From quantum Hall to Hubbard physics in twisted bilayer graphene
2024-05-10 10:00 - 11:30 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Eslam Khalaf (Harvard)
Title: From quantum Hall to Hubbard physics in twisted bilayer graphene

Abstract: Early on it was noticed that twisted bilayer graphene (TBG) has elements in common with two paradigmatic examples of strongly correlated physics: Hubbard physics and quantum Hall physics. On the one hand, TBG hosts flat topological Landau-level-like bands which realize quantum anomalous Hall states and orbital ferromagnetism under the right conditions. On the other hand, these bands are characterized by concentrated charge density and show experimental signs of fluctuating magnetism, and unconventional superconductivty; all characteristics of Hubbard-model-like physics. The emergence of fluctuating moments is particularly surprising, as localized Wannier states do not exist in topological bands. I will discuss a phenomenological model for the flat bands in TBG that centers the concentration of charge density and, relatedly, the concentration of Berry flux. The bands obtained have excellent quantitative agreement with the Bistritzer-Macdonald model for realistic parameters. I will show that, rather remarkably, the model hosts decoupled flavor moments which despite being only power-law delocalized with infinite localization length, have parametrically small overlap with each other. I will show how this unifies many of the observations in TBG and leads to a novel Mott semimetal phase for intermediate temperatures where moments are thermally disordered but charge fluctuations are mostly frozen.

On the landscape of 4d N=2 SCFTs
2024-05-10 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Robert Moscrop, Harvard CMSA
Title: On the landscape of 4d N=2 SCFTs
Abstract: Four-dimensional conformal field theories with sufficient (N = 2) supersymmetry are highly constrained. So much so, there has been an ongoing effort to classify them using only information about their moduli space of vacua. In this talk, I will review recent progress in this classification before detailing a subclass of theories for which the classification problem is particularly tractable.

An analogue of non-interacting quantum field theory in Riemannian signature
2024-05-13 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Mikhail Molodyk, Stanford
Title: An analogue of non-interacting quantum field theory in Riemannian signature
Abstract: Recent advances using microlocal tools have led to constructions, for wave operators on various classes of spacetimes, of four distinguished Fredholm inverses which have the singular behavior required of retarded, advanced, Feynman, and anti-Feynman propagators in QFT. Vasy and Wrochna have used these to define a QFT on asymptotically Minkowski spacetimes, for which they construct Hadamard states described by asymptotic data at infinity. I will describe an analogue of this construction on Riemannian manifolds with two asymptotically conic ends, defining quantum fields satisfying the Helmholtz equation and using scattering data to construct states satisfying a wavefront mapping-property version of the Hadamard condition. The absence of a spacetime interpretation lends itself to a sharper focus on the theory’s analytic structure, from whose perspective the Feynman propagators are no less natural than the advanced/retarded ones. I will also highlight some differences between Feynman propagators defined as distinguished inverses and as time-ordered expectations. Based on joint work with András Vasy.

Improving Mean-Field Theory for Quantum Magnets
2024-05-13 13:30 - 14:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Junyi Zhang, Johns Hopkins University
Title: Improving Mean-Field Theory for Quantum Magnets
Abstract: Frustrated magnets have garnered significant attention because of their potential to host exotic spin liquids, while many real material candidates exhibit magnetic orders. Due to their proximity to spin liquid phases, the semiclassical descriptions of these magnetic orders often fall short in capturing their intricate quantum behaviors. In this talk, I will introduce an improved mean-field method, named density-matrix mean-field theory (DMMFT), for these quantum magnets. Using local reduced density matrix, DMMFT can systematically incorporate the quantum fluctuations beyond conventional mean-field theories. Notably, it not only quantitatively evaluates the renormalization of order parameters induced by quantum fluctuations but also has the capability to detect the topological order of quantum phases. DMMFT offers an efficient approach to explore phases displaying unconventional quantum orders, particularly beneficial for investigating frustrated spin systems in high spatial dimensions.
Ref: https://arxiv.org/abs/2401.06236

Errors and Correction in Cumulative Knowledge
2024-05-13 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Madhu Sudan, Harvard University
Title: Errors and Correction in Cumulative Knowledge
Abstract: Societal accumulation of knowledge is a complex, and arguably error-prone, process. The correctness of new units of knowledge depends not only on the correctness of the new reasoning, but also on the correctness of old units that the new one builds on. If left unchecked, errors could completely ruin the validity of most of this knowledge so there must some error-correcting going on. What are the error-corrections processes employed in nature and how effective are they? In this talk, we describe our attempts to model such phenomena using probablistic models – we describe models for growth of cumulative knowledge, emergence of errors and methods to check for errors and eliminate them. We then analyze in this compound model, when effects of errors may survive, and when they are totally eliminated.
The central discovery in our work is the following optimistic statement: If we do checking correctly (most of the time) investing just a constant factor (<1) of our effort in checking (and saving the remaining constant factor towards deriving new units of knowledge), then effects of errors can be kept in check. Notably the amount of effort expended on checking does not scale with the volume of total knowledge or the depth of dependencies in the new units of knowledge, either of which would be overwhelming.
Based on the papers:
Is this correct? Let’s check!
Omri Ben-Eliezer, Dan Mikulincer, Elchanan Mossel, Madhu Sudan
arXiv:2211.12301
Errors are Robustly Tamed in Cumulative Knowledge Processes
Anna Brandenberger, Cassandra Marcussen, Elchanan Mossel, Madhu Sudan
arXiv:2309.05638

Quasilocal mass for general domains in space
2024-05-14 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Member Seminar
Speaker: Jue Liu
Title: Quasilocal mass for general domains in space
Abstract: Diffeomorphism-invariant quasilocal mass in classical general relativity has been studied for decades, but it is still an open problem how to define quasi-local mass for general domains with multiple boundaries in space. Using the Hamiltonian formulation, we will provide a new way to define the nonnegative quasi-local mass, and give recent progress in overcoming the difficulties.

Mathematical Aspects of Scattering Amplitudes Lecture
2024-05-14 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: James Drummond, University of Southampton

Title: Multiple light-like Wilson loops in N=4 super Yang-Mills theory

Mathematical Aspects of Scattering Amplitudes Lecture
2024-05-15 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: Matteo Parisi, Harvard CMSA
Title: The amplituhedron and cluster algebras

Mirror symmetry and log del Pezzo surfaces
2024-05-16 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Franco Rota, University of Glasgow
Title: Mirror symmetry and log del Pezzo surfaces
Abstract: The homological mirror symmetry conjecture predicts a duality, expressed in terms of categorical equivalences, between the complex geometry of a variety X (the B side) and the symplectic geometry of its mirror object Y (the A side). Motivated by this, we study a series of singular surfaces (called log del Pezzo). I will describe the category arising in the B side, using the McKay correspondence and explicit birational geometry. I will discuss early results on the A side, using the language of pseudolattices to focus on the special case of a smooth degree 2 del Pezzo surface. This is joint work with Giulia Gugiatti.

Controlling chaotic advection in 2D active nematics
2024-05-16 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar
Speaker: Kevin Mitchell, University of California, Merced
Title: Controlling chaotic advection in 2D active nematics
Abstract: Recent years have seen a surge of interest in active materials, in which energy injected at the microscale gives rise to mesoscale coherent motion. One prominent example is an active 2D “liquid crystal” composed of microtubules in the nematic phase. The activity is generated by molecular motors that consume ATP to generate local shearing between the microtubules. The resulting 2D fluid flow exhibits self-generated mesoscale chaotic dynamics with a characteristic folding and stretching pattern. We analyze this dynamics from the perspective of chaotic advection, in which the fluid can be viewed as “stirred” by topological defects in the nematic order parameter. Typically, these defects move in an irregular, chaotic pattern. We explore conditions, both theoretically and in experiments, under which the topological defects can be coaxed to perform regular periodic motion, thus bringing some degree of order to the chaos.

Geometry of Generalized Lienard Equations, Contact Geometry and Metriplectic Structure
2024-05-16 15:00 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Partha Guha, Khalifa University, Abu Dhabi
Title: Geometry of Generalized Lienard Equations, Contact Geometry and Metriplectic Structure
Abstract: We start with a self-contained brief review of the construction of non-standard Lagrangian and Hamiltonian structures using the Jacobi Last Multiplier for the (generalized) reduced Lienard equations satisfying Chiellini integrability condition, we then explore its connection with the contact Hamiltonian mechanics. We also explore Lienard equations, of reaction-diffusion class and conformal (or contact) Hamiltonian Structures. Finally, we present the geometric formulation of the generalized Lienard equations in terms of metriplectic structure.
Joint work with Jose Carinena and Anindya Ghose Choudhury

Love and Naturalness
2024-05-17 10:30 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Mikhail Ivanov (MIT)
Title: Love and Naturalness
Abstract: Recent progress in gravitational wave astronomy has spurred the development of efficient tools to describe gravitational binary dynamics. One such tool is classical worldline effective field theory (EFT). In the first part of my talk, I will show how to use this EFT for systematic studies of tidal heating and deformations (Love numbers) of compact objects. I will present a gauge-invariant definition of Love numbers and show how to extract them in a coordinate-independent way from scattering amplitudes of the gravitational Raman process. I will show that the worldline EFT exhibits strong fine-tuning when applied to black holes. This gives rise to a naturalness paradox associated with the vanishing of black hole static Love numbers. In the second part of my talk, I will present a new symmetry of black holes (Love symmetry) that elegantly resolves this paradox. The Love symmetry is tightly connected to isometries of extremal black holes that appear in many holographic constructions. It also provides a curious example of IR/UV mixing, which may give insights for other hierarchy problems.

Mathematical Aspects of Scattering Amplitudes Lecture
2024-05-21 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: 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

Mathematical Aspects of Scattering Amplitudes Lecture
2024-05-22 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematical Aspects of Scattering Amplitudes Lecture
Speaker: Chaim Even-Zohar, Technion
Title: Amplituhedron tiles and twistor polynomials

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

Corks for exotic diffeomorphisms
2024-06-04 14:00 - 15:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Slava Krushkal, University of Virginia
Title: Corks for exotic diffeomorphisms
Abstract: Exotic smooth structures on simply-connected 4-manifolds are known to be related by cork twists: cutting out and re-gluing certain smooth contractible submanifolds. Work in progress, joint with A. Mukherjee, M. Powell, and T. Warren, provides a localization result for exotic diffeomorphisms of 4-manifolds. I will also discuss applications to known examples of exotic diffeomorphisms.

Can embedding problems be used to distinguish S^4 from other (possible) homotopy 4-spheres?
2024-06-04 16:00 - 17:00 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Michael Freedman, Harvard CMSA
Title: Can embedding problems be used to distinguish S^4 from other (possible) homotopy 4-spheres?
Abstract: There are approaches in the literature (using Khovanov homology) to detecting a homotopy 4-sphere, via the 4-ball genus of knots. I’d like to suggest moving from surfaces to 3-manifolds, that is approaching the problem by considering the which closed 3-manifolds embed. Embedding in the actual S^4 implies a curious condition on the possible Heegaard diagrams for the 3-manifold. I’ll explain this condition and speculate on how it might be exploited.

Phases and Phase Transitions of Spin Chains with Non-invertible Symmetries
2024-06-07 14:30 - 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Arkya Chatterjee (MIT)
Title: Phases and Phase Transitions of Spin Chains with Non-invertible Symmetries
Abstract: Non-invertible symmetries are often emergent at low-energies in gapless states of quantum matter. It is useful to construct lattice models that have these as exact symmetries in order to provide a UV-complete setting in which they are well-controlled. To that end, we propose to study one-dimensional Hamiltonians defined on tensor product Hilbert spaces with finite on-site dimension — referred to as “spin chains” in short — with exact non-invertible symmetries. We focus on two concrete examples: a spin chain with (invertible) S_3 symmetry and one with (non-invertible) Rep(S_3) symmetry. These models are largely analytically tractable and demonstrate all spontaneous symmetry breaking (SSB) phases of these symmetries. With the aid of tensor network algorithms, we systematically study the phase transitions between these SSB phases. Both models possess (intrinsically) non-invertible self-duality symmetries, for which we provide sequential circuit implementations. On the self-dual manifold in parameter space, we discover an incommensurate gapless phase with an anomalous U(1) symmetry which emanates from lattice translation.

Landscape of Tensor Network States Preparable from Measurement
2024-06-21 14:00 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter in Mathematics and Physics Seminar
Speaker: Rahul Sahay (Harvard)

Title: Landscape of Tensor Network States Preparable from Measurement

Abstract: Measurements and feedback have emerged as powerful resources for creating many-body quantum states. However, a detailed understanding of what is possible is restricted to fixed-point representatives of phases of matter. In this talk, we go beyond this, characterizing more general patterns of many-body entanglement that can be deterministically created from measurement. In 1D, a complete framework is developed for the case where a single round of measurements is the only entangling operation. Specifically, we completely classify the space of 1D preparable quantum states (forming a strict subset of all matrix product states), and characterize their physical constraints. In doing so, we find an intriguing physical trade-off between the richness of the preparable entanglement spectrum and correlation functions, naturally implying a powerful no-go theorem for preparing certain quantum states. Moreover, our classification enables one to search for and engineer preparable quantum states with a range of desired correlation lengths and entanglement properties. We conclude by charting out generalizations, such as higher dimensional examples, considering multiple rounds of measurements, and implementing matrix product operators. At a high level, our work offers a resource-theoretic perspective on preparable quantum entanglement and shows how to systematically create states of matter, away from their fixed points, in quantum devices. This is based on two recent works with Ruben Verresen [arXiv:2404.17087; arXiv:2404.16753].

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

Compactified Hyperboloidal Evolution in Numerical Relativity
2024-08-15 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Shalabh Gautam, International Center for Theoretical Sciences, Tata Institute of Fundamental Research
Title: Compactified Hyperboloidal Evolution in Numerical Relativity
Abstract: In the field of numerical relativity, a major challenge is to incorporate future null infinity into the computational domain using a well-posed formulation. Achieving this goal would not only allow us to accurately simulate any astrophysical system, such as binary black holes, and extract the resulting signal, like gravitational waves, at future null infinity, but it would also facilitate the study of various fundamental phenomena. One proposed solution is to use hyperboloidal slices. In this presentation, I will introduce an alternative approach to numerical relativity that leverages these slices in spherical symmetry. I will also discuss our ongoing efforts to establish a well-posed formulation of the Einstein Field Equations on these slices, and propose numerical schemes to ensure stability and convergence for linear hyperbolic systems over long durations. Additionally, I will present our method for constructing initial data corresponding to an asymptotically flat spacetime. A natural extension of our work will be to generalize these methods to the Einstein Field Equations with appropriate initial data in the full 3d.

Advances in Probability Theory and Interacting Particle Systems
2024-08-26 09:00 - 2024-08-28 17:00 | Harvard Geological Lecture Hall, 24 Oxford St, Cambridge, 02138, United States
Advances in Probability Theory and Interacting Particle Systems

A conference in honor of S. R. Srinivasa Varadhan.

August 26 – August 28, 2024

Harvard Geological Lecture Hall

Conference Website: www.math.harvard.edu/event/math-conference-honoring-srinivasa-varadhan

Speakers

Ines Armendariz, Universidad de Buenos Aires
Yuri Bakhtin, Courant Institute
Gérard Ben Arous, Courant Institute
Sourav Chatterjee, Stanford University
Amir Dembo, Stanford University
Peter K. Friz, TU-Berlin
Nina Holden, Courant Institute
Jiaoyang Huang, University of Pennsylvania
Elena Kosygina, City University of New York
Claudio Landim, IMPA
Eyal Lubetzky, Courant Institute
Chiranjib Mukherjee, Uni Münster
Stefano Olla, Université Paris Dauphine
Jeremy Quastel, University of Toronto
Kavita Ramanan, Brown University
Alejandro Ramirez, NYU Shanghai
Fraydoun Rezakhanlou, Berkeley
Sunder Sethuraman, University of Arizona
Scott Sheffield, MIT
Ofer Zeitouni, Weizmann Institute

Organizers: Paul Bourgade (New York University, Courant Institute) and Horng-Tzer Yau (Harvard University).

Sponsored by Harvard University Department of Mathematics and the Center of Mathematical Studies and Applications (CMSA).

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.

Instanton in Lattice QCD from Higher Categories and Higher Anafunctors
2024-08-28 14:00 - 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Jing-Yuan Chen, Tsinghua University
Title: Instanton in Lattice QCD from Higher Categories and Higher Anafunctors

Abstract: Putting continuum QFT (not just TQFT) on the lattice is important for both fundamental understandings and practical numerics. The traditional way of doing so, based on simple intuitions, however, does not admit natural definitions for general topological operators of continuous-valued fields—one such example is the long standing problem in lattice QCD of lacking a natural definition for Yang-Mills instantons.
In this talk, I will explain a more systematic way to relate continuum and lattice QFT, based on higher categories and higher anafunctors, so that the topological operators in the continuum can be naturally defined on the lattice. The idea, though formulated formally, is physically very intuitive—we want to effectively capture the different possibilities of how a lattice field may interpolate into the continuum, so the higher categories that are employed to study higher homotopy theory should be naturally involved. Via this formalism, we resolve the long-standing problem of defining instanton (as well as Chern-Simons term) in lattice Yang-Mills theory, in terms of multiplicative bundle gerbes. Moreover, when the fields become discrete, our formalism can recover the Dijkgraaf-Witten and Turaev-Viro theory, so we hope this formalism to be a good starting point towards (in the very long term) a comprehensive categorical understanding of QFT that encompass both continuous and discrete degrees of freedom, applicable both to IR and to UV.

Mathematics and Machine Learning Program
2024-09-03 09:00 - 2024-11-01 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematics and Machine Learning Program
Dates: September 3 – November 1, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge, MA 0213
Machine learning and AI are increasingly important tools in all fields of research. Recent milestones in machine learning for mathematics include data-driven discovery of theorems in knot theory and representation theory, the discovery and proof of new singular solutions of the Euler equations, new counterexamples and lower bounds in graph theory, and more. Rigorous numerical methods and interactive theorem proving are playing an important part in obtaining these results. Conversely, much of the spectacular progress in AI has a surprising simplicity at its core. Surely there are remarkable mathematical structures behind this, yet to be elucidated.
The program will begin and end with two week-long workshops, and will feature focus weeks on number theory, knot theory, graph theory, rigorous numerics in PDE, and interactive theorem proving, as well as a course on geometric aspects of deep learning.

September 3–5, 2024: Opening Workshop: AI for Mathematicians, with Leon Bottou, François Charton, David McAllester, Adam Wagner and Geordie Williamson. A series of six lectures covering logic and theorem proving, AI methods, theory of machine learning, two lectures on case studies in math-AI, and a lecture and discussion on open problems and the ethics of AI in science.
Opening Workshop Youtube Playlist

September 6–7, 2024: Big Data Conference

September 9–13, 2024: Applying Machine Learning to Math, with François Charton and Geordie Williamson
Public Lecture September 12, 2024: Geordie Williamson, University of Sydney: Can AI help with hard mathematics? (Youtube link)
The focus of this week will be on practical examples and techniques for the mathematics researcher keen to explore or deepen their use of AI techniques. We will have talks showcasing easily stated problems, on which machine learning techniques can be employed profitably. These provide excellent toy examples for generating intuition. We will also have expert talks on some of the technical subtleties which arise. There are several instances where the accepted heuristics emerging from the study of large language models (LLM) and image recognition don’t appear to apply on mathematics problems, and we will try to highlight these subtleties.
Applying Machine Learning to Math Youtube Playlist

September 16–20, 2024: Number theory, with Drew Sutherland
The focus of this week will be on the use of ML as a tool for finding and understanding statistical patterns in number-theoretic datasets, using the recently discovered (and still largely unexplained) “murmurations” in the distribution of Frobenius traces in families of elliptic curves and other arithmetic L-functions as a motivating example.
Number Theory Youtube Playlist

September 23–27, 2024: Knot theory, with Sergei Gukov
Knot theory is a great source of labeled data that can be synthetically generated. Moreover, many outstanding problems in knot theory and low-dimensional topology can be formulated as decision and classification tasks, e.g. “Is the knot 123_45 slice?” or “Can two given Kirby diagrams be related by a sequence of Kirby moves?” During this focus week we will explore various ways in which AI can be applied to problems in knot theory and how, based on these applications, mathematical reasoning can advance development of AI algorithms. Another goal will be to develop formal knot theory libraries (e.g. contributions to mathlib) and to apply AI models to formal proof systems, in particular in the context of knot theory.
Knot Theory Youtube Playlist

September 30: Teaching and Machine Learning Panel Discussion, 3:30-5:30 pm ET

September 30–October 4, 2024: Graph theory and combinatorics, with Adam Wagner
This week, we will consider how machine learning can help us solve problems in combinatorics and graph theory, broadly interpreted, in practice. The advantage of these fields is that they deal with finite objects that are simple to set up using computers, and programs that work for one problem can often be adapted to work for several other related problems as well. Many times, the best constructions for a problem are easy to interpret, making it simpler to judge how well a particular algorithm is performing. On the other hand, there are lots of open conjectures that are simple to state, for which the best-known constructions are counterintuitive, making it perhaps more likely that machine learning methods can spot patterns that are difficult to understand otherwise.
Graph Theory and Combinatorics Youtube Playlist

October 7–11, 2024: More number theory, with Drew Sutherland
The focus of this week will be on the use of AI as a tool to search for and/or construct interesting or extremal examples in number theory and arithmetic geometry, using LLM-based genetic algorithms, generative adversarial networks, game-theoretic methods, and heuristic tree pruning as alternatives to conventional local search strategies.
More Number Theory Youtube Playlist

October 14 –18, 2024: Interactive theorem proving
This week we will discuss the use of interactive theorem proving systems such as Lean, Coq and Isabelle in mathematical research, and AI systems which prove theorems and translate between informal and formal mathematics.
Interactive Theorem Proving Youtube Playlist

October 21–25, 2024: Numerical Partial Differential Equations (PDE), with Tristan Buckmaster and Javier Gomez-Serrano
The focus of this week will be on constructing solutions to partial differential equations and dynamical systems (finite and infinite dimensional) more broadly defined. We will discuss several toy problems and comment on issues like sampling strategies, optimization algorithms, ill-posedness, or convergence. We will also outline strategies about further developing machine-learning findings and turn them into mathematical theorems via computer-assisted approaches.
Numerical PDEs Youtube Playlist

October 28–Nov. 1, 2024: Closing Workshop: The closing workshop will provide a forum for discussing the most current research in these areas, including work in progress and recent results from program participants.
Math and Machine Learning Closing Workshop Youtube Playlist

September 3–Nov. 1: Graduate topics in deep learning theory (Boston College) taught by Eli Grigsby, held at the CMSA Tuesdays and Thursdays 2:30–3:45 pm Eastern Time. Course website (link).
Graduate Topics in Deep Learning Youtube Playlist
Course description: This is a course on geometric aspects of deep learning theory. Broadly speaking, we’ll investigate the question: How might human-interpretable concepts be expressed in the geometry of their data encodings, and how does this geometry interact with the computational units and higher-level algebraic structures in various parameterized function classes, especially neural network classes? During the portion of the course Sep. 3-Nov. 1, the course will be presented as part of the Math and Machine Learning program at the CMSA in Cambridge. During that portion, we will focus on the current state of research on mechanistic interpretability of transformers, the architecture underlying large language models like Chat-GPT.

Prerequisites: This course is targeted to graduate students and advanced undergraduates in mathematics and theoretical computer science. No prior background in machine learning or learning theory will be assumed, but I will assume a degree of mathematical maturity (at the level of–say—the standard undergraduate math curriculum+ first-year graduate geometry/topology sequence)

Program Organizers

Francois Charton (Meta AI)
Michael R. Douglas (Harvard CMSA)
Michael Freedman (Harvard CMSA)
Fabian Ruehle (Northeastern)
Geordie Williamson (Univ. of Sydney)

Program Schedule
Monday
10:30–noon
Open Discussion
Room G10
12:00–1:30 pm
Group lunch
CMSA Common Room
Tuesday
2:30–3:45 pm
Topics in deep learning theory
Room G10
4:00–5:00 pm
Open Discussion/Tea
CMSA Common Room
Wednesday
10:30 am–12:00 pm
Open Discussion
Room G10
2:00–3:00 pm
New Technologies in Mathematics Seminar
Room G10
Thursday
2:30–3:45 pm
Topics in deep learning theory
Room G10
Friday
10:30 am–12:00 pm
Open Discussion
Room G10

Harvard CMSA thanks Mistral AI for a generous donation of computing credit.

Mathematics and Machine Learning Program Opening Workshop
2024-09-03 10:00 - 2024-09-05 16:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Mathematics and Machine Learning Program Opening Workshop
Dates: September 3 – 5, 2024
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA & via Zoom Webinar
AI for Mathematicians, with Leon Bottou, François Charton, David McAllester, Adam Wagner, Boris Hanin, and Geordie Williamson. A series of 6 tutorial lectures introducing concepts of AI and of theorem proving, with many case studies of AI applied to mathematics, and including lectures and discussion sessions on open questions, future prospects, and ethical questions.

Speakers

Leon Bottou (Meta AI)
François Charton (Meta AI)
Michael R. Douglas (Harvard CMSA)
Boris Hanin (Princeton)
David McAllester (TTIC)
Adam Wagner (WPI)
Geordie Williamson (University of Sydney)

Schedule (link to downloadable pdf)
Videos from the Opening Workshop (Youtube Link)

Tuesday Sep. 3, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
Mike Douglas: Overview of AI for mathematics
Slides (pdf)

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
David McAllester: Logic and formal methods
Slides (pdf)

2:30–3:00 pm
Coffee break

3:00–4:00 pm
Panel Discussion: Automated mathematical discovery

Wednesday Sep. 4, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
Boris Hanin: Theory of Machine Learning
Slides (pdf)

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
Adam Wagner: Case studies I: Reinforcement learning and pattern finding
Slides (pdf)

2:30–3:00 pm
Coffee break

Thursday Sep. 5, 2024

9:30–10:00 am
Morning refreshments

10:00–11:30 am
François Charton (slides-pdf)
and Geordie Williamson: Case studies II

11:30 am–12:00 pm
Discussion

12:00–1:30 pm
Break

1:30–2:30 pm
Leon Bottou: Open questions in AI
Slides (pdf)

2:30–3:00 pm
Coffee break

3:00–4:00 pm
Panel Discussion: How might AI change mathematics?

CMSA/Math Fall Gathering
2024-09-05 16:00 - 17:00 | CMSA 20 Garden Street Cambridge, Massachusetts 02138 United States
September 5, 2024
4:00 pm
CMSA Common Room, 20 Garden Street, Cambridge MA
All CMSA and Math affiliates are invited.

Big Data Conference 2024
2024-09-06 09:00 - 2024-09-07 17:00 | 20 Garden Street, Cambridge, MA 02138, MA, MA, 02138, United States
Youtube Playlist
On September 6-7, 2024, the CMSA hosted the tenth 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.
Location: Harvard University CMSA, 20 Garden Street, Cambridge & via Zoom

Speakers:

Tianxi Cai, Harvard Chan School
Raj Chetty, Harvard
Bianca Dumitrascu, Columbia
Boris Hanin, Princeton
Peter Hull, Brown
Jamie Morgenstern, U Washington
Kavita Ramanan, Brown
Neil Thompson, MIT
Melanie Weber, Harvard
Kun-Hsing Yu, Harvard Medical School

Organizers:

Rediet Abebe, Harvard Society of Fellows
Morgane Austern, Harvard University Statistics
Michael R. Douglas, Harvard CMSA
Yannai Gonczarowski, Harvard University Economics and Computer Science
Sam Kou, Harvard University Statistics

SCHEDULE (downloadable pdf)
Friday, Sep. 6, 2024
9:00 am: Breakfast
9:30 am: Introductions
9:45–10:45 am
Speaker: Peter Hull, Brown University
Title: Measuring Discrimination in Multi-Phase Systems, with an Application to Child Protection
Abstract: Large racial disparities have been documented in many high-stakes settings—such as employment, health care, housing, and criminal justice—raising concerns of discrimination by individual decision-makers. At the same time, there is growing understanding that a focus on individual decisions can yield an incomplete view of discrimination; an extensive theoretical literature shows how discrimination can arise and compound across multiple decision-makers in interconnected systems. We develop new empirical tools for studying discrimination in such multi-phase systems and apply them to the setting of foster care placement by child protective services. Leveraging the quasi-random assignment of two sets of decision-makers—initial hotline call screeners and subsequent investigators—we study how unwarranted racial disparities arise and propagate through this system. Using a sample of over 200,000 maltreatment allegations, we find that calls involving Black children are 55% more likely to result in foster care placement than calls involving white children with the same potential for future maltreatment in the home. Call screeners account for up to 19% of this unwarranted disparity, with the remainder due to investigators. Unwarranted disparity is concentrated in cases with potential for future maltreatment, suggesting that white children may be harmed by “underplacement” in high-risk situations.
10:45–11:00 am: Break
11:00 am –12:00 pm
Speaker: Jamie Morgenstern, U Washington
Title: What governs predictive disparity in modern machine learning applications?
Abstract: The deployment of statistical models in impactful environments is far from new—simple correlations have been used to guide decisions throughout the sciences, health care, political campaigns, and in pricing financial instruments and other products for decades. Many such models, and the decisions they supported, were known to have different degrees of predictive power for different demographic groups. These differences had numerous sources, including: limited expressiveness of the statistical models; limited availability of data from marginalized populations; noisier measurements of both features and targets from certain populations; and features with less mutual information about the prediction target for some populations than others.
Modern decision systems which use machine learning are more ubiquitous than ever, as are their differences in performance for different populations of people. In this talk, I will discuss some similarities and differences in the sources of differing performance in contemporary ML systems including facial recognition systems and those incorporating generative AI.
12:00–1:30 pm: Lunch Break
1:30–2:30 pm
Speaker: Kavita Ramanan, Brown University
Title: Understanding High-dimensional Stochastic Dynamics on Realistic Networks
Abstract: Large collections of randomly evolving particles that interact locally with respect to an underlying network model a variety of phenomena ranging from magnetism, the spread of diseases, neural and neuronal networks, opinion dynamics and load balancing on computer networks. Due to their high-dimensional nature, these systems are typically intractable to analyze exactly. Classical work, falling under the rubric of mean-field approximations, has mostly focused on the case when this interaction graph is dense. However, most real-world networks are sparse and often random. We describe a new approach to develop principled approximations for dynamics on realistic networks that beats the curse of dimensionality, and illustrate its efficacy on a class of epidemiological models. This is based on joint works with Michel Davydov, Ankan Ganguly and Juniper Cocomello.
2:30–2:45 pm: Break
2:45–3:45 pm
Speaker: Raj Chetty, Harvard University
Title: The Science of Economic Opportunity: New Insights from Big Data
Abstract: How can we improve economic opportunities for children growing up in low-income families? This talk will present findings from a recent set of studies that use various sources of big data — ranging from anonymized tax records to social network data — to understand the science of economic opportunity. Among other topics, the talk will discuss how and why children’s chances of climbing the income ladder vary across neighborhoods, the drivers of racial disparities in economic mobility, how highly selective colleges may amplify the persistence of privilege, and the role of social capital as a driver of upward mobility. The talk will conclude by giving examples of how academic research using big data is informing policy decisions from the local to federal level to expand opportunities for all.
3:45–4:00 pm: Break
4:00–5:00 pm
Speaker: Neil Thompson
Title: How Algorithmic Progress is driving progress in Big Data and AI
Abstract: Algorithm improvement is one of the purest forms of innovation: it allows the same computational task to be achieved with far fewer resources by proposing clever new ways to do that computation. In this talk, I will discuss the work that my lab has done tracking and quantifying progress across decades of algorithm research and practice. As I will show, this algorithmic progress has often outpaced hardware improvement as the most important driver of progress in Big Data and AI.

Saturday, Sep. 7, 2024
9:00 am: Breakfast
9:30 am: Introductions
9:45–10:45 am
Speaker: Tianxi Cai, Harvard Chan School
Title: Crowdsourcing with Multi-institutional EHR to Improve Reliability of Real World Evidence – Opportunities and Challenges
Abstract: The wide adoption of electronic health records (EHR) systems has led to the availability of large clinical datasets available for discovery research. EHR data, linked with bio- repository, is a valuable new source for deriving real-word, data-driven prediction models of disease risk and progression. Yet, they also bring analytical difficulties especially when aiming to leverage multi-institutional EHR data. Synthesizing information across healthcare systems is challenging due to heterogeneity and privacy. Statistical challenges also arise due to high dimensionality in the feature space. In this talk, I’ll discuss analytical approaches for mining EHR data to improve the reliability and generalizability of real world evidence generated from the analyses. These methods will be illustrated using EHR data from Mass General Brigham and Veteran Health Administration.
10:45–11:00 am: Break
11:00 am–12:00 pm
Speaker: Bianca Dumitrascu, Columbia Data Science Institute
Title: Statistical machine learning for learning representations of embryonic development
Abstract: During embryonic development, single cells read in local information from their environments and use this information to move, divide and specialize. As a result, the environments themselves change. However, it remains unclear how gene expression programs interact with cell morphology and mechanical forces to orchestrate organogenesis in early embryos. Recent advances in single cell techniques and in toto imaging enable unique venues in exploring this link between genomics and biophysics, which dynamically maps cells to organisms.
In this talk, I will describe statistical machine learning frameworks aimed at understanding how tissue level mechanical and morphometric information impact gene expression patterns in spatio-temporal contexts. We use these tools to understand boundary formation in the early development of mouse embryos and to align data from light sheet recordings of pre-gastrulation development.
12:00–1:30 pm: Lunch Break
1:30–2:30 pm
Speaker: Melanie Weber, Harvard Mathematics
Title: Data and Model Geometry in Deep Learning
Abstract: Data with geometric structure is ubiquitous in machine learning. Often such structure arises from fundamental symmetries in the domain, such as permutation-invariance in graphs and sets, and translation-invariance in images. In this talk we discuss implications of this structure on the design and complexity of neural networks. Equivariant architectures, which encode symmetries as inductive bias, have shown great success in applications with geometric data, but can suffer from instabilities as their depths increases. We propose a new architecture based on unitary group convolutions, which allows for deeper networks with less instability. In the second part of the talk we discuss the impact of data and model geometry on the learnability of neural networks. We discuss learnability in several geometric settings, including equivariant neural networks, as well as learnability with respect to the geometry of the input data manifold.
2:30–2:45 pm: Break
2:45–3:45 pm
Speaker: Boris Hanin, Princeton University
Title: Scaling Limits of Neural Networks
Abstract: Neural networks are often studied analytically through scaling limits: regimes in which taking some structural network parameters (e.g. depth, width, number of training datapoints, and so on) to infinity results in simplified models of learning. I will motivative and discuss recent results using several such approaches. I will emphasize both new theoretical insights into how model, training data, and optimizer impact learning and their practical implications for hyperparameter transfer.
3:45–4:00 pm: Break
4:00–5:00 pm
Speaker: Kun-Hsing Yu, Harvard Medical School
Title: Foundation Models for Real-Time Cancer Diagnosis
Abstract: Artificial intelligence (AI) is transforming the landscape of medical research and practice. Recent advances in microscopic image digitization, foundation models, and scalable computing infrastructure have opened new avenues for AI-enhanced cancer diagnosis. In this talk, I will highlight recent breakthroughs in multi-modal AI systems for cancer pathology evaluation, discuss integrative biomedical informatics methods that link cell morphology with molecular profiles, and outline critical challenges in developing robust medical AI systems.

Information about the 2023 Big Data Conference can be found here.

Combinatorics and geometry of the amplituhedron
2024-09-09 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Lauren Williams, Harvard University
Title: Combinatorics and geometry of the amplituhedron
Abstract: The amplituhedron is a geometric object introduced by Arkani-Hamed and Trnka to compute scattering amplitudes in N=4 super Yang Mills theory. It generalizes interesting objects such as cyclic polytopes and the positive Grassmannian. It has connections to tropical geometry, cluster algebras, and combinatorics (plane partitions, Catalan numbers). I’ll give a gentle introduction to the amplituhedron, then survey some recent progress on some of the main conjectures about the amplituhedron: the Magic Number Conjecture, the BCFW tiling conjecture, and the Cluster Adjacency conjecture. Based on joint works withEvan-Zohar, Lakrec, Parisi, Sherman-Bennett, and Tessler.

Topics in Deep Learning Theory
2024-09-10 14:30 - 15:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topics in Deep Learning Theory
Eli Grigsby

Math and Machine Learning Program Open Discussion/Tea
2024-09-10 16:00 - 17:00 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Open Discussion/Tea
