookhaven National Lab)

Tuesday, March 26–Friday, March 29: Arithmetic Quantum Field Theory
Speakers

Anne-Marie Aubert (IMJ-PRG)
Roman Bezrukavnikov (MIT)
Sasha Braverman (Toronto / Perimeter)
Alejandra Castro (Cambridge)
YoungJu Choie (POSTECH)
Pavel Etingof (MIT)
Davide Gaiotto (Perimeter)
Axel Kleinschmidt (Max Planck Institute for Gravitational Physics)
Kobi Kremnitzer (Oxford)
Spencer Leslie (Boston College)
David Nadler (Berkeley)
Bảo Châu Ngô (U Chicago)
George Pappas (Michigan State)
Sam Raskin (Yale)
Peng Shan (Tsinghua)
Zhiwei Yun (MIT)

Conference Schedule
Arithmetic Quantum Field Theory Conference
March 25–29, 2024
Download Program (pdf)

Monday, March 25, 2024 – Women in Math and Physics

8:30 – 9:00 am

Breakfast

9:00 – 10:00 am

Melanie Matchett Wood (Harvard)
Title: Statistics of Number fields, function fields, and 3-manifolds
Abstract: Motivated by conjectures of Cohen, Lenstra, and Martinet on the distribution of class groups of number fields, we describe the analogous questions of understanding distributions of class groups and fundamental groups of curves over finite fields, and the distribution of fundamental groups of 3-manifolds. We describe results on these distributions in the cases of curves over finite fields and 3-manifolds, joint with Liu, Zureick-Brown, and Sawin, and discuss how ideas have passed back and forth between the number field, curves over finite fields, and 3-manifold settings.

10:00 – 10:20 am

Coffee break

10:20 – 11:20 am

Charlotte Chan (U Michigan)
Title: Generic character sheaves on parahoric subgroups
Abstract: Lusztig’s theory of character sheaves for connected reductive groups is one of the most important developments in representation theory in the last few decades. I will give an overview of this theory and explain the need, from the perspective of the representation theory of p-adic groups, of a theory of character sheaves on jet schemes. Recently, R. Bezrukavnikov and I have developed the “generic” part of this desired theory. In the simplest nontrivial case, this resolves a conjecture of Lusztig and produces perverse sheaves on jet schemes compatible with parahoric Deligne–Lusztig induction. This talk is intended to describe in broad strokes what we know about these generic character sheaves, especially within the context of the Langlands program.

11:30 – 12:30 pm

Kim Klinger–Logan (Kansas State)
Title: Connections between special values of L-functions and scattering amplitudes
Abstract: In this talk we will attempt make a connection between zeros and special values of L-functions and scattering amplitudes. The connection is best seen through solutions to differential equations of the form $(Delta-lambda)f = S$ on $X=SL(2,Z)SL(2,R)/SO(2,R)$ for $Delta=y^2(partial_x^2+partial_y^2)$ and $H^{-infty}(X)cup M$ where $M$ is the space of moderate growth functions. Recently, Bombieri and Garrett (following work of Hass, Hejhal, and Colin de Verdiere) laid out the possibly connection with eigenvalue solutions to equations of this form with zeros of L-functions. On the other hand, physicists such as Green, Russo, Vanhove found that eigenfunction solutions to equations of this form give coefficients of the 4-graviton scattering amplitude. We will elaborate on these connections and discuss some recent work on finding solutions for such equations. This work is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko and Don Zagier.
Slides (pdf)

12:30 – 2:15 pm

Lunch

2:15 – 3:15 pm

Fei Yan (Brookhaven National Lab)
Title: Topological defects on the lattice
Abstract: Topological defects, endowed with a rich mathematical structure, play important roles in condensed matter physics, high energy theory and quantum information science. Realization of such defects on the lattice not only has interesting theoretical consequences, but also opens the pathway to quantum simulation of physical systems. In this talk, I will discuss lattice realizations of topological defects in simple (1+1)-d systems, taking the transverse field Ising model and the three-state Potts model as examples. Time permitting, I will also briefly comment on topological defects in non-equilibrium systems, such as periodically-driven Floquet systems.

3:15 – 3:30 pm

Coffee break

3:30 – 4:30 pm

Sarah Harrison (Northeastern)
Title: Liouville Theory and Weil-Petersson Geometry
Abstract: Two-dimensional conformal field theory is a powerful tool to understand the geometry of surfaces. Liouville conformal field theory in the classical (large central charge) limit encodes the geometry of the moduli space of Riemann surfaces. I describe an efficient algorithm to compute the Weil–Petersson metric to arbitrary accuracy using Zamolodchikov’s recursion relation for conformal blocks, focusing on examples of a sphere with four punctures and generalizations to other one-complex-dimensional moduli spaces. Comparison with analytic results for volumes and geodesic lengths finds excellent agreement. In the case of M_{0,4}, I discuss numerical results for eigenvalues of the Weil-Petersson Laplacian and connections with random matrix theory. Based on work with K. Coleville, A. Maloney, K. Namjou, and T. Numasawa.
Slides (pdf)

Tuesday, March 26, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

Roman Bezrukavnikov (MIT)
Title: From affine Hecke category to invariant distributions
Abstract: By a result of Ben-Zvi, Nadler and Preygel the cocenter of the affine Hecke category can be identified with coherent sheaves on the appropriate stack of commuting pairs in the Langlands dual group. In a joint work (in progress) with Ciubotaru, Kazhdan and Varshavsky we recover the space of unipotent invariant distributions on the p-adic group from that category and develop applications to harmonic analysis, including endoscopic property of unipotent L-packets. Time permitting, I will explain how a part of this result can be recovered from a geometric realization of Lusztig’s asymptotic affine Hecke algebra J (joint with Karpov and Krylov), and present a conjecture generalizing the story to other depth zero representations; another special case of this generalization appears in a joint work with Varshavsky.

10:30 – 11:00am

Coffee break

11:00 – 12:00 pm

Sasha Braverman (Toronto/Perimeter)
Title: Hecke operators for algebraic curves over local non-archimedian fields: a survey of some recent results
Abstract: The main goal of this talk is to discuss Hecke operators and Hecke eigen-functions for the moduli space of G-bundles on a smooth projective algebraic curve X defined over a local non-archimedian field K (possibly with level structures at finitely many points). The plan is to discuss the following subjects: 1) Definition of Hecke operators and the space on which they act 2) Relation to “classical story” – i.e. eigen-functions of Hecke operators for curves over a finite field. 3) Detailed discussion of the examples when X has genus zero and we consider bundles with trivialization at two points. In this case we’ll discuss the relation to classical representation theory of p-adic groups and two representation theory of Cherednik algebras. Based on joint works with P. Etingof, D.Kazhdan, and A. Polishchuk.

12:00 – 12:15 pm

Group photo.

12:15 – 1:30 pm

Lunch

1:30 – 2:30 pm

Peng Shan (Tsinghua)
Title: Modularity for W-algebras, affine Springer fibres and associated variety
Abstract: I will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. I will also explain how to match the modular group action on the characters of representations with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. If time permits, I will discuss some extension of these results to non-admissible levels and some conjectures about their associated varieties. This is based on joint work with Dan Xie, Wenbin Yan, and Qixian Zhao.

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Bảo Châu Ngô (U Chicago)
Title: On the nonabelian Fourier kernel and the Lafforgue transform
Abstract: In the case of SL2, we present an analytic formula for the nonabelian Fourier kernel responsible for the functional equation of automorphic L-functions. We use the Gelfand-Graev formula for Langlands’ stable transfer factor and a linear map between the Bernstein center and the cocenter that we call the Lafforgue transform. This is a joint work with Zhilin Luo.

Wednesday, March 27, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

YoungJu Choie (POSTECH)
Title: Schubert Eisenstein series and Poisson summation for Schubert varieties
Abstract: Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety has been studied. In the case of GL3 over Q it was proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit the conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ngˆo, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. This is a joint work with Jayce Getz.
Slides (pdf)

10:30 – 11:00 am

Coffee break

11:00 – 12:00 pm

Axel Kleinschmidt (MPI)
Title: Automorphic representations in string amplitudes
Abstract: I will review how automorphic representations arise in the low-energy expansion of string scattering amplitudes, highlighting the connection found by Green/Miller/Vanhove between wavefront sets and BPS conditions. To study the wavefront sets I will present reduction principles for the calculation of Fourier coefficients. String theory also predicts new types of automorphic objects that are characterised by lacking finiteness under the center of the universal enveloping algebra.
Slides (pdf)

12:00 – 1:30 pm

Lunch

1:30 – 2:30 pm

Pavel Etingof (MIT)
Title: Analytic Langlands correspondence over C and R
Abstract: I will review the analytic component of the geometric Langlands correspondence, developed recently in my joint work with E. Frenkel and D. Kazhdan (based on previous works by other authors), with a special focus on archimedian local fields, especially R. This is based on our work with E. Frenkel and D. Kazhdan and insights shared by D. Gaiotto and E. Witten.
Slides (pdf)

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Davide Gaiotto (Perimeter)
Title: Unexpected Unitarity
Abstract: Much of the mathematical content of Supersymmetric Quantum Field Theories can be extracted through “twisted theories”: simplified QFTs which are topological (or holomorphic) in a derived sense and often amenable of a rigorous mathematical treatment. The twisting procedure destroys or obfuscates certain properties of the underlying SQFTs, including unitarity. I will discuss a variety of situations where some form of unitarity can be restored, endowing the twisted theories with unexpected structures. This includes the recently developed Analytic Langlands program, an analytic version of Symplectic Duality, an A-model description of quantization (as opposed to deformation quantization) and other constructions of Hodge-theoretic or twistorial flavour.

Thursday, March 28, 2024

8:30 – 9:00 am

Breakfast

9:00 – 10:00 am

Spencer Leslie (Boston College)
Title: Relative Langlands and endoscopy
Abstract: Spherical varieties play an important role in the study of periods of automorphic forms. But very closely related varieties can lead to very distinct arithmetic problems. Motivated by applications to relative trace formulas, we discuss the natural question of distinguishing different forms of a given spherical variety in arithmetic settings, giving a solution for symmetric varieties. It turns out that the answer is intimately connected with the construction of the dual Hamiltonian variety associated with the symmetric variety by Ben-Zvi, Sakellaridis, and Venkatesh. I will explain the source of these questions in the theory of endoscopy for symmetric varieties, with application to the (pre)-stabilization of relative trace formulas.

10:00 – 10:30 am

Coffee break

10:30 – 11:30 am

Anne-Marie Aubert (IMJ–PRG)
Title: The Local Langlands correspondence: from extended quotients to affine Hecke algebras
Abstract: We will introduce the notion of extended quotient, illustrate it on examples, and show how it can be used to construct the local Langlands correspondence in the nonarchimedean case. Next, we will connect extended quotients, that are attached to the Bernstein decomposition of the category of smooth representations of p-adic groups, and their Langlands duals, to representations of affine Hecke algebras in order to get a description of the LLC as a correspondence between the categories of modules of two collections of such algebras.
Slides (pdf)

11:45 – 12:45 pm

Kobi Kremnitzer (Oxford)
Title: Functional analysis over the integers, L-functions and global Hodge theory
Abstract: In this talk I will explain how using bornological methods one can develop functional analysis over the integers unifying Archimedean and non-Archimedean analysis. I will give examples of algebras of functions and distributions defined over the integers which base change to the usual algebras over the reals and p-adics. Using these it is possible to write some version of L-functions over the integers. I will then introduce an analytic stack over the integers for which the category of quasi-coherent sheaves gives global Hodge structures. I will relate the integral L-functions to trivialisations of line bundles on this stack. I will also explain how to define a cohomology theory for schemes valued in global Hodge structures (possibly related to q-deRham) and speculate on the relation between the determinant of cohomology and L-functions. This is work in progress joint with Federico Bambozzi and Jack Kelly.

12:45 – 2:00 pm

Lunch

2:00 – 3:00 pm

David Nadler (Berkeley)
Title: Going to the boundary
Abstract: I’ll describe several situations where degenerating a marked smooth curve to a marked nodal curve leads to interesting structures on automorphic moduli spaces. In particular, I’ll discuss its implications for the cocenter of the affine Hecke category, real-symmetric duality in relative Langlands, and some other conjectural situations. The talk will borrow from joint work with various authors including D. Ben-Zvi, T.-H. Chen, P. Li, and Z. Yun.

3:00 – 3:30 pm

Coffee break

Friday, March 29, 2024

9:00 – 9:30 am

Breakfast

9:30 – 10:30 am

George Pappas (Michigan State)
Title: Finite and p-adic Chern-Simons type invariants
Abstract: We will define arithmetic invariants of Galois covers and of ‘etale local systems which are inspired by the classical constructions of Dijkgraaf-Witten and Chern-Simons. We will discuss various conjectures and recent results about these invariants.

10:30 – 11:00 am

Coffee break

11:00 – 12:00 pm

Sam Raskin (Yale)
Title: The geometric Langlands conjecture
Abstract: I will describe the main ideas that go into the proof of the (unramified, global) geometric Langlands conjecture. All of this work is joint with Gaitsgory and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, Lin, and Rozenblyum.

12:00 – 1:30 pm

Lunch

1:30 – 2:30 pm

Alejandra Castro (Cambridge)
Title: The light we can see: Extracting black holes from weak Jacobi forms
Abstract: Modular forms play a pivotal role in the counting of black hole microstates. The underlying modular symmetry of counting formulae was key in the precise match between the Bekenstein-Hawking entropy of supersymmetric black holes and Cardy’s formula for the asymptotic growth of states. The goal of this talk is to revisit the connection between modular forms and black hole entropy, and tie it with other consistency conditions of AdS/CFT. We will focus our attention on weak Jacobi forms.
I will quantify how constraints on polar states affect the asymptotic growth of non-polar states in weak Jacobi forms. The constraints I’ll consider are sparseness conditions on the Fourier coefficients of these forms, which are necessary to interpret them as gravitational path integrals. In short, the constraints will leave an imprint on the subleading corrections to the asymptotic growth of heavy states. With this we will revisit the UV/IR connection that relates black hole microstate counting to modular forms. In particular, I’ll provide a microscopic interpretation of the logarithmic corrections to the entropy of supersymmetric black holes and tie it to consistency conditions in AdS_3/CFT_2.

2:30 – 3:00 pm

Coffee break

3:00 – 4:00 pm

Zhiwei Yun (MIT)
Title: Theta correspondence and relative Langlands
Abstract: A reductive dual pair (such as a symplectic group and an orthogonal group) acting on the tensor product of their standard representations is an example of hyperspherical varieties, and is the geometric avatar for theta correspondence. I will explain two geometric results/constructions motivated by the theta correspondence over finite fields. The first one describes how principal series representations behave under theta correspondence using Springer correspondence (joint with Jiajun Ma, Congling Qiu and Jialiang Zou). The second one is a definition of character sheaves in the setup of theta correspondence (joint with Shamgar Gurevich). I will speculate how the first result fits into relative Langlands duality.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to cstillman@math.harvard.eduno later than Sunday, February 25, 2024.
Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

This event will be co-funded by the National Science Foundation.
The conference is part of the Arithmetic Quantum Field Theory Program, Feb 4-March 29, 2024.

2024 Ding Shum Lecture: Yann LeCun: Objective-Driven AI: Towards AI systems that can learn, remember, reason, and plan
2024-03-28 16:30 - 17:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
LECTURE SLIDES (pdf)
On March 28, 2024, the CMSA will host the fifth annual Ding Shum Lecture, given by Yann LeCun.
Time: 4:30–5:30 pm ET
Speaker: Yann Lecun, New York University & META
Location: Harvard Science Center Hall A & via Zoom Webinar
Title: Objective-Driven AI: Towards AI systems that can learn, remember, reason, and plan

Abstract:
How could machines learn as efficiently as humans and animals?
How could machines learn how the world works and acquire common sense?
How could machines learn to reason and plan?
Current AI architectures, such as Auto-Regressive Large Language Models fall short. I will propose a modular cognitive architecture that may constitute a path towards answering these questions. The centerpiece of the architecture is a predictive world model that allows the system to predict the consequences of its actions and to plan a sequence of actions that optimize a set of objectives. The objectives include guardrails that guarantee the system’s controllability and safety. The world model employs a Hierarchical Joint Embedding Predictive Architecture (H-JEPA) trained with self-supervised learning. The JEPA learns abstract representations of the percepts that are simultaneously maximally informative and maximally predictable. The corresponding working paper is available here: https://openreview.net/forum?id=BZ5a1r-kVsf

This event is made possible by the generous funding of Ding Lei and Harry Shum.

Current Developments in Mathematics Conference 2024
2024-04-05 14:00 - 2024-04-06 17:00 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
CURRENT DEVELOPMENTS IN MATHEMATICS 2024
APRIL 5-6, 2024
HARVARD UNIVERSITY SCIENCE CENTER
LECTURE HALL C
https://www.math.harvard.edu/event/current-developments-in-mathematics-2024/

Speakers:
Daniel Cristofaro-Gardiner – University of Maryland
Samit Dasgupta – Duke University
Jiaoyang Huang – University of Pennsylvania
Daniel Litt – University of Toronto
Lisa Piccirillo – MIT/University of Texas

Download PDF for a detailed schedule of lectures and events.

Friday, April 5

Saturday, April 6

1:30 p.m. – 2:20 p.m. Part 1
2:20 p.m. – 2:30 p.m. Break
2:30 p.m. – 3:20 p.m. Part 2

Jiaoyang Huang
Random Matrix Statistics and Airy Line Ensembles

9:05 a.m. – 9:55 a.m. Part 1
9:55 a.m. – 10:05 a.m. Break
10:05 a.m. – 10:55 a.m. Part 2

Daniel Litt
Motives, mapping class groups, and monodromy

3:20 p.m. – 3:35 p.m.
Break

10:55 a.m. – 11:10 a.m.
Break

3:35 p.m. – 4:25 p.m. Part 1
4:25 p.m. – 4:35 p.m. Break
4:35 p.m. – 5:25 p.m. Part 2

Lisa Piccirillo
Exotic phenomena in dimension 4

11:10 a.m. – 12 p.m. Part 1
12 p.m. – 1:30 p.m. Lunch
1:30 p.m. – 2:20 p.m. Part 2

Samit Dasgupta
Stark’s conjectures and explicit class field theory

2:20 p.m. – 2:35 p.m.
Break

2:35 p.m. – 3:25 p.m. Part 1
3:25 p.m. – 3:35 p.m. Break
3:35 p.m. – 4:25 p.m. Part 2

Dan Cristofaro-Gardiner
Low-dimensional topology and dynamics

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).
Sponsored by the National Science Foundation, Harvard University Mathematics, and the Massachusetts Institute of Technology.
Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here.

Program on Mathematical Aspects of Scattering Amplitudes
2024-04-15 09:00 - 2024-05-24 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Program
Dates: April 15 – May 24, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 02138
This program will bring together and foster interaction between theoretical physicists and mathematicians working on various topics connected to recent developments in our understanding of scattering amplitudes in quantum field theory. The field has advanced considerably since 2019 when the CMSA hosted the program “Spacetime and Quantum Mechanics, Total Positivity and Motives.” Recent developments have primed this area for further significant advances, which will be facilitated by bringing together many of the relevant experts for intensive discussion and collaboration.
The program will feature a weekly seminar series.
Tuesday, April 16, 2024
4:15 pm
Sabrina Pasterski, Perimeter Institute
Title: Radiation in Holography
[Physics Talk]
Wednesday, April 17, 2024
4:30 pm – Cruft 309
Ana-Maria Raclariu, King’s College London
Thursday, April 18, 2024
4:15 pm
Lionel Mason, University of Oxford
Title: Hidden symmetries of SD Poincare Einstein metrics in split signature
[Physics Talk]
Tuesday, April 23, 2024
4:30 pm – Jefferson 256
Juan Maldacena, Institute for Advanced Study
Thursday, April 25, 2024
4:15 pm
Tomasz Taylor, Northeastern University
Title: Progress in Yang-Mills-Liouville Theory
[CMSA Colloquium]
Monday, April 29, 2024
4:30 – 5:30 pm
Lance Dixon, Stanford
Title: The DNA of Particle Scattering
Tuesday, April 30, 2024
9:00 am- Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 1
1:00 pm – Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 2
4:00 pm – Jefferson 453
Nima Arkani-Hamed, IAS
Title: Surfaceology and the Real World Part 3
Wednesday, May 1, 2024
11:00 am – Science Center 507
Jaroslav Trnka, UC Davis
Title: Loops of loops expansion in the Amplituhedron
3:00 pm
Yu-tin Huang, National Taiwan University
Title: Loop in trees: Chambers in amplitudes and correlation functions
4:00 pm
Livia Ferro, University of Hertfordshire
Title: Scattering Amplitudes from Null-cone Geometry
5:00 pm
Stephan Stieberger, Max Planck Institute
Title: One-loop Double Copy Relation in String Theory and Twisted (Co)homology
Thursday, May 2, 2024
11:00 am – Science Center 310
Daniil Rudenko, Chicago
Title: Introduction to Cluster Polylogarithms
Friday, May 3, 2024
11:00 am
Andrew McLeod, Edinburgh
Title: Genealogical Constraints on Feynman Integrals
Tuesday, May 7, 2024
3:00 pm
Jacob Bourjaily, Penn State
Title: The Algebraic and Transcendental Structure of Perturbative QFT
Wednesday, May 8, 2024
3:00 pm
Ruth Britto, Trinity
Title: Cuts and Symbols
Tuesday, May 14, 2024
3:00 pm
James Drummond, University of Southampton
Title: Multiple light-like Wilson loops in N=4 super Yang-Mills theory
Wednesday, May 15, 2024
3:00 pm
Matteo Parisi, Harvard CMSA
Title: The amplituhedron and cluster algebras
Tuesday, May 21, 2024
11:00 am
Michael Borinsky, ETH Zurich
Title: On the Euler characteristic of the commutative graph complex and the top-weight cohomology of the moduli space of curves
Wednesday, May 22, 2024
11:00 am
Chaim Even-Zohar, Technion
Title: Amplituhedron tiles and twistor polynomials

Organizers:

Nima Arkani-Hamed (Institute for Advanced Study)
Marcus Spradlin (Brown University)
Andrew Strominger (Harvard University)
Anastasia Volovich (Brown University)
Lauren Williams (Harvard University)

Participants:

Michael Borinsky, ETH Zurich
Jacob Bourjaily, Pennsylvania State University
Ruth Britto, Trinity College
Lance Dixon, Stanford Linear Accelerator Center
James Drummond, University of Southampton
Chaim Even-Zohar, Technion
Livia Ferro, University of Hertfordshire
Carolina Figueiredo, Princeton University
Hadleigh Frost, Oxford University
Bruno Gimenez, University of Southampton
Omer Gurdogan, University of Southampton
Xuhua He, Chinese University of Hong Kong
Paul Heslop, Durham University
Yu-Tin Huang, National Taiwan University
Dani Kaufman, University of Copenhagen
Jianrong Li, University of Vienna
Tomasz Lukowski, University of Hertfordshire
Yelena Mandelshtam, University of California, Berkeley
Lionel Mason, University of Oxford
Andrew McLeod, University of Edinburgh
Natalie Paquette, University of Washington
Matteo Parisi, Harvard University
Sabrina Pasterski, Perimeter Institute
Dmitri Pavlov, Max Planck Institute for Mathematics in the Sciences, Leipzig
Lizzie Pratt, University of California, Berkeley
Claudia Rella, University of Geneva
Daniil Rudenko, University of Chicago
Giulio Salvatori, Max Planck Institute for Physics
Melissa Sherman-Bennett, Massachusetts Institute of Technology
Jonah Stalknecht, University of Hertfordshire
Stephan Stieberger, Max Planck Institute
Tomasz Taylor, Northeastern University
Ran Tessler, Weizmann Institute of Science
Hugh Thomas, Université du Québec à Montréal
Jaroslav Trnka, University of California, Davis
Cristian Vergu, Pennsylvania State University

Workshop on Global Categorical Symmetries
2024-04-29 09:00 - 2024-05-03 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
The CMSA will be hosting a Workshop on Global Categorical Symmetries from April 29–May 3, 2024.
Participation in the workshop is by invitation.
The workshop will hold three Symmetry Colloquia open to the community on Thursday, May 2, 2024.
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 2:00 – 2:50 pm
Speaker: Clay Còrdova, University of Chicago
Title: Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry
Abstract: We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 3:00 – 3:50 pm
Speaker: Thomas Dumitrescu, UCLA
Title: Symmetries, Invertible Field Theories, and Gauge Theory Phases
Abstract: I will start with a brief overview of gauge theory phases in 3+1 dimensions through the lens of higher symmetries — in particular the realization of 1-form symmetries acting on loop order parameters. I will then review recent progress in refining this characterization using invertible field theories, or equivalently symmetry protected topological phases (SPTs). This refinement leads to new results in gauge theories with fundamental matter, such as quantum chromodynamics (QCD), which do not possess 1-form symmetries. I will explain why these theories must sometimes undergo a phase transition between their confining and Higgs regimes, despite the fact that classic results and standard lore say they should be continuously connected.

Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Time: 4:30 – 5:20 pm
Speaker: Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
Title: The Universal Target Category
Abstract: Hilbert’s Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. In other words, sVec (and not Vec!) is “algebraically closed”. These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, …. This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et al.

Organizers:
Dan Freed (Harvard CMSA & Math)
Constantin Teleman (UC Berkeley)
This event is co-sponsored by the Simons Foundation.

Symmetry Colloquia - Global Categorical Symmetries
2024-05-02 14:00 - 14:50 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Symmetry Colloquia – Global Categorical Symmetries
May 2, 2024
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Speaker: Clay Còrdova, University of Chicago
Title: Particle-Soliton Degeneracies from Spontaneously Broken Non-Invertible Symmetry
Abstract: We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading Z2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Symmetry Colloquia - Global Categorical Symmetries
2024-05-02 15:00 - 15:50 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Symmetry Colloquia – Global Categorical Symmetries
May 2, 2024
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Speaker: Thomas Dumitrescu, UCLA
Title: Symmetries, Invertible Field Theories, and Gauge Theory Phases
Abstract: I will start with a brief overview of gauge theory phases in 3+1 dimensions through the lens of higher symmetries — in particular the realization of 1-form symmetries acting on loop order parameters. I will then review recent progress in refining this characterization using invertible field theories, or equivalently symmetry protected topological phases (SPTs). This refinement leads to new results in gauge theories with fundamental matter, such as quantum chromodynamics (QCD), which do not possess 1-form symmetries. I will explain why these theories must sometimes undergo a phase transition between their confining and Higgs regimes, despite the fact that classic results and standard lore say they should be continuously connected.

Symmetry Colloquia - Global Categorical Symmetries
2024-05-02 16:30 - 17:20 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Symmetry Colloquia – Global Categorical Symmetries
May 2, 2024
Location: Room G-10, CMSA, 20 Garden Street, Cambridge MA 02138
Speaker: Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
Title: The Universal Target Category
Abstract: Hilbert’s Nullstellensatz says that the complex numbers C satisfy a universal property among all R-algebras: every not-too-large nonzero commutative R-algebra maps to C. Deligne proved a similar statement in categorical dimension 1: every not-too-large symmetric monoidal category over R maps to the category sVec of super vector spaces. In other words, sVec (and not Vec!) is “algebraically closed”. These statements help explain why quantum field theory requires imaginary numbers and fermions. I will describe the universal symmetric monoidal higher category that extends the sequence C, sVec, …. This is joint work in progress with David Reutter, and builds on closely-related work by GCS collaborators Freed, Scheimbauer, and Teleman and Schlank et al.

Amplituhedra, Cluster Algebras, and Positive Geometry
2024-05-29 09:00 - 2024-05-31 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Amplituhedra, Cluster Algebras, and Positive Geometry
Dates: May 29-31, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge MA 02138 & via Zoom
In recent years, a remarkable paradigm shift has occurred in understanding quantum observables in particle physics and cosmology, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron—the first and major example of a positive geometry. Building on the work of Lusztig and Postnikov on the positive Grassmannian, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that “explains” the so-called BCFW recurrence for scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously, cluster algebras, originally introduced by Fomin and Zelevinsky to study total positivity, have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus, one can use ideas from quantum field theory (QFT) to connect cluster algebras to positive geometries, and in particular to the amplituhedron. Additionally, QFT can also be used to discover new examples of positive geometries. The conference will bring together a wide range of mathematicians and physicists both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT.
The conference features: Introductory Lectures, an Open Problems Forum, Emerging Scholars Talks, and talks by experts in the fields.

Conference Videos (Youtube Playlist)

Confirmed Speakers:

Evgeniya Akhmedova, Weizmann Institute of Science
Nima Arkani-Hamed, IAS
Paolo Benincasa, MPI
Nick Early, Weizmann Institute of Science
Carolina Figueiredo, Princeton University
Yu-tin Huang, National Taiwan University
Dani Kaufman, University of Copenhagen
Chia-Kai Kuo, National Taiwan University
Thomas Lam, University of Michigan
Yelena Mandelshtam, UC Berkeley
Shruti Paranjape, UC Davis
Lizzie Pratt, UC Berkeley
Lecheng Ren, Brown University
Sebastian Seemann, KU Leuven
Khrystyna Serhiyenko, University of Kentucky
Melissa Sherman-Bennett, MIT & UC Davis
Marcus Spradlin, Brown University
Ran Tessler, Weizmann Institute of Science
Hugh Thomas, Université du Québec à Montréal
Jaroslav Trnka, UC Davis
Anastasia Volovich, Brown University

Organizers:

Matteo Parisi, Harvard CMSA
Lauren Williams, Harvard Mathematics

Participants (PDF)
This event is co-funded by the National Science Foundation.
Limited funding to help defray travel expenses is available for graduate students and recent PhDs. If you are a graduate student or postdoc and would like to apply for support, please register above and send an email to amplituhedra@cmsa.fas.harvard.edu no later than Friday, April 19, 2024.
Please include your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research. If you are a postdoc, please include a copy of your CV.

SCHEDULE (pdf download)
Wednesday, May 29, 2024
8:30 – 9:00 am
Registration and Breakfast
9:00 – 10:00 am
Jaroslav Trnka, UC Davis
Title: Amplituhedron
Abstract: I will review basics of the Amplituhedron, connection to the positive Grassmannian on the mathematical side, and the scattering amplitudes on the physics side.
10:00 – 10:15 am
Coffee Break
10:15 – 11:15 am
via Zoom
Khrystyna Serhiyenko, University of Kentucky
Title: Introduction to Cluster Algebras
Abstract: Cluster algebras is a class of commutative rings with an intricate combinatorial structure. They were introduced by Fomin and Zelevinsky in 2002 to study total positivity and canonical basis in Lie Theory, but quickly evolved into a highly active research area with surprising connections to numerous other areas of mathematics and physics.
In this course we will introduce cluster algebras and discuss their basic properties including positivity and Laurent phenomenon. We will also review cluster structures coming from coordinate rings of Grassmannians and the combinatorics of plabic graphs.
11:15 – 11:30 am
Coffee Break
11:30 – 12:30 pm
Thomas Lam, University of Michigan
Title: Introductory Lecture on Positive Geometries
Abstract: Positive geometries are semialgebraic spaces that appear in the study of scattering amplitudes. Examples include polytopes, totally nonnegative parts of flag varieties, and conjecturally, the amplituhedron. We will give a broad introduction to positive geometries, and to their canonical forms.
12:30 – 2:00 pm
Lunch Break
2:00 – 2:50 pm
Anastasia Volovich, Brown University
Title: Scattering Amplitudes and Cluster Algebras
Abstract: I will review some of the deep connections between cluster algebras and the (loop level) scattering amplitudes in N=4 super Yang-Mills theory, focusing on the cases of n=6 and 7 particles where the corresponding Grassmannian cluster algebras Gr(4,n) are finite and certain features of the amplitudes are known or believed to be true to all loop order.
2:50 – 3:00 pm
Coffee Break
3:00 – 3:50 pm
Marcus Spradlin, Brown University
Title: Scattering Amplitudes, Positive Geometry and the Amplituhedron
Abstract: I will review the status of (loop level) scattering amplitudes in N=4 super Yang-Mills theory for n>7, where the corresponding Grassmannian cluster algebras Gr(4,n) are infinite and novel features emerge, notably the appearance of certain “marginally positive” algebraic functions of cluster variables.
3:50 – 4:00 pm
Coffee Break
4:00 – 4:30 pm
Carolina Figueiredo, Princeton University
Title: All-order splits and multi-soft limits for particle and string amplitude
Abstract: The most important aspects of scattering amplitudes have long been thought to be associated with their poles. Recently a very different sort of “split” factorizations for a wide range of particle and string tree amplitudes have been discovered away from poles. In this talk, I will explain how natural properties of the binary geometry of the curve integral formulation for scattering amplitudes for Tr$(Phi^3)$ theory give a simple, conceptual origin for these splits, that generalizes them to all orders in the topological expansion. I will also explain how the splits allow us to access and compute loop-integrated multi-soft limits for particle and string amplitudes in Tr$(Phi^3)$ theory, the non-linear sigma model and Yang-Mills theory.
4:30 – 5:15 pm
Yelena Mandelshtam, UC Berkeley
Title: Combinatorics of m=1 Grasstopes
Abstract: A Grasstope is a linear projection of the totally nonnegative Grassmannian to a smaller Grassmannian. This is a generalization of the amplituhedron, a geometric object of great importance to calculating scattering amplitudes in physics. The amplituhedron is a Grasstope arising from a totally positive linear map. While amplituhedra are relatively well-studied, much less is known about general Grasstopes. In this talk, I will discuss combinatorics and geometry of Grasstopes in the m=1 case. In particular, I will show that they can be characterized as unions of cells of a hyperplane arrangement satisfying a certain sign variation condition and argue that amplituhedra are (in a certain sense) minimal Grasstopes. This is based on joint work with Dmitrii Pavlov and Lizzie Pratt.
5:30 – 6:30 pm
Welcome Reception

Thursday, May 30, 2024
8:30 – 9:00 am
Breakfast
9:00 – 10:00 am
Nima Arkani-Hamed, IAS
Title: Surface Kinematics and THE all-loop integrand for gluon amplitudes
10:00 – 10:30 am
Coffee Break
10:30 – 11:20 am
Hugh Thomas, Université du Québec à Montréal
Title: u-equations from finite dimensional algebras
Abstract: In this talk, I will explain how to write down and solve a system of u-equations associated to any finite dimensional algebra with finitely many indecomposable representations. These vastly generalize the system of equations written down by Koba and Nielsen in 1969, which from our point of view are associated to the representation theory of a Dynkin type A quiver. I will discuss features of the resulting solution spaces, including connections to tau-tilting theory, and the relationships that exist among different spaces of solutions. I will also say something about how different choices of finite-dimensional algebra put us in (i) the setting of cluster algebras, (ii) the Grassmannian combinatorics of non-kissing complexes, or (iii) the curves-on-surfaces model directly relevant to amplitudes. This talk reports on joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.
11:20 – 11:30 am
Coffee Break
11:30 – 12:20 pm
Dani Kaufman, University of Copenhagen
Title: Affine Cluster Algebras
Abstract: Affine cluster algebras form the simplest examples of non-finite type cluster algebras. While they have infinitely many clusters, they are still mutation finite and have essentially one mutation sequence which produces infinitely many clusters. I will give an introduction to these cluster algebras by comparing them with finite cluster algebras. I will also show how some structures similar to finite type cluster algebras appear “in the limit” along this infinite mutation sequence. If time I will also mention how the “infinite cluster variables” which live in the limit are related to the algebraic letters appearing in the symbol alphabet for 8 particle N=4 SYM amplitudes.
12:30-12:45 pm
Group Photo, 20 Garden Street, front entrance stairs outside building
12:45 – 2:00 pm
Lunch Break
2:00 – 2:50 pm
via Zoom
Ran Tessler, Weizmann Institute of Science
Title: The magic number for the m=2 amplituhedron
Abstract: We will start by reviewing the amplituhedron and its tilings.
We will then show that all tilings of the m=2 amplituhedron have the same cardinality (“the magic number”), proving the m=2 case of a conjecture that the same holds for all even-m amplituhedra. If time permits we will discuss related results and consequences.
Based on a joint work with Parisi, Sherman-Bennett and Williams.
2:50 – 3:00 pm
Coffee Break
3:00 – 3:50 pm
Melissa Sherman-Bennett, MIT & UC Davis
Title: Cluster algebras and tilings of amplituhedra
Abstract: Physicists Arkani-Hamed and Trnka introduced the amplituhedron to better understand scattering amplitudes in N=4 super Yang-Mills theory. The amplituhedron is the image of the totally nonnegative Grassmannian under the “amplituhedron map”. Examples of amplituhedra include cyclic polytopes, the totally nonnegative Grassmannian itself, and cyclic hyperplane arrangements. Of primary interest to physics are tilings of amplituhedra, which are roughly analogous to subdivisions of polytopes. I will discuss joint work with Even-Zohar, Lakrec, Parisi, Tessler and Williams on BCFW tilings of m=4 amplituhedra and the surprising connection between these tilings and the cluster algebra structure of the Grassmannian.
3:50 – 4:00 pm
Coffee Break
4:00 – 5:30 pm
Open Problems Forum
6:00 – 8:00 pm
Conference Dinner (by invitation)

Friday, May 31, 2024
8:30 – 9:00 am
Breakfast
9:00 – 10:00 am
Yu-tin Huang, National Taiwan University
Title: Chambers and all loop geometry for four-point correlators
Abstract: The all loop amplituhedron for N=4 SYM (and ABJM theory) can be recast into the notion of loop fibration over tree geometry. This leads to a further dissection of the tree geometry into “chambers”, whose boundaries represents when the associated loop-form changes. In this talk I will present a new geometry associated with the all loop four-point correlator of N=4 SYM, where similar description is present. Interestingly, at four-loops, this gives a first example where the chamber form is rational even though it’s loop form contains elliptic integrals.
10:00 – 10:15 am
Coffee Break
10:15 – 12:30 am
Emerging Scholar Talks
10:15 – 10:40 am
Evgeniya Akhmedova, Weizmann Institute of Science
Title: The tropical amplituhedron
Abstract: The Amplituhedron is a geometric object discovered recently by Arkani-Hamed and Trnka, that provides a completely new direction for calculating scattering amplitudes in quantum field theory. We define a tropical analogue of this object, the tropicial amplituhedron and study its structure and boundaries. It can be considered as both the tropical limit of the amplituhedron and a generalization of the tropical positive Grassmannian.
10:40 – 11:10 am
Lizzie Pratt, UC Berkeley
Title: The Chow-Lam Form
Abstract: The classical Chow form encodes any projective variety by one equation. We introduce the Chow-Lam form for subvarieties of a Grassmannian. By evaluating the Chow-Lam form at twistor coordinates, we obtain universal projection formulas, which were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra. This is joint work with Bernd Sturmfels.
11:10– 11:30 am
Sebastian Seemann, KU Leuven
Title: Vandermonde cells as positive geometries
Abstract: Vandermonde cells represent semialgebraic subsets of R^n, characterized as the image of a simplex under the Vandermonde map. However, within the realm of positive geometry, several challenges arise in establishing canonical forms for these cells. These include issues such as non-normal boundaries, non-transversal intersections, and singularities of boundary curves. Even more difficulties appear when considing the limiting Vandermonde cell, which is not semi-algebraic and thus doesn’t fit within the standard framework of positive geometries. In this presentation, I will first review the notion of Polypols and their canonical forms, examining the complexities encountered when dealing with Vandermonde cells. In particular, I will explain what goes wrong in the case of Vandermonde cells and which obstructions we can deal with.
11:30 – 11:40 am
Coffee break
11:40 – 12:10 pm
Chia-Kai Kuo, National Taiwan University
Title: Geometric transition from maximal SYM to ABJM
Abstract: Recently, the ABJM amplituhedron has been proposed, encoding all-loop and all-multiplicity ABJM amplitudes. It is constructed by slightly modifying the original definition. In this talk, I will explore the significance of these modifications in transitioning theoretical models from super Yang-Mills theory to ABJM theory. A key focus will be on how symplectic reduction and the overall sign change in the positivity conditions ensure the consistency of ABJM amplitudes. Additionally, I will discuss some distinct features of this geometry.
12:10– 12:30 pm
Lecheng Ren, Brown University
Title: Symbol alphabets from tensor diagrams
Abstract: We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar N= 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope C(4,9) encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.
12:30 – 2:00 pm
Lunch Break
2:00 – 2:50 pm
via Zoom
Paolo Benincasa, MPI
Title: Cosmological Polytopes & Beyond
Abstract: Together with being the source of the most profound questions in fundamental physics, cosmology turns out to be an arena from where novel combinatorial structures emerge. In this talk, I will give a gentle introduction to the cosmological polytopes, describing the so-called Bunch-Davies wavefunction for a large class of scalar theories, and how it can be used to define and characterize less conventional objects, named optical polytopes and weighted cosmological polytopes, which provide examples of non-convex and weighted geometries respectively.
2:50 – 3:00 pm
Coffee Break
3:00 – 3:45 pm
Shruti Paranjape, UC Davis
Title: Loops in a loop expansion
Abstract: In a paper by Arkani-Hamed, Henn and Trnka, it was shown that the amplituhedron construction of N=4 SYM can be recast in terms of negative geometries with a certain hierarchy of loops (closed cycles) in the space of loop momentum twistors. Furthermore, using differential equation methods, it was possible to calculate and resum integrated expressions and obtain strong coupling results. In this talk, we provide a more general framework for the loops of loops expansion and outline a powerful method for the determination of differential forms for higher-order geometries. In particular, we will focus on the case of 1 closed cycle in loop space and select integrated results.
3:45 – 4:30 pm
Nick Early, Weizmann Institute of Science
Title: Minimal Kinematics on $mathcal{M}_{0,n}$, and beyond
Abstract: Minimal Kinematics (MK) identifies kinematic degenerations of the CHY scattering potential where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh; they are specified combinatorially by 2-trees. On the other hand, Planar Kinematics (PK) identifies the locus in $M_{0,n}$ which is fixed by cyclic permutation. Combining MK and PK realizes a maximally thin relative of the associahedron known as the PK polytope; it is a reflexive polytope, and its polar dual, the root polytope, has volume a Catalan number. In this talk, we start by exploring MK and PK on the moduli space $M_{0,n}$. We explain how this story generalizes to moduli spaces $X(k,n)$ of points in projective space $mathbb{P}^{k-1}$, to CEGM amplitudes and beyond.
4:30 – 5:00 pm
Coffee and Farewell

About the image:

Left: the 3-dimensional associahedron, Fomin and Zelevinsky

Center: artistic depiction of the amplituhedron, Gilmore
Right: Schlegel diagram of a hypersimplex, Ziegler

Workshop on Fibration and Degeneration in Calabi-Yau Geometry
2024-06-24 08:00 - 2024-06-26 17:00 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
Workshop on Fibration and Degeneration in Calabi-Yau Geometry
Dates: June 24-26, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 02138
Organizer: Chuck Doran, Harvard CMSA

Calabi-Yau manifolds occupy a central place in geometry. Their critical role as the cut-case between basic Fano building blocks and the zoo of General Type manifolds is key to the wide variety of important applications of Calabi-Yau geometry to theoretical physics. In turn, ideas from theoretical physics, such as Mirror Symmetry, help shape investigations in Calabi-Yau geometry
This workshop focuses on a structural feature of Calabi-Yau geometry identified a decade ago by Doran, Harder, and Thompson. It is an organizing principle that conjecturally underlies any and all constructions of mirror pairs of Calabi-Yau manifolds. Put simply, the DHT Mirror Symmetry slogan is: “Degeneration is mirror to fibration.”

Confirmed Speakers:

David Favero (University of Minnesota)
Andrew Harder (Lehigh University)
Jesse Huang (University of Alberta)
Mohsen Karkheiran* (University of Alberta)
Matt Kerr* (Washington University in St. Louis)
Thorsten Schimannek* (Utrecht University)
Michael Schultz (Virginia Tech)
Alan Thompson (Loughborough University)
Fenglong You (University of Nottingham & ETH Zurich)

*= via Zoom

Schedule
Monday, June 24, 2024
9:30 – 10:00 am: Breakfast
10:00 – 11:00 am
Speaker: Alan Thompson, Loughborough University
Title: Mirror symmetry for fibrations and degenerations of K3 surfaces
Abstract: I will describe recent progress, joint with Luca Giovenzana, on the DHT problem for K3 surfaces. I will give an lattice-theoretic definition for when a Tyurin degeneration of K3 surfaces and an elliptically-fibred K3 surface, with an appropriate splitting of the base, form a mirror pair. I will then explain how this definition is compatible with lattice polarised mirror symmetry for K3 surfaces and with Fano-LG mirror symmetry for (quasi) del Pezzo surfaces. The upshot will be a concrete statement of the DHT conjecture for K3 surfaces.
12:00 – 1:00: Lunch
1:00 – 2:00 pm
Speaker: David Favero, University of Minnesota
Title: Homotopy Path Algebras and Resolutions
Abstract: A homotopy path algebra is like a directed version of the group ring on a fundamental group. One can imagine a directed graph (quiver) embedded in a topological space and considering the path algebra up to homotopy. Alternatively, one can think of homotopy classes of directed paths in a stratified topological space. I will introduce homotopy path algebras and describe their connections to mirror symmetry and resolutions of coherent sheaves on toric varieties.
3:00 – 4:00 pm
Speaker: Andrew Harder, Lehigh University
Title: Tropical Hodge theory for hypersurfaces and Clarke duality
Abstract: Results of Itenberg, Katzarkov, Mikhalkin, and Zharkov (IKMZ) show that if a projective variety admits a smooth tropicalization, then there is a collection of sheaves on its tropicalization that can be used to compute its Hodge numbers. However, smooth tropicalizations fail to exist even in the case of toric hypersurfaces. In work with Sukjoo Lee, we show that for any toric hypersurface, an analogue of IKMZ’s result holds. I’ll discuss this sheaf, and how this allows us to prove that Clarke dual pairs of Landau-Ginzburg models satisfy a particular Hodge number duality. This is a vast generalization of work of Batyrev and Borisov from the 90s.
4:00 – 4:30 pm: Coffee/Tea

Tuesday, June 25, 2024
9:30 – 10:00 am: Breakfast
10:00 – 11:00 am
Speaker: Matt Kerr, Washington University in St. Louis
Title: Hypergeometric families and Beilinson’s conjectures
Abstract: I will describe the construction of motivic cohomology classes on hypergeometric families of Calabi-Yau 3-folds using Hadamard convolutions. These are analogous to elements of the Mordell-Weil group for families of elliptic curves, and produce solutions to certain inhomogeneous Picard-Fuchs equations. This is part of a joint project with Vasily Golyshev in which we numerically verify Beilinson’s conjectures in some new cases.
12:00 – 1:00: Lunch
1:00 – 2:00 pm
Speaker: Fenglong You, University of Nottingham & ETH Zurich
Title: Theta functions in mirror symmetry
Abstract: To obtain a mirror of a Calabi—Yau manifold using Gross—Siebert’s intrinsic mirror symmetry, one considers a maximally unipotent monodromy degeneration of the Calabi—Yau and take proj of the degree zero part of a relative quantum cohomology ring associated with the degeneration. Theta functions form a canonical basis of the degree zero part of the relative quantum cohomology ring. Theta functions can also be defined in terms of punctured invariants of the broken line type. I will explain a variant of intrinsic mirror symmetry using orbifold invariants, theta functions for general snc pairs and a relation with the DHT conjecture.
3:00 – 4:00 pm
Speaker: Mohsen Karkheiran, University of Alberta
Title: Emergence of Heterotic-Type II duality from DHT conjecture
Abstract: The duality between Heterotic and Type IIA strings was conjectured in mid-90’s based on the properties of 4D N=2 field theories and solitonic strings in 6D. Here, we show that this duality can also emerge from the DHT conjecture. We assume both IIA and IIB strings are compactified over toric Calabi-Yau threefolds which admit K3-fibrations with arbitrary polarizations. Then by applying the Hori-Vafa mirror symmetry to the “pieces” of these Calabi-Yau manifolds, we will be able to derive the defining data for Heterotic strings. This approach works for any gauge group on the Heterotic side, and we will show how it can be practically useful to derive the Heterotic dual for any toric Calabi-Yau threefolds in Type IIA or F-theory.
4:00 – 4:30 pm: Coffee/Tea

Wednesday, June 26, 2024
9:30 – 10:00 am: Breakfast
10:00 – 11:00 am
Speaker: Thorsten Schimannek, Utrecht University
Title: Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
Abstract: Smooth M_m-polarized K3-fibered Calabi-Yau (CY) 3-folds have been classified in [DHNT] and [KT] in terms of the choice of a generalized functional invariant (GFI) and, in the case m=1, a generalized homological invariant (GHI). The resulting geometries generally exhibit a small number of complex structure moduli greater or equal to two. I will start my talk by discussing a concrete choice of these invariants that realizes (almost all of) the geometries with exactly two moduli and describe the structure of the corresponding moduli spaces. The corresponding variations of Hodge structure are entirely determined by the regular periods, for which we obtain a generic expression in terms of m and three integers i,j,s. Using the form of this period and Batyrev-Borisov mirror symmetry I will then explicitly construct the corresponding mirror CY 3-folds with two Kaehler moduli and show consistency with the DHT conjecture. In the cases with s=0, the mirror CY 3-folds are again K3-fibered but with a 2m-polarization. The generic form of the periods allows us to derive generic modular expressions for the A-model topological string free energies and we argue that those are a consequence of a Tyurin degeneration of the GFI with the central fiber being an M_m-polarized K3.
The talk is based on work in progress with Charles Doran and Boris Pioline.
12:00 – 1:00: Lunch
1:00 – 2:00 pm
Speaker: Michael Schultz, Virginia Tech
Title: Mirror Symmetry from Irrationality Proofs and a Proposal for Local Invariants
Abstract: While Apéry’s original proof of the irrationality of ζ(3) stunned the mathematics community in 1978, subsequent generations of mathematicians (including a number of those at this workshop) have discovered geometric and modular structures underlying these irrationality proofs that are arguably even more striking. One such well known example are connections to modular pencils of elliptic curves and K3 surfaces and their Picard-Fuchs operators, which exhibit maximally unipotent monodromy. These objects are respectively mirror dual to anticanonical divisors in certain del Pezzo surfaces and Fano threefolds, and their Picard-Fuchs operators to the A-side connection on small quantum cohomology for these varieties. Although the Yukawa couplings calculated in classical mirror symmetry for elliptic curves and K3 surfaces are trivial, I will show in this talk how a blend of the perspectives above allows one to define “virtual” Yukawa couplings for these families that are not trivial. It will be proposed that the utility of this perspective is in computing local invariants related to the mirror, which recovers some known results in the literature and utilizes connections to work on the DHT conjecture and the twist construction of Doran & Malmendier.
3:00 – 4:00 pm
Speaker: Jesse Huang, University of Alberta
Title: An invitation to global toric mirror symmetry
4:00 – 4:30 pm: Coffee/Tea
