CMSA

Unveiling Correlated Topological Insulators through Fermionic Tensor Network States
2023-11-01 10:30 - 11:30 (America/New_York) | 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.

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.

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.

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.

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.

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.

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.

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?

Resolving memory in numerical relativity, and fixing BMS frames for modeling
2023-10-24 11:00 - 12:00
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.

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

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.

Black Holes as Quantum Systems
2023-10-20 12:00 - 13:00 | 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.

CMSA Director Dan Freed featured in Harvard Gazette
2023-10-20 09:00
CMSA Director Dan Freed featured in Harvard Gazette
“Center welcomes new director, a Harvard alum who will explore ‘beautiful, deep’ interactions between mathematics, science.”
Link to article.
Story by Anne J. Manning, Harvard Staff Writer
Photo courtesy of Niles Singer, Harvard University.

Geometry of the doubly periodic Aztec dimer model
2023-10-18 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Tomas Berggren (MIT)
Title: Geometry of the doubly periodic Aztec dimer model
Abstract: Random dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit, we see three types of macroscopic regions — known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.

Physics of Language Models: Knowledge Storage, Extraction, and Manipulation
2023-10-18 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Yuanzhi Li, CMU Dept. of Machine Learning and Microsoft Research
Title: Physics of Language Models: Knowledge Storage, Extraction, and Manipulation
Abstract: Large language models (LLMs) can memorize a massive amount of knowledge during pre-training, but can they effectively use this knowledge at inference time? In this work, we show several striking results about this question. Using a synthetic biography dataset, we first show that even if an LLM achieves zero training loss when pretraining on the biography dataset, it sometimes can not be finetuned to answer questions as simple as “What is the birthday of XXX” at all. We show that sufficient data augmentation during pre-training, such as rewriting the same biography multiple times or simply using the person’s full name in every sentence, can mitigate this issue. Using linear probing, we unravel that such augmentation forces the model to store knowledge about a person in the token embeddings of their name rather than other locations.
We then show that LLMs are very bad at manipulating knowledge they learn during pre-training unless a chain of thought is used at inference time. We pretrained an LLM on the synthetic biography dataset, so that it could answer “What is the birthday of XXX” with 100% accuracy. Even so, it could not be further fine-tuned to answer questions like “Is the birthday of XXX even or odd?” directly. Even using Chain of Thought training data only helps the model answer such questions in a CoT manner, not directly.
We will also discuss preliminary progress on understanding the scaling law of how large a language model needs to be to store X pieces of knowledge and extract them efficiently. For example, is a 1B parameter language model enough to store all the knowledge of a middle school student?

Composite fermions and the fractional quantum anomalous Hall effect
2023-10-18 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Hart Goldman, University of Chicago
Title: Composite fermions and the fractional quantum anomalous Hall effect
Abstract: Recent experiments have revealed evidence for fractional quantum anomalous Hall (FQAH) states at zero magnetic field in a growing number of moire materials. In this talk, I will argue that a composite fermion description, already a unifying framework for the phenomenology of 2d electron gases at high magnetic fields, provides a similarly powerful perspective in this new zero-field context. In particular, a central prediction of the composite fermion framework is a non-Fermi liquid metal of composite fermions at even-denominator fillings. To this end, I will present exact diagonalization evidence for such composite Fermi liquid states at zero magnetic field in twisted MoTe2 bilayers, at fillings n = 1/2 and n = 3/4. Dubbing these states anomalous composite Fermi liquids (ACFLs), I will argue that they play a central organizing role in the FQAH phase diagram. I will also develop a long wavelength theory for this ACFL state, which offers concrete experimental predictions that I will discuss in relation to current measurements. For example, upon doping the composite Fermi sea, one obtains a Jain sequence of FQAH states consistent with those observed experimentally, as well as a new type of commensurability oscillations originating from the superlattice potential intrinsic to the system. Finally, I will discuss opportunities for new physics not possible in quantum Hall systems at finite magnetic field.

An exploration of infinite games—infinite Wordle and the Mastermind numbers
2023-10-16 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Joel D. Hamkins (Notre Dame and Oxford)
Title: An exploration of infinite games—infinite Wordle and the Mastermind numbers
Abstract: Let us explore the nature of strategic reasoning in infinite games, focusing on the cases of infinite Wordle and infinite Mastermind. The familiar game of Wordle extends naturally to longer words or even infinite words in an idealized language, and Mastermind similarly has natural infinitary analogues. What is the nature of play in these infinite games? Can the codebreaker play so as to win always at a finite stage of play? The analysis emerges gradually, and in the talk I shall begin slowly with some easy elementary observations. By the end, however, we shall engage with sophisticated ideas in descriptive set theory, a kind of infinitary information theory. Some assertions about the minimal size of winning sets of guesses, for example, turn out to be independent of the Zermelo-Fraenkel ZFC axioms of set theory. Some questions remain open.

Breaking ergodicity: quantum scars and regular eigenstates
2023-10-16 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Ceren Dag, Harvard
Title: Breaking ergodicity: quantum scars and regular eigenstates
Abstract: Quantum many-body scars (QMBS) consist of a few low-entropy eigenstates in an otherwise chaotic many-body spectrum and can weakly break ergodicity resulting in robust oscillatory dynamics. The notion of QMBS follows the original single-particle scars introduced within the context of quantum billiards, where scarring manifests in the form of a quantum eigenstate concentrating around an underlying classical unstable periodic orbit (UPO). A direct connection between these notions remains an outstanding problem. Here, we study a many-body spinor condensate that, owing to its collective interactions, is amenable to the diagnostics of scars. We characterize the system’s rich dynamics, spectrum, and phase space, consisting of both regular and chaotic states. The former are low in entropy, violate the Eigenstate Thermalization Hypothesis (ETH), and can be traced back to integrable effective Hamiltonians, whereas most of the latter are scarred by the underlying semiclassical UPOs, while satisfying ETH. We outline an experimental proposal to probe our theory in trapped spin-1 Bose-Einstein condensates. If time permits, I will also mention our latest efforts in introducing spatial dimension to this model with a true semiclassical limit, and how quantum scars persist to exist in a many-body system. Reference: arXiv 2306.10411, in peer review.

Moduli of boundary polarized Calabi-Yau pairs
2023-10-16 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Pre-talk Speaker: Rosie Shen (Harvard): 10:00-10:30 am
Pre-talk Title: Introduction to the singularities of the MMP

Speaker: Dori Bejleri (Harvard Math & CMSA)
Title: Moduli of boundary polarized Calabi-Yau pairs
Abstract: The theories of KSBA stability and K-stability furnish compact moduli spaces of general type pairs and Fano pairs respectively. However, much less is known about the moduli theory of Calabi-Yau pairs. In this talk I will present an approach to constructing a moduli space of Calabi-Yau pairs which should interpolate between KSBA and K-stable moduli via wall-crossing. I will explain how this approach can be used to construct projective moduli spaces of plane curve pairs. This is based on joint work with K. Ascher, H. Blum, K. DeVleming, G. Inchiostro, Y. Liu, X. Wang.

On the Breakdown of Einstein's Gravity
2023-10-13 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Puskar Mondal (CMSA)
Title: On the Breakdown of Einstein’s Gravity
Abstract: It is important to understand under which conditions, the solutions of non-linear hyperbolic PDEs break down in finite time. In the context of Einstein’s gravity, this is very closely tied to naked singularity formation and Penrose’s weak cosmic censorship conjecture. In this talk, I will give sharp estimates on the relevant geometric entities that allow one to continue the solutions of Einstein’s equations indefinitely in the future in a ‘time’ direction without forming a naked singularity.

Contractility, structure formation and fluctuations in active gels, with and without molecular motors
2023-10-12 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Fred MacKintosh (Rice University)
Title: Contractility, structure formation and fluctuations in active gels, with and without molecular motors
Abstract: Various processes in living cells depend on contractile forces that are often generated by myosin motors in concert with polar actin filaments. A textbook example of this is the actomyosin contractile ring that forms during cell division. Recent evidence, however, has begun to suggest alternate or redundant mechanisms that do not depend on myosin. Experiments on simplified, reconstituted systems also point to contractility and structure formation in disordered, apolar arrays of filaments. We propose a motor-free mechanism that can generate contraction in biopolymer networks without the need for motors such as myosin or polar filaments such as actin. This mechanism is based on active binding and unbinding of cross-linkers that breaks the principle of detailed balance, together with the asymmetric force-extension response of semiflexible biopolymers. We discuss the resulting force-velocity relation and other implications of this, as well as possible evidence for non-motor force generation.

Non-invertible symmetries, leptons, quarks, and why multiple generations
2023-10-11 16:30 - 18:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Quantum Matter Seminar
Speaker: Seth Koren (Notre Dame)
Title: Non-invertible symmetries, leptons, quarks, and why multiple generations
Abstract: Generalized global symmetries are present in theories of particle physics, and understanding their structure can give insight into these theories and UV completions thereof. After discussing the generalized symmetries of the Standard Model, we will go Beyond and show that the identification of a non-invertible symmetry of Z’ models of L_µ – L_τ reveals the existence of non-Abelian horizontal gauge theories which naturally produce exponentially small Dirac neutrino masses. Next we will uncover a subtler non-invertible symmetry in horizontal gauge theories of the quark sector which will lead us to a massless down-type quarks solution to strong CP in color-flavor unification. Intriguingly, this theory works by virtue of the SM having the same numbers of colors and generations.

LeanDojo: Theorem Proving with Retrieval-Augmented Language Models
2023-10-11 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speaker: Alex Gu, MIT Dept. of EE&CS
Title: LeanDojo: Theorem Proving with Retrieval-Augmented Language Models
Abstract: Large language models (LLMs) have shown promise in proving formal theorems using proof assistants such as Lean. However, existing methods are difficult to reproduce or build on, due to private code, data, and large compute requirements. This has created substantial barriers to research on machine learning methods for theorem proving. We introduce LeanDojo: an open-source Lean playground consisting of toolkits, data, models, and benchmarks. LeanDojo extracts data from Lean and enables interaction with the proof environment programmatically. It contains fine-grained annotations of premises in proofs, providing valuable data for premise selection: a key bottleneck in theorem proving. Using this data, we develop ReProver (Retrieval-Augmented Prover): the first LLM-based prover that is augmented with retrieval for selecting premises from a vast math library. It is inexpensive and needs only one GPU week of training. Our retriever leverages LeanDojo’s program analysis capability to identify accessible premises and hard negative examples, which makes retrieval much more effective. Furthermore, we construct a new benchmark consisting of 96,962 theorems and proofs extracted from Lean’s math library. It features a challenging data split requiring the prover to generalize to theorems relying on novel premises that are never used in training. We use this benchmark for training and evaluation, and experimental results demonstrate the effectiveness of ReProver over non-retrieval baselines and GPT-4. We thus provide the first set of open-source LLM-based theorem provers without any proprietary datasets and release it under a permissive MIT license to facilitate further research.

CMSA Q and A Seminar 10/10/23
2023-10-10 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Dan Freed (Harvard Math and CMSA) and Sunghyuk Park (CMSA)
Topics:
Dan Freed: What is Dijkgraaf-Witten theory?
Sunghyuk Park: What happened at the Clay Math Institute Workshop?

Tidal Squeezing of Black Holes
2023-10-10 11:00 - 12:00
General Relativity Seminar
Speaker: Maria Rodriguez (Utah)
Title: Tidal Squeezing of Black Holes
Abstract: Recent developments indicate that Kerr black holes do not deform when perturbed by a static external gravitational field. Relying on hidden symmetries, compelling progress has been achieved to explain that Love numbers for Kerr black holes vanish. How does the phenomenon of tidal squeezing manifest in broader contexts? An elementary presentation of dynamical tidal squeezing of Kerr black holes will be given.

Random matrices and large deviations
2023-10-06 12:00 - 13:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Benjamin McKenna
Title: Random matrices and large deviations
Abstract: We give a generalist overview of random matrices and their (a)typical behaviors. In recent years, classical results have been complemented by a variety of new ones, in both the math and physics literatures, whose proofs leverage connections with special integrals over matrix groups. Some of these models exhibit interesting transition points, whose motivating relationships to eigenvector (de)localization are not yet fully understood. Based on joint work with Jonathan Husson.

Dipolar and modulated symmetry protected topological phases
2023-10-04 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Ho Tat Lam, MIT
Title: Dipolar and modulated symmetry protected topological phases
Abstract: Modulated symmetries are symmetries whose symmetry generators exhibit spatial modulations. We will discuss one-dimensional symmetry protected topological (SPT) phases protected by modulated symmetries. We will present a simple recipe for constructing modulated SPT models by generalizing the concept of decorated domain walls. We will then focus on the simplest modulated SPT protected by dipolar symmetries, classify them using matrix product states and construct their response field theories using twisted finite tensor gauge theories.

CMSA Q and A Seminar 10/3/2023
2023-10-03 12:30 - 13:30 | Common Room, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA Q and A Seminar
Speakers: Dan Freed (Harvard Math & CMSA) and Dan Berwick-Evans (University of Illinois at Urbana-Champaign)

Topics:
Dan Freed: What is framing anomaly? How is it different from other anomalies?
Dan Berwick-Evans: What is Atiyah-Singer index theorem?

A Smooth Horizon without a Smooth Horizon
2023-10-03 11:00 - 12:00
General Relativity Seminar
Speaker: Chethan Krishnan (IISc Bangalore)
Title: A Smooth Horizon without a Smooth Horizon
Abstract: I will talk about some work that is about to appear, where we note one precise way in which the stretched horizon can simulate a smooth horizon. I will also make an effort to put things in some perspective (brickwalls, fuzzballs, Type I algebras,…)

Gravitational Instantons
2023-10-02 16:30 - 17:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Yu-Shen Lin (Boston University)
Title: Gravitational Instantons
Abstract: Gravitational instantons were introduced by Hawking as building blocks of his Euclidean quantum gravity theory back in the 1970s. These are non-compact Calabi-Yau surfaces with L2 curvature and thus can be viewed as the non-compact analogue of K3 surfaces. K3 surfaces are 2-dimensional Calabi-Yau manifolds and are usually the testing stone before conquering the general Calabi-Yau problems. The moduli space of K3 surfaces and its compactification on their own form important problems in various branches in geometry. In this talk, we will discuss the Torelli theorem of gravitational instantons, how the cohomological invariants of a gravitational instanton determine them. As a consequence, this leads to a description of the moduli space of gravitational instantons.
