Black Holes as Quantum Systems
2023-10-20 12:00 - 13:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Daniel Kapec (CMSA)
Title: Black Holes as Quantum Systems
Abstract: To an outside observer, a black hole appears to be an ordinary quantum mechanical system with finite entropy and highly chaotic internal dynamics. Nevertheless, the low-temperature thermodynamics of the Kerr black hole presents several puzzles. For instance, the leading order semiclassical approximation to the black hole density of states predicts a surprisingly large ground state degeneracy, while poorly understood quantum corrections are known to become increasingly important at low temperatures. I will review the modern picture of black holes as quantum systems and then discuss a recent result on the leading correction to the low-temperature thermodynamics of the Kerr black hole that resolves many of the old puzzles.

Gauged Linear Sigma Models and Cohomological Field Theories
2023-10-23 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Speaker: David Favero, University of Minnesota

Title: Gauged Linear Sigma Models and Cohomological Field Theories
Abstract: This talk is dedicated to the memory of my friend and collaborator Bumsig Kim and based on joint work with Ciocan-Fontanine-Guere-Kim-Shoemaker. Gauged Linear Sigma Models (GLSMs) serve as a means of interpolating between Kahler geometry and singularity theory. In enumerative geometry, they should specialize to both Gromov-Witten and Fan-Jarvis-Ruan-Witten theory. In joint work with Bumsig Kim (see arXiv:2006.12182), we constructed such enumerative invariants for GLSMs. Furthermore, we proved that these invariants form a Cohomological Field Theory. In this lecture, I will describe GLSMs and Cohomological Field Theories, review the history of their development in enumerative geometry, and discuss the construction of these general invariants. Briefly, the invariants are obtained by forming the analogue of a virtual fundamental class which lives in the twisted Hodge complex over a certain “moduli space of maps to the GLSM”. This virtual fundamental class roughly comes as the Atiyah class of a “virtual matrix factorization” associated to the GLSM data.

On Provable Copyright Protection for Generative Model
2023-10-23 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Boaz Barak (Harvard)
Title: On Provable Copyright Protection for Generative Model
Abstract: There is a growing concern that learned conditional generative models may output samples that are substantially similar to some copyrighted data C that was in their training set. We give a formal definition of near access-freeness (NAF) and prove bounds on the probability that a model satisfying this definition outputs a sample similar to C, even if C is included in its training set.
Roughly speaking, a generative model p is k-NAF if for every potentially copyrighted data C, the output of p diverges by at most k-bits from the output of a model q that did not access C at all. We also give generative model learning algorithms, which efficiently modify the original generative model learning algorithm in a black box manner, that output generative models with strong bounds on the probability of sampling protected content. Furthermore, we provide promising experiments for both language (transformers) and image (diffusion) generative models, showing minimal degradation in output quality while ensuring strong protections against sampling protected content.
Joint work with Nikhil Vyas and Sham Kakade. Paper appeared in ICML 2023 and is on https://arxiv.org/abs/2302.10870

Resolving memory in numerical relativity, and fixing BMS frames for modeling
2023-10-24 11:00 - 12:00 | MA
General Relativity Seminar
Speaker: Leo Stein (Mississippi)
Title: Resolving memory in numerical relativity, and fixing BMS frames for modeling
Abstract: Numerical relativity waveforms serve as ground truth for detection and parameter estimation of binary black hole mergers. Most NR waveforms to date miss memory effects, as they were extracted from simulations using an approximation called extrapolation. I will report on the SXS collaboration’s capacity to resolve memory effects in production NR simulations using Cauchy-characteristic evolution (CCE), and in the future with Cauchy-characteristic matching (CCM). I will further report on how BH perturbation and post-Newtonian theory furnish natural BMS frames. With these BMS frames, we can extract well-defined remnant quantities, perform precision ringdown modeling, and build complete surrogate waveform models that capture memory effects.

CMSA Q and A Seminar 10/24/23
2023-10-24 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Xi Yin (Harvard Physics) and Peter Kronheimer (Harvard Math)

Topics:
Xi Yin: What is string field theory?
Peter Kronheimer: What is Seiberg-Witten theory?

Llemma: an open language model for mathematics
2023-10-25 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Sean Welleck, CMU, Language Technologies Institute
Title: Llemma: an open language model for mathematics
Abstract: We present Llemma: 7 billion and 34 billion parameter language models for mathematics. The Llemma models are initialized with Code Llama weights, then trained on the Proof-Pile II, a 55 billion token dataset of mathematical web data, code, and scientific papers. The resulting models show improved mathematical capabilities, and can be adapted to various tasks. For instance, Llemma outperforms the unreleased Minerva model suite on an equi-parameter basis, and is capable of tool use and formal theorem proving without any further fine-tuning. We openly release all artifacts, including the Llemma models, the Proof-Pile II, and code to replicate our experiments. We hope that Llemma serves as a platform for new research and tools at the intersection of generative models and mathematics.

Tail estimates for stationary KPZ models
2023-10-25 15:30 - 16:30 | Virtual
Probability Seminar
Speaker: Benjamin Landon (University of Toronto)
Title: Tail estimates for stationary KPZ models
Abstract: The limiting distributions of the KPZ universality class exhibit tail exponents of 3/2 and 3. In this talk we will review recent work studying the upper tail exponent 3/2 in the moderate deviations regime of several KPZ models at finite size, including the stochastic six vertex model, the ASEP and a class of non-integrable interacting diffusions.
Joint work with Christian Noack and Phil Sosoe.

Scaling behavior and control of nuclear wrinkling
2023-10-26 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Nicolas Romeo (UChicago)
Title: Scaling behavior and control of nuclear wrinkling
Abstract: The cell nucleus is enveloped by a complex membrane, whose wrinkling has been implicated in disease and cellular aging. The biophysical dynamics and spectral evolution of nuclear wrinkling during multicellular development remain poorly understood due to a lack of direct quantitative measurements. We characterize the onset and dynamics of nuclear wrinkling during egg development in the fruit fly when nurse cell nuclei increase in size and display stereotypical wrinkling behaviour. A spectral analysis of three-dimensional high-resolution live-imaging data from several hundred nuclei reveals a robust asymptotic power-law scaling of angular fluctuations consistent with renormalization and scaling predictions from a nonlinear elastic shell model. We further demonstrate that nuclear wrinkling can be reversed through osmotic shock and suppressed by microtubule disruption, providing tunable physical and biological control parameters for probing the mechanical properties of the nuclear envelope, highlighting in passing the importance of nonlinear response to biological robustness.

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.

Extension of pluricanonical forms in positive and mixed characteristics
2023-10-27 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Iacopo Brivio (CMSA)
Title: Extension of pluricanonical forms in positive and mixed characteristics
Abstract: The geometry of a complex manifold $X$ is to a large extent determined by its pluricanonical forms, i.e. global sections of $(Omega^{dim X}_X)^{otimes m}$ for $mgeq 0$. A famous theorem of Siu states that when $Xto D$ is a smooth projective family of complex manifolds, then every pluricanonical form on $X_0$ extends to the whole of $X$. Both this theorem and the tools used in its proof had a deep impact in higher dimensional birational geometry and moduli theory. In this talk I am going to give an overview of the extension problem for pluricanonical forms when $D$ is the spectrum of a positive or mixed characteristic discrete valuation ring.

Homotopy categories of rings: some properties and consequences in module categories
2023-10-30 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Manuel Cortés-Izurdiaga (University of Malaga)
Title: Homotopy categories of rings: some properties and consequences in module categories
Abstract: Given a non-necessarily commutative ring with unit and an additive subcategory of the category of right modules, one can consider complexes of modules in the subcategory and the corresponding homotopy category. Sometimes, these homotopy categories are the first step in studying other (algebraic) homotopy categories, such as those associated to a scheme. To study these categories, one can use results from the category of modules or the category of complexes. In the first part of the talk, we will see how some results of homotopy categories of complexes extend to homotopy categories of N-complexes, for a natural number N greater than or equal to 2, using some techniques from module categories, such us the deconstruction of a class of modules.
Another approximation is to use other methods for studying homotopy categories, like those coming from triangulated categories. In some cases, the results obtained in homotopy categories imply some consequences in the category of modules. In the second part of the talk, we will see how to prove the existence of Gorenstein-projective precovers for some specific rings using this approach.

Unveiling Correlated Topological Insulators through Fermionic Tensor Network States
2023-11-01 10:30 - 11:30 | Virtual
Topological Quantum Matter Seminar
Speaker: Shenghan Jiang, Kavli Institute for Theoretical Sciences UCAS
Title: Unveiling Correlated Topological Insulators through Fermionic Tensor Network States
Abstract: The study of topological band insulators has revealed fascinating phases characterized by band topology indices, harboring extraordinary boundary modes protected by anomalous symmetry actions. In strongly correlated systems, it has been established that topological insulator phases persist as stable phases. However, due to the inability to express the ground states of such systems as Slater determinants, the formulation of generic variational wavefunctions for numerical simulations is highly desirable.
In this talk, we tackle this challenge by developing a comprehensive framework with fermionic tensor network states. Starting from simple assumptions, we write down tensor equations, construct edge theories and extract quantum anomaly data for topological insulators. By exhaustively exploring all possible sets of equations, we achieve a systematic classification of topological insulator phases. Imposing the solutions of a given set of equations onto local tensors, we obtain generic variational wavefunctions for corresponding topological insulator phases. Our methodology provides a crucial first step towards simulating topological insulators in strongly correlated systems.

Universality of max-margin classifiers
2023-11-01 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Youngtak Sohn (MIT)
Title: Universality of max-margin classifiers
Abstract: Many modern learning methods, such as deep neural networks, are so complex that they perfectly fit the training data. Despite this, they generalize well to the unseen data. Motivated by this phenomenon, we consider high-dimensional binary classification with linearly separable data. First, we consider Gaussian covariates and characterize linear classification problems for which the minimum norm interpolating prediction rule, namely the max-margin classification, has near-optimal prediction accuracy. Then, we discuss universality of max-margin classification. In particular, we characterize the prediction accuracy of the non-linear random features model, a two-layer neural network with random first layer weights. The spectrum of the kernel random matrices plays a crucial role in the analysis. Finally, we consider the wide-network limit, where the number of neurons tends to infinity, and show how non-linear max-margin classification with random features collapse to a linear classifier with a soft-margin objective.
Joint work with Andrea Montanari, Feng Ruan, Jun Yan, and Basil Saeed.

Solving spin systems, the Babylonian way
2023-11-02 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Nicola Kistler (Johann Wolfgang Goethe-Universität Frankfurt am Main)
Title: Solving spin systems, the Babylonian way
Abstract: The replica method, together with Parisi’s symmetry breaking mechanism, is an extremely powerful tool to compute the limiting free energy of virtually any mean field disordered system. Unfortunately, the tool is dramatically flawed from a mathematical point of view. I will discuss a truly elementary procedure which allows to rigorously implement two (out of three) steps of the replica method, and conclude with some remarks on the relation between this new point of view and old work by Mezard and Virasoro on the microstructure of ultrametricity, the latter being the fundamental yet unjustified Ansatz in the celebrated Parisi solution. We are still far from a clear understanding of the issues, but quite astonishingly, evidence is mounting that Parisi’s ultrametricity assumption, the onset of scales and the universal hierarchical self-organisation of random systems in the infinite volume limit, is intimately linked to hidden geometrical properties of large random matrices which satisfy rules reminiscent of the popular SUDOKU game.

Landscape of quantum phases in quantum materials
2023-11-02 16:30 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Joint Quantum Matter in Mathematics and Physics and Topological Quantum Matter Seminar
Speaker: Liujun Zou (Perimeter Institute)
Title: Landscape of quantum phases in quantum materials

Abstract: A central goal of condensed matter physics is to understand which quantum phases of matter can emerge in a quantum material. For this purpose, one should be able to not only describe the quantum phases using some effective field theories, but also capture the important microscopic information of the material via mathematical formulation. In this talk, I will present a framework to classify quantum phases in quantum materials, where the microscopic information of a material is encoded in its quantum anomaly. I will talk about the application of this framework to classify various exotic quantum phases of matter in different lattice systems. Using our framework, we have obtained many results unexpected from the previous literature.

Deformations of Landau-Ginzburg models and their fibers
2023-11-03 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Zi Yang Kang
Title: Deformations of Landau-Ginzburg models and their fibers
Abstract: In mirror symmetry, the dual object to a Fano variety is a Landau-Ginzburg model. Broadly, a Landau-Ginzburg model is quasi-projective variety Y with a superpotential function w, but not all such pairs correspond to Fano varieties under mirror symmetry, so a very natural question to ask is: Which Landau-Ginzburg models are mirror to Fano varieties? In this talk, I will discuss a cohomological characterization of mirrors of (semi-)Fano varieties, focusing on the case of threefolds. I’ll discuss how this characterization relates to the deformation and Hodge theory of (Y,w), and in particular, how the classification of (semi-)Fano threefolds is related to questions about moduli spaces of lattice polarized K3 surfaces.

Deformations of Landau-Ginzburg models and their fibers
2023-11-06 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Andrew Harder, Lehigh University
Title: Deformations of Landau-Ginzburg models and their fibers
Abstract: In mirror symmetry, the dual object to a Fano variety is a Landau-Ginzburg model. Broadly, a Landau-Ginzburg model is quasi-projective variety Y with a superpotential function w, but not all such pairs correspond to Fano varieties under mirror symmetry, so a very natural question to ask is: Which Landau-Ginzburg models are mirror to Fano varieties? In this talk, I will discuss a cohomological characterization of mirrors of (semi-)Fano varieties, focusing on the case of threefolds. I’ll discuss how this characterization relates to the deformation and Hodge theory of (Y,w), and in particular, how the classification of (semi-)Fano threefolds is related to questions about moduli spaces of lattice polarized K3 surfaces.

Impossibility results in classical dynamical systems
2023-11-06 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Matthew Foreman (UC Irvine)
Title: Impossibility results in classical dynamical systems
Abstract: In 1932, motivated by questions in statistical and celestial mechanics, von Neumann proposed classifying the statistical behavior of dynamical systems. In the 1960’s, motivated by work of Poincaré, Smale proposed classifying the qualitative behavior of dynamical systems. These questions laid the groundwork for enormous amounts of work, but the fundamental questions remain open. This talk shows that they are impossible to answer in a rigorous sense. The talk will discuss various kinds of impossibility results and describe how they apply to von Neumann’s program and Smale’s program.

Fluid stabilization in slowly expanding cosmological spacetime
2023-11-07 11:00 - 12:00 | Hybrid – G10
General Relativity Seminar
Speaker: David Fajman (Vienna)
Title: Fluid stabilization in slowly expanding cosmological spacetime
Abstract: Relativistic fluids are known to form shocks during their evolution from near-homogeneous initial data. In expanding spacetimes, shock formation is suppressed, if the expansion is sufficiently strong. We refer to this effect as fluid stabilization. The occurrence of this phenomenon depends on features of the fluid and has implications for our understanding of structure formation and cosmological evolution. While the effect is well studied in the regime of accelerated expansion, in recent years it has been shown that fluid stabilization occurs as well in spacetimes with slower expansion rates. In this talk we present different recent results on fluid stabilization in slowly expanding spacetimes and aspects of the methods involved in the respective proofs.

Talk via Zoom, broadcast in G10

CMSA Q and A Seminar 11/7/2023
2023-11-07 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Michael Hopkins (Harvard Math) and Minhea Popa (Harvard Math)
Topics:
Mike Hopkins: What are topological modular forms?
Mihnea Popa: What is the abundance conjecture and why should it to be true?

A Long Exact Sequence in Symmetry Breaking
2023-11-07 16:30 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Cameron Krulewski (MIT) and Leon Liu (Harvard)
Title: A Long Exact Sequence in Symmetry Breaking
Abstract: We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations that sometimes occur at the cores of these objects. These are topologically protected by an ’t Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the “symmetry breaking long exact sequence” (SBLES).
The mathematical backbone of the SBLES is the Smith homomorphism, which gives a family of maps between twisted bordism groups. Though many examples have been studied, we give the first completely general account of the Smith homomorphism. We lift it to a map of Thom
spectra and identify the cofiber, producing a long exact sequence of twisted bordism groups; the SBLES is the Anderson dual of that long exact sequence. Our work develops further the theory of higher Berry phase and its bulk-boundary correspondence and serves as a new computational tool for classifying symmetry protected topological phases.

Peano: Learning Formal Mathematical Reasoning Without Human Data
2023-11-08 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Gabriel Poesia, Dept. of Computer Science, Stanford University
Title: Peano: Learning Formal Mathematical Reasoning Without Human Data
Abstract: Peano is a theorem proving environment in which a computational agent can start tabula rasa in a new domain, learn to solve problems through curiosity-driven exploration, and create its own higher level actions. Gabriel will describe the system, present case studies on learning to solve simple algebra problems from the Khan Academy platform, and describe work on progress on learning the Natural Number Game, a popular introduction to theorem proving in Lean for mathematicians.

Fitting ellipsoids to random points
2023-11-08 15:30 - 16:30 | Virtual
Probability Seminar
Speaker: Antoine Maillard (ETH Zürich)
Title: Fitting ellipsoids to random points
Abstract: We consider the problem of exactly fitting an ellipsoid (centered at 0) to n standard Gaussian random vectors in dimension d, for very large n and d. This problem has connections to questions in statistical learning and theoretical computer science, and is conjectured to undergo a sharp transition: with high probability, it has a solution if n < d^2/4, while it is not satisfiable if n > d^2/4. In this talk we will discuss the origin of this conjecture, and highlight some recent progress, in three different directions:

A proof that the problem is feasible for n < d^2 / C, for some (large) constant C, significantly improving over previously-known bounds.
A non-rigorous characterization of the conjecture, as well as significant generalizations, using analytical methods of statistical physics.
A rigorous proof of a satisfiability transition exactly at n = d^2 / 4 in a slightly relaxed version of the problem, the first rigorous result characterizing the expected phase transition in ellipsoid fitting. The proof is inspired by the non-rigorous characterization discussed above.

This talk is based on the three manuscripts: arXiv:2307.01181, arXiv:2310.01169, arXiv:2310.05787, which are joint works with A. Bandeira, Tim Kunisky, Shahar Mendelson and Elliot Paquette.

Nuclear chromodynamics: non-equilibrium phase transition in the nucleus of a living cell
2023-11-09 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Alexander Grosberg (NYU)

Title: Nuclear chromodynamics: non-equilibrium phase transition in the nucleus of a living cell
Abstract: Nucleus of a living cell houses a cell genome – a polymer called chromatin, which is a functional form of DNA. It is very long, e.g., 2 meters long for every human cell. Nucleus is also an arena of incessant energy-driven activity. Experiments show that chromatin undergoes large scale motions sustained over long times of order seconds. In the talk, after reviewing the phenomenology, I will show how these flows may arise due to a phase transition in which chromatin-driving motors, such as RNA polymerase, form a polar (“ferromagnetic”) order controlled by hydrodynamic interactions. The talk is based on the joint work with I.Eshghi and A.Zidovska.

Uniqueness of Landau levels and their analogs with higher Chern numbers
2023-11-09 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Bruno Mera, Instituto Superior Tecnico
Title: Uniqueness of Landau levels and their analogs with higher Chern numbers
Abstract: Lowest Landau level wavefunctions are eigenstates of the Hamiltonian of a charged par- ticle in two dimensions under a uniform magnetic field. They are known to be holomorphic both in real and momentum spaces, and also exhibit uniform, translationally invariant, geometrical properties in momentum space. In this talk, using the Stone-von Neumann the- orem, we show that lowest Landau level wavefunctions are indeed the only possible states with unit Chern number satisfying these conditions. We also prove the uniqueness of their direct analogs with higher Chern numbers and provide their expressions.
Ref: Bruno Mera and Tomoki Ozawa. Uniqueness of Landau levels and their analogs with higher Chern numbers. arXiv:2304.00866, 2023. arXiv:2304.00866.

Stacky small resolutions of determinantal octic double solids and noncommutative Gopakumar-Vafa invariants
2023-11-13 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar

Speaker: Sheldon Katz, UIUC
Title: Stacky small resolutions of determinantal octic double solids and noncommutative Gopakumar-Vafa invariants
Abstract: A determinantal octic double solid is the double cover X of P^3 branched along the degree 8 determinant of a symmetric matrix of homogeneous forms on P^3. These X are nodal CY threefolds which do not admit a projective small resolution. B-model techniques can be applied to compute GV invariants up to g le 32. This raises the question: what is the geometric meaning of these invariants?
Evidence suggests that these enumerative invariants are associated with moduli stacks of coherent sheaves of modules over a sheaf B of noncommutative algebras on X constructed by Kuznetsov. One of these moduli stacks is a stacky small resolution X’ of X itself. This leads to another geometric interpretation of the invariants as being associated with moduli of sheaves on X’ twisted by a Brauer class. Geometric computations based on these working definitions always agree with the B-model computations.
This talk is based on joint work with Albrecht Klemm, Thorsten Schimannek, and Eric Sharpe.

Koszul duality in QFT
2023-11-13 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Brian Williams (Boston University)
Title: Koszul duality in QFT
Abstract: We will describe appearances of the algebraic phenomena of Koszul duality in the context of boundary conditions and defects in quantum field theory. Primarily motivated by topological string theory, this point of view was pioneered by Costello and Li in their proposal for a twisted version of the AdS/CFT correspondence. Since then, many important examples of (twisted) holographic dualities in string and M-theory have been studied in work of Costello, Gaiotto, Paquette and many others. I will survey some of these examples and some current work with Raghavendran and Saberi which uses this formalism to predict exceptional symmetries present in M-theory.

Quasi-Local Mass in a Binary Black Hole Merger
2023-11-14 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Daniel Kolb (Max Planck Institute)
Title: Quasi-Local Mass in a Binary Black Hole Merger
Abstract: One of the major open problems in classical general relativity is how one should define the mass of a finite region of space. In this talk, we will investigate a promising definition proposed by Wang and Yau in 2009. A closed 2-surface bounding the region of interest is embedded isometrically into Minkowski space. The mass is then calculated by comparing the extrinsic geometries. The Wang-Yau mass has many desirable properties, but it has previously not been calculated for surfaces in dynamical spacetimes. To remedy this, we will discuss how the Wang-Yau mass can be computed in practice and extend the definition to surfaces important in black hole dynamics: their quasi-local horizons. Finally, we look at how this mass behaves in a merger of black holes.

A bulk gap in the presence of edge states for a truncated Haldane pseudopotential
2023-11-15 10:30 - 11:30 | Virtual
Topological Quantum Matter Seminar
Speaker: Amanda Young, UIUC
Title: A bulk gap in the presence of edge states for a truncated Haldane pseudopotential
Abstract: Haldane pseudopotentials were first introduced as Hamiltonian models for the fractional quantum Hall effect, and it has been long expected that they should exhibit the characteristic properties of this exotic phase of matter, including a spectral gap above the ground state energy. We will discuss recent work that verified this gap conjecture for a truncated version of the 1/3-filled Haldane pseudopotential in the cylinder geometry. Numerical evidence suggested that for open boundary conditions the gap of the truncated model closes as the cylinder radius converges to zero and that this closure is due to the presence of edge modes; in contrast, for periodic boundary conditions, the gap remains robustly order one in the same radius limit. The standard scheme for applying spectral gap estimating techniques to the model with periodic boundary conditions, though, produces a lower bound on the bulk gap that still reflects the energy of the edge modes. To obtain an estimate on the bulk gap that reflects its true behavior, a new gap estimating strategy was developed. By customizing the spectral gap method to key invariant subspaces of the Hamiltonian, we are able to successfully avoid the edge states and produce a more accurate lower bound on the bulk gap. In this talk, we discuss this invariant subspace strategy for proving bulk gaps in the presence of edge states. This is based off joint work with S. Warzel.

On the Power of Forward pass through Transformer Architectures
2023-11-15 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Abhishek Panigrahi, Dept. of Computer Science, Princeton University
Title: On the Power of Forward pass through Transformer Architectures
Abstract: Highly trained transformers are capable of interesting computations as they infer for an input. The exact mechanism that these models use during forward passes is an interesting area of study. This talk studies two interesting phenomena.
In the first half, we explore how and why pre-trained language models, specifically BERT of moderate sizes, can effectively learn linguistic structures like parse trees during pre-training. Specifically, using synthetic data through PCFGs, we show how moderate-sized transformers can perform forward-backward parsing, also known as the inside-outside algorithm, during inference. We further understand the role of the pre-training loss for the model to learn to parse during pre-training.
In the second half, we consider in-context learning of large language models, where they learn to reason on the fly. An ongoing hypothesis is that transformers simulate gradient descent at inference to perform in-context learning. We propose the Transformer in Transformer (TinT) framework, which creates explicit transformer architectures that can simulate and fine-tune a small pre-trained transformer model during inference. E.g. a 1.3B parameter TINT model can simulate and fine-tune a 125 million parameter model in a single forward pass. This framework suggests that large transformers might execute intricate sub-routines during inference, and provides insights for enhancing their capabilities through intelligent design considerations.
