CMSA

Knowledge graph representation: From recent models towards a theoretical understanding
2021-01-27 15:00 - 16:00 (America/New_York) | MA
Speaker: Carl Allen and Ivana Balažević – University of Edinburgh School of Informatics
Title: Knowledge graph representation: From recent models towards a theoretical understanding
Abstract: Knowledge graphs (KGs), or knowledge bases, are large repositories of facts in the form of triples (subject, relation, object), e.g. (Edinburgh, capital_of, Scotland). Many models have been developed to succinctly represent KGs such that known facts can be recalled (question answering) and, more impressively, previously unknown facts can be inferred (link prediction). Subject and object entities are typically represented as vectors in R^d and relations as mappings (e.g. linear transformations) between them. Such representation can be interpreted as positioning entities in a space such that relations are implied by their relative locations. In this talk we give an overview of knowledge graph representation including select recent models; and, by drawing a connection to word embeddings, explain a theoretical model for how semantic relationships can correspond to geometric structure.

A Mathematical Exploration of Why Language Models Help Solve Downstream Tasks
2021-02-10 15:00 - 16:00 | MA
Speaker: Nikunj Saunshi, Dept. of Computer Science, Princeton University
Title: A Mathematical Exploration of Why Language Models Help Solve Downstream Tasks
Abstract: Autoregressive language models pretrained on large corpora have been successful at solving downstream tasks, even with zero-shot usage. However, there is little theoretical justification for their success. This paper considers the following questions: (1) Why should learning the distribution of natural language help with downstream classification tasks? (2) Why do features learned using language modeling help solve downstream tasks with linear classifiers? For (1), we hypothesize, and verify empirically, that classification tasks of interest can be reformulated as next word prediction tasks, thus making language modeling a meaningful pretraining task. For (2), we analyze properties of the cross-entropy objective to show that eps-optimal language models in cross-entropy (log-perplexity) learn features that are O(sqrt{eps}) good on such linear classification tasks, thus demonstrating mathematically that doing well on language modeling can be beneficial for downstream tasks. We perform experiments to verify assumptions and validate our theoretical results. Our theoretical insights motivate a simple alternative to the cross-entropy objective that performs well on some linear classification tasks.

A Mathematical Language
2021-02-24 15:00 - 16:00 | MA
Speaker: Thomas Hales, Univ. of Pittsburgh Dept. of Mathematics
Title: A Mathematical Language
Abstract: A controlled natural language for mathematics is an artificial language that is designed in an explicit way with precise computer-readable syntax and semantics. It is based on a single natural language (which for us is English) and can be broadly understood by mathematically literate English speakers. This talk will describe the design of a controlled natural language for mathematics that has been influenced by the Lean theorem prover, by TeX, and by earlier controlled natural languages. The semantics are provided by dependent type theory.

Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
2021-03-03 15:00 - 16:00 | Virtual
Speaker: Jason Rute, CIBO Technologies
Title: Neural Theorem Proving in Lean using Proof Artifact Co-training and Language Models
Abstract: Labeled data for imitation learning of theorem proving in large libraries of formalized mathematics is scarce as such libraries require years of concentrated effort by human specialists to be built. This is particularly challenging when applying large Transformer language models to tactic prediction, because the scaling of performance with respect to model size is quickly disrupted in the data-scarce, easily-overfitted regime. We propose PACT ({bf P}roof {bf A}rtifact {bf C}o-{bf T}raining), a general methodology for extracting abundant self-supervised data from kernel-level proof terms for co-training alongside the usual tactic prediction objective. We apply this methodology to Lean, an interactive proof assistant which hosts some of the most sophisticated formalized mathematics to date. We instrument Lean with a neural theorem prover driven by a Transformer language model and show that PACT improves theorem proving success rate on a held-out suite of test theorems from 32% to 48%.

The Ramanujan Machine: Using Algorithms for the Discovery of Conjectures on Mathematical Constants
2021-03-10 15:00 - 16:00 | Virtual
Speaker: Ido Kaminer, Technion – Israel Institute of Technology, Faculty of Electrical Engineering
Title: The Ramanujan Machine: Using Algorithms for the Discovery of Conjectures on Mathematical Constants
Abstract: In the past, new conjectures about fundamental constants were discovered sporadically by famous mathematicians such as Newton, Euler, Gauss, and Ramanujan. The talk will present a different approach – a systematic algorithmic approach that discovers new mathematical conjectures on fundamental constants. We call this approach “the Ramanujan Machine”. The algorithms found dozens of well-known formulas as well as previously unknown ones, such as continued fraction representations of π, e, Catalan’s constant, and values of the Riemann zeta function. Part of the conjectures were in retrospect simple to prove, whereas others remained so far unproved. We will discuss these puzzles and wider open questions that arose from this algorithmic investigation – specifically, a newly-discovered algebraic structure that seems to generalize all the known formulas and connect between fundamental constants. We will also discuss two algorithms that proved useful in finding conjectures: a variant of the meet-in-the-middle algorithm and a gradient descent algorithm tailored to the recurrent structure of continued fractions. Both algorithms are based on matching numerical values; consequently, they conjecture formulas without providing proofs or requiring prior knowledge of the underlying mathematical structure. This way, our approach reverses the conventional usage of sequential logic in formal proofs; instead, using numerical data to unveil mathematical structures and provide leads to further mathematical research.

Word and Graph Embeddings for Machine Learning
2021-03-24 15:00 - 16:00 | MA
Speaker: Steve Skiena, Dept. of Computer Science and AI Insititute, Stony Brook University
Title: Word and Graph Embeddings for Machine Learning
Abstract: DeepWalk is an approach we have developed to construct vertex embeddings: vector representations of vertices which be applied to a very general class of problems in data mining and information retrieval. DeepWalk exploits an appealing analogy between sentences as sequences of words and random walks as sequences of vertices to transfer deep learning (unsupervised feature learning) techniques from natural language processing to network analysis. It has become extremely popular, having been cited by over 4600 research papers since its publication at KDD 2014. In this talk, I will introduce the notion of graph embeddings, and demonstrate why they make such powerful features for machine learning applications. I will focus on more recent efforts concerning (1) fast embedding methods for very large networks, (2) techniques for embedding dynamic graphs, and (3) embedding spaces as models for knowledge generation.

Doing Mathematics with Simple Types: Infinitary Combinatorics in Isabelle/HOL
2021-03-31 15:00 - 16:00 | MA
Speaker: Lawrence Paulson, University of Cambridge Computer Laboratory
Title: Doing Mathematics with Simple Types: Infinitary Combinatorics in Isabelle/HOL
Abstract: Are proof assistants relevant to mathematics? One approach to this question is to explore the breadth of mathematical topics that can be formalised. The partition calculus was introduced by Erdös and R. Rado in 1956 as the study of “analogues and extensions of Ramsey’s theorem”. Highly technical results were obtained by Erdös-Milner, Specker and Larson (among many others) for the particular case of ordinal partition relations, which is concerned with countable ordinals and order types. Much of this material was formalised last year (with the assistance of Džamonja and Koutsoukou-Argyraki). Some highlights of this work will be presented along with general observations about the formalisation of mathematics, including ZFC, in simple type theory.

Type theory from the perspective of artificial intelligence
2021-04-07 15:00 - 16:00 | MA
Speaker: David McAllester – Toyota Technological Institute at Chicago
Title: Type theory from the perspective of artificial intelligence
Abstract: This talk will discuss dependent type theory from the perspective of artificial intelligence and cognitive science. From an artificial intelligence perspective it will be argued that type theory is central to defining the “game” of mathematics — an action space and reward structure for pure mathematics. From a cognitive science perspective type theory provides a model of the grammar of the colloquial (natural) language of mathematics. Of particular interest is the notion of a signature-axiom structure class and the three fundamental notions of equality in mathematics — set-theoretic equality between structure elements, isomorphism between structures, and Birkoff and Rota’s notion of cryptomorphism between structure classes. This talk will present a version of type theory based on set-theoretic semantics and the 1930’s notion of structure and isomorphism given by the Bourbaki group of mathematicians. It will be argued that this “Bourbaki type theory” (BTT) is more natural and accessible to classically trained mathematicians than Martin-Löf type theory (MLTT). BTT avoids the Curry-Howard isomorphism and axiom J of MLTT. The talk will also discuss BTT as a model of MLTT. The BTT model is similar to the groupoid model in that propositional equality is interpreted as isomorphism but different in various details. The talk will also briefly mention initial thoughts in defining an action space and reward structure for a game of mathematics.

A Bayesian neural network predicts the dissolution of compact planetary systems
2021-04-14 15:00 - 16:00 | Virtual
Speaker: Miles Cranmer – Princeton University
Title: A Bayesian neural network predicts the dissolution of compact planetary systems
Abstract: Despite over three hundred years of effort, no solutions exist for predicting when a general planetary configuration will become unstable. I will discuss our deep learning architecture (arxiv:2101.04117) which pushes forward this problem for compact systems. While current machine learning algorithms in this area rely on scientist-derived instability metrics, our new technique learns its own metrics from scratch, enabled by a novel internal structure inspired from dynamics theory. The Bayesian neural network model can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on three-planet configurations, the model demonstrates robust generalization to five-planet systems, even outperforming models designed for that specific set of integrations. I will also discuss some work on recovering symbolic representations of such models using arxiv:2006.11287.

Homotopy type theory and the quest for extensionality
2021-04-21 15:00 - 16:00 | Virtual
Speaker: Michael Shulman – Dept. of Mathematics, University of San Diego
Title: Homotopy type theory and the quest for extensionality
Abstract: Over the past decades, dependent type theory has proven to be a powerful framework for verified software and formalized mathematics. However, its treatment of equality has always been somewhat uncomfortable. Recently, homotopy type theory has made progress towards a more useful notion of equality, which natively implements both isomorphism-invariance in mathematics and representation-independence in programming. This progress is based on ideas from abstract homotopy theory and higher category theory, and with the development of cubical type theories it can be implemented as a true programming language. In this talk, I will survey these developments and their potential applications, and suggest some directions for further improvement.

The complexity of matrix multiplication approached via algebraic geometry and representation theory
2021-09-09 14:00 - 15:00 | MA
Speaker: JM Landsberg, Texas A&M
Title: The complexity of matrix multiplication approached via algebraic geometry and representation theory
Abstract: In 1968 V. Strassen discovered the way we usually multiply matrices is not the most efficient possible, and after considerable work by many authors, it is generally conjectured by computer scientists that as the size of matrices becomes large, it becomes almost as easy to multiply them as it is to add them. I will give a brief history of the problem, explain how this conjecture is naturally understood in the framework of classical algebraic geometry and representation theory, and conclude by describing recent advances using more sophisticated tools from algebraic geometry. For most of the talk, no knowledge of algebraic geometry or representation theory will be needed.

Threshold phenomena in random graphs and hypergraphs
2021-09-10 09:30 - 10:30 | MA
Member Seminar
Speaker: Michael Simkin
Title: Threshold phenomena in random graphs and hypergraphs
Abstract: In 1959 Paul Erdos and Alfred Renyi introduced a model of random graphs that is the cornerstone of modern probabilistic combinatorics. Now known as the “Erdos-Renyi” model of random graphs it has far-reaching applications in combinatorics, computer science, and other fields.
The model is defined as follows: Given a natural number $n$ and a parameter $p in [0,1]$, let $G(n;p)$ be the distribution on graphs with $n$ vertices in which each of the $binom{n}{2}$ possible edges is present with probability $p$, independent of all others. Despite their apparent simplicity, the study of Erdos-Renyi random graphs has revealed many deep and non-trivial phenomena.
A central feature is the appearance of threshold phenomena: For all monotone properties (e.g., connectivity and Hamiltonicity) there is a critical probability $p_c$ such that if $p >> p_c$ then $G(n;p)$ possesses the property with high probability (i.e., with probability tending to 1 as $n to infty$) whereas if $p << p_c$ then with high probability $G(n;p)$ does not possess the property. In this talk we will focus on basic properties such as connectivity and containing a perfect matching. We will see an intriguing connection between these global properties and the local property of having no isolated vertices. We will then generalize the Erdos-Renyi model to higher dimensions where many open problems remain.

Hyperbolic Geometry and Quantum Invariants
2021-09-15 09:30 - 10:30 | Virtual
Speaker: Tian Yang (Texas A&M University)
Title: Hyperbolic Geometry and Quantum Invariants
Abstract: There are two very different approaches to 3-dimensional topology, the hyperbolic geometry following the work of Thurston and the quantum invariants following the work of Jones and Witten. These two approaches are related by a sequence of problems called the Volume Conjectures. In this talk, I will explain these conjectures and present some recent joint works with Ka Ho Wong related to or benefited from this relationship.

CMSA Colloquium 9/15/2021 - 5/25/2022
2021-09-15 09:30 - 2022-05-25 10:30 | CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
During the 2021–22 academic year, the CMSA will be hosting a Colloquium, organized by Du Pei, Changji Xu, and Michael Simkin. It will take place on Wednesdays at 9:30am – 10:30am (Boston time). The meetings will take place virtually on Zoom. All CMSA postdocs/members are required to attend the weekly CMSA Members’ Seminars, as well as the weekly CMSA Colloquium series. The schedule below will be updated as talks are confirmed.
Spring 2022

Date
Speaker
Title/Abstract

1/26/2022
Samir Mathur (Ohio State University)
Title: The black hole information paradox
Abstract: In 1975, Stephen Hawking showed that black holes radiate away in a manner that violates quantum theory. Starting in 1997, it was observed that black holes in string theory did not have the form expected from general relativity: in place of “empty space will all the mass at the center,” one finds a “fuzzball” where the mass is distributed throughout the interior of the horizon. This resolves the paradox, but opposition to this resolution came from groups who sought to extrapolate some ideas in holography. In 2009 it was shown, using some theorems from quantum information theory, that these extrapolations were incorrect, and the fuzzball structure was essential for resolving the puzzle. Opposition continued along different lines, with a postulate that information would leak out through wormholes. Recently, it was shown that this wormhole idea had some basic flaws, leaving the fuzzball paradigm as the natural resolution of Hawking’s puzzle.
Video

2/2/2022
Adam Smith (Boston University)
Title: Learning and inference from sensitive data
Abstract: Consider an agency holding a large database of sensitive personal information—say, medical records, census survey answers, web searches, or genetic data. The agency would like to discover and publicly release global characteristics of the data while protecting the privacy of individuals’ records.
I will discuss recent (and not-so-recent) results on this problem with a focus on the release of statistical models. I will first explain some of the fundamental limitations on the release of machine learning models—specifically, why such models must sometimes memorize training data points nearly completely. On the more positive side, I will present differential privacy, a rigorous definition of privacy in statistical databases that is now widely studied, and increasingly used to analyze and design deployed systems. I will explain some of the challenges of sound statistical inference based on differentially private statistics, and lay out directions for future investigation.

2/8/2022
Wenbin Yan (Tsinghua University)
(special time: 9:30 pm ET)
Title: Tetrahedron instantons and M-theory indices
Abstract: We introduce and study tetrahedron instantons. Physically they capture instantons on $mathbb{C}^{3}$ in the presence of the most general intersecting codimension-two supersymmetric defects. In this talk, we will review instanton moduli spaces, explain the construction, moduli space and partition functions of tetrahedron instantons. We will also point out possible relations with M-theory index which could be a generalization of Gupakuma-Vafa theory.
Video

2/16/2022
Takuro Mochizuki (Kyoto University)
Title: Kobayashi-Hitchin correspondences for harmonic bundles and monopoles
Abstract: In 1960’s, Narasimhan and Seshadri discovered the equivalence
between irreducible unitary flat bundles and stable bundles of degree $0$ on compact Riemann surfaces. In 1980’s, Donaldson, Uhlenbeck and Yau generalized it to the equivalence between irreducible Hermitian-Einstein bundles
and stable bundles on smooth projective varieties. This is a surprising bridge connecting differential geometry and algebraic geometry. Since then, many interesting generalizations have been studied.
In this talk, we would like to review a stream in the study of such correspondences for Higgs bundles, integrable connections, $D$-modules and periodic monopoles.

2/23/2022
Bartek Czech (Tsinghua University)
Title: Holographic Cone of Average Entropies and Universality of Black Holes
Abstract: In the AdS/CFT correspondence, the holographic entropy cone, which identifies von Neumann entropies of CFT regions that are consistent with a semiclassical bulk dual, is currently known only up to n=5 regions. I explain that average
entropies of p-partite subsystems can be checked for consistency with a semiclassical bulk dual far more easily, for an arbitrary number of regions n. This analysis defines the “Holographic Cone of Average
Entropies” (HCAE). I conjecture the exact form of HCAE, and find that it has the following properties: (1) HCAE is the simplest it could be, namely it is a simplicial cone. (2) Its extremal rays represent stages of thermalization (black hole formation). (3) In a time-reversed picture, the extremal rays of HCAE represent stages of unitary black hole evaporation, as stipulated by the island solution of the black hole information paradox. (4) HCAE is bound by a novel, infinite family of holographic entropy inequalities. (5) HCAE is the simplest it could be also in its dependence on the number of regions n, namely its bounding inequalities are n-independent. (6) In a precise sense I describe, the bounding inequalities of HCAE unify (almost) all previously discovered holographic inequalities and strongly constrain future inequalities yet to be discovered. I also sketch an interpretation of HCAE in terms of error correction and the holographic Renormalization Group. The big lesson that HCAE seems to be teaching us is about the universality of black hole physics.

3/2/2022
Richard Kenyon (Yale University)

3/9/2022
Richard Tsai (UT Austin)

3/23/2022
Joel Cohen (University of Maryland)

3/30/2022
Rob Leigh (UIUC)

4/6/2022
Johannes Kleiner (LMU München)

4/13/2022
Yuri Manin (Max-Planck-Institut für Mathematik)

4/20/2022
TBA

4/27/2022
TBA

5/4/2022
Melody Chan (Brown University)

5/11/2022
TBA

5/18/2022
TBA

5/25/2022
Heeyeon Kim (Rutgers University)

Fall 2021

Date
Speaker
Title/Abstract

9/15/2021
Tian Yang, Texas A&M
Title: Hyperbolic Geometry and Quantum Invariants
Abstract: There are two very different approaches to 3-dimensional topology, the hyperbolic geometry following the work of Thurston and the quantum invariants following the work of Jones and Witten. These two approaches are related by a sequence of problems called the Volume Conjectures. In this talk, I will explain these conjectures and present some recent joint works with Ka Ho Wong related to or benefited from this relationship.

9/29/2021
David Jordan, University of Edinburgh
Title: Langlands duality for 3 manifolds
Abstract: Langlands duality began as a deep and still mysterious conjecture in number theory, before branching into a similarly deep and mysterious conjecture of Beilinson and Drinfeld concerning the algebraic geometry of Riemann surfaces. In this guise it was given a physical explanation in the framework of 4-dimensional super symmetric quantum field theory by Kapustin and Witten. However to this day the Hilbert space attached to 3-manifolds, and hence the precise form of Langlands duality for them, remains a mystery.
In this talk I will propose that so-called “skein modules” of 3-manifolds give natural candidates for these Hilbert spaces at generic twisting parameter Psi , and I will explain a Langlands duality in this setting, which we have conjectured with Ben-Zvi, Gunningham and Safronov.
Intriguingly, the precise formulation of such a conjecture in the classical limit Psi=0 is still an open question, beyond the scope of the talk.

10/06/2021
Piotr Sulkowski, U Warsaw
Title: Strings, knots and quivers
Abstract: I will discuss a recently discovered relation between quivers and knots, as well as – more generally – toric Calabi-Yau manifolds. In the context of knots this relation is referred to as the knots-quivers correspondence, and it states that various invariants of a given knot are captured by characteristics of a certain quiver, which can be associated to this knot. Among others, this correspondence enables to prove integrality of LMOV invariants of a knot by relating them to motivic Donaldson-Thomas invariants of the corresponding quiver, it provides a new insight on knot categorification, etc. This correspondence arises from string theory interpretation and engineering of knots in brane systems in the conifold geometry; replacing the conifold by other toric Calabi-Yau manifolds leads to analogous relations between such manifolds and quivers.

10/13/2021
Alexei Oblomkov, University of Massachusetts
Title: Knot homology and sheaves on the Hilbert scheme of points on the plane.
Abstract: The knot homology (defined by Khovavov, Rozansky) provide us with a refinement of the knot polynomial knot invariant defined by Jones. However, the knot homology are much harder to compute compared to the polynomial invariant of Jones. In my talk I present recent developments that allow us to use tools of algebraic geometry to compute the homology of torus knots and prove long-standing conjecture on the Poincare duality the knot homology. In more details, using physics ideas of Kapustin-Rozansky-Saulina, in the joint work with Rozansky, we provide a mathematical construction that associates to a braid on n strands a complex of sheaves on the Hilbert scheme of n points on the plane. The knot homology of the closure of the braid is a space of sections of this sheaf. The sheaf is also invariant with respect to the natural symmetry of the plane, the symmetry is the geometric counter-part of the mentioned Poincare duality.

10/20/2021
Peng Shan, Tsinghua U
Title: Categorification and applications
Abstract: I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.

10/27/2021
Karim Adiprasito, Hebrew University and University of Copenhagen
Title: Anisotropy, biased pairing theory and applications
Abstract: Not so long ago, the relations between algebraic geometry and combinatorics were strictly governed by the former party, with results like log-concavity of the coefficients of the characteristic polynomial of matroids shackled by intuitions and techniques from projective algebraic geometry, specifically Hodge Theory. And so, while we proved analogues for these results, combinatorics felt subjugated to inspirations from outside of it.
In recent years, a new powerful technique has emerged: Instead of following the geometric statements of Hodge theory about signature, we use intuitions from the Hall marriage theorem, translated to algebra: once there, they are statements about self-pairings, the non-degeneracy of pairings on subspaces to understand the global geometry of the pairing. This was used to establish Lefschetz type theorems far beyond the scope of algebraic geometry, which in turn established solutions to long-standing conjectures in combinatorics.
I will survey this theory, called biased pairing theory, and new developments within it, as well as new applications to combinatorial problems. Reporting on joint work with Stavros Papadaki, Vasiliki Petrotou and Johanna Steinmeyer.

11/03/2021
Tamas Hausel, IST Austria
Title: Hitchin map as spectrum of equivariant cohomology
Abstract: We will explain how to model the Hitchin integrable system on a certain Lagrangian upward flow as the spectrum of equivariant cohomology of a Grassmannian.

11/10/2021
Peter Keevash, Oxford
Title: Hypergraph decompositions and their applications
Abstract: Many combinatorial objects can be thought of as a hypergraph decomposition, i.e. a partition of (the edge set of) one hypergraph into (the edge sets of) copies of some other hypergraphs. For example, a Steiner Triple System is equivalent to a decomposition of a complete graph into triangles. In general, Steiner Systems are equivalent to decompositions of complete uniform hypergraphs into other complete uniform hypergraphs (of some specified sizes). The Existence Conjecture for Combinatorial Designs, which I proved in 2014, states that, bar finitely many exceptions, such decompositions exist whenever the necessary ‘divisibility conditions’ hold. I also obtained a generalisation to the quasirandom setting, which implies an approximate formula for the number of designs; in particular, this resolved Wilson’s Conjecture on the number of Steiner Triple Systems. A more general result that I proved in 2018 on decomposing lattice-valued vectors indexed by labelled complexes provides many further existence and counting results for a wide range of combinatorial objects, such as resolvable designs (the generalised form of Kirkman’s Schoolgirl Problem), whist tournaments or generalised Sudoku squares. In this talk, I plan to review this background and then describe some more recent and ongoing applications of these results and developments of the ideas behind them.

11/17/2021
Andrea Brini, U Sheffield
Title: Curve counting on surfaces and topological strings
Abstract: Enumerative geometry is a venerable subfield of Mathematics, with roots dating back to Greek Antiquity and a present inextricably linked with developments in other domains. Since the early 90s, in particular, the interaction with String Theory has sent shockwaves through the subject, giving both unexpected new perspectives and a remarkably powerful, physics-motivated toolkit to tackle several traditionally hard questions in the field.
I will survey some recent developments in this vein for the case of enumerative invariants associated to a pair (X, D), with X a complex algebraic surface and D a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to (X, D), including the log Gromov-Witten invariants of the pair, the Gromov-Witten invariants of an associated higher dimensional Calabi-Yau variety, the open Gromov-Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed Gopakumar-Vafa-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.

12/01/2021
Richard Wentworth, University of Maryland
Title: The Hitchin connection for parabolic G-bundles
Abstract: For a simple and simply connected complex group G, I will discuss some elements of the proof of the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points. Under the isomorphism with the bundle of conformal blocks, this connection is equivalent to the one constructed by conformal field theory. This is joint work with Indranil Biswas and Swarnava Mukhopadhyay.

12/08/2021
Maria Chudnovsky, Princeton
Title: Induced subgraphs and tree decompositions
Abstract: Tree decompositions are a powerful tool in both structural
graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph.
Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

12/15/21
Constantin Teleman (UC Berkeley)
Title: The Kapustin-Rozanski-Saulina “2-category” of a holomorphic integrable system
Abstract: I will present a construction of the object in the title which, applied to the classical Toda system, controls the theory of categorical representations of compact Lie groups, along with applications (some conjectural, some rigorous) to gauged Gromov-Witten theory. Time permitting, we will review applications to Coulomb branches and the categorified Weyl character formula.

Why abstraction is the key to intelligence, and what we’re still missing
2021-09-15 15:00 - 16:00 | MA
Speaker: Francois Chollet, Google
Title: Why abstraction is the key to intelligence, and what we’re still missing
Abstract: This talk provides a personal perspective on the way forward towards more human-like and more intelligent artificial systems. Traditionally, symbolic and probabilistic methods have dominated the domains of concept formation, abstraction, and automated reasoning. More recently, deep learning-based approaches have led to significant breakthroughs, including successes in games and combinatorial search tasks. However, the resulting systems are still limited in scope and capabilities — they remain brittle, data-hungry, and their generalization capabilities are limited. We will address a set of questions: why is conceptual abstraction essential for intelligence? What is the nature of abstraction, and its relationship to generalization? What kind of abstraction can deep learning models generate, and where do they fail? What are the methods that are currently successful in generating strong conceptual abstraction? Finally, we will consider how to leverage a hybrid approach to reinforce the strength of different approaches while compensating for their respective weaknesses.

Geometry, Entanglement and Quasi Local Data
2021-09-17 09:30 - 10:30 | MA
Member Seminar
Speaker: Itamar Shamir
Title: Geometry, Entanglement and Quasi Local Data
Abstract: I will review some general ideas about gravity as motivation for an approach based on quasi local quantities.

Stability and convergence issues in mathematical cosmology
2021-09-24 09:30 - 10:30 | MA
Member Seminar
Speaker: Puskar Mondal
Title: Stability and convergence issues in mathematical cosmology
Abstract: The standard model of cosmology is built on the fact that while viewed on a sufficiently coarse-grained scale the portion of our universe that is accessible to observation appears to be spatially homogeneous and isotropic. Therefore this observed `homogeneity and isotropy’ of our universe is not known to be dynamically derived. In this talk, I will present an interesting dynamical mechanism within the framework of the Einstein flow (including physically reasonable matter sources) which suggests that many closed manifolds that do not support homogeneous and isotropic metrics at all will nevertheless evolve to be asymptotically compatible with the observed approximate homogeneity and isotropy of the physical universe. This asymptotic spacetime is naturally isometric to the standard FLRW models of cosmology. In order to conclude to what extent the asymptotic state is physically realized, one needs to study its stability properties. Therefore, I will briefly discuss the stability issue and its consequences (e.g., structure formation, etc).

Constructions in combinatorics via neural networks
2021-09-29 15:00 - 16:00 | MA
Speaker: Adam Wagner, Tel Aviv University
Title: Constructions in combinatorics via neural networks
Abstract: Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In this talk I will give a very basic introduction to neural networks and reinforcement learning algorithms. I will also indicate how these methods can be adapted to the ““game” of trying to find a counterexample to a mathematical conjecture, and show some examples where this approach was successful.

Langlands duality for 3 manifolds
2021-09-29 17:15 - 18:15 | Virtual
Speaker: David Jordan (U Edinburgh)
Title: Langlands duality for 3 manifolds
Abstract: Langlands duality began as a deep and still mysterious conjecture in number theory, before branching into a similarly deep and mysterious conjecture of Beilinson and Drinfeld concerning the algebraic geometry of Riemann surfaces. In this guise it was given a physical explanation in the framework of 4-dimensional super symmetric quantum field theory by Kapustin and Witten. However to this day the Hilbert space attached to 3-manifolds, and hence the precise form of Langlands duality for them, remains a mystery.
In this talk I will propose that so-called “skein modules” of 3-manifolds give natural candidates for these Hilbert spaces at generic twisting parameter Psi , and I will explain a Langlands duality in this setting, which we have conjectured with Ben-Zvi, Gunningham and Safronov.
Intriguingly, the precise formulation of such a conjecture in the classical limit Psi=0 is still an open question, beyond the scope of the talk.

Instability of naked singularities in general relativity
2021-10-01 09:30 - 10:30 | MA
Member Seminar
Speaker: Jue Liu
Title: Instability of naked singularities in general relativity
Abstract: One of the fundamental problems in mathematical relativity is the weak cosmic censorship conjecture, proposed by Penrose, which roughly states that for generic physical spacetime, the singularities (if existed) must be hidden behind the black holes. Unfortunately, the singularities visible to faraway observers, which are called by naked singularities, indeed exist. The first example constructed by Christodoulou in 1994 is a family of self-similar spherically symmetric spacetime, in which the naked singularity forms due to a self-gravitating scalar field. Therefore the suitable censorship conjecture should be reduced to prove the instability of the naked singularities. In 1999 Christodoulou succeeded to prove the weak cosmic censorship conjecture in spherically symmetric cases, and recently the co-author and I found that the corresponding results have a big probability to be extended to spacetime without symmetries. In this talk I will discuss how to prove the instability of naked singularities using the energy method, and it is this wild method that helps us to extend some results to the asymmetric cases.

Strings, knots and quivers
2021-10-06 09:30 - 10:30 | Virtual
Speaker: Piotr Sułkowski (University of Warsaw)
Title: Strings, knots and quivers
Abstract: I will discuss a recently discovered relation between quivers and knots, as well as – more generally – toric Calabi-Yau manifolds. In the context of knots this relation is referred to as the knots-quivers correspondence, and it states that various invariants of a given knot are captured by characteristics of a certain quiver, which can be associated to this knot. Among others, this correspondence enables to prove integrality of LMOV invariants of a knot by relating them to motivic Donaldson-Thomas invariants of the corresponding quiver, it provides a new insight on knot categorification, etc. This correspondence arises from string theory interpretation and engineering of knots in brane systems in the conifold geometry; replacing the conifold by other toric Calabi-Yau manifolds leads to analogous relations between such manifolds and quivers.

New results in Supergravity via ML Technology
2021-10-06 15:00 - 16:00 | MA
Speaker: Thomas Fischbacher, Google
Title: New results in Supergravity via ML Technology
Abstract: The infrastructure built to power the Machine Learning revolution has many other uses beyond Deep Learning. Starting from a general architecture-level overview over the lower levels of Google’s TensorFlow machine learning library, we review how this has recently helped us to find all the stable vacua of SO(8) Supergravity in 3+1 dimensions, has allowed major progress on other related questions about M theory, and briefly discuss other applications in field theory and beyond.

Knowledge Graph Embeddings and Inference
2021-10-08 09:30 - 10:30 | MA
Member Seminar
Speaker: Michael Douglas
Title: Knowledge Graph Embeddings and Inference
Abstract: A knowledge graph (KG) is a data structure which represents entities and relations as the vertices and edges of a directed graph. Two examples are Wikidata for general knowledge and SemMedDB for biomedical data.
A popular KG representation method is graph embedding, which facilitates question answering, inferring missing edges, and logical reasoning tasks. In this talk we introduce the topic and explain relevant mathematical results on graph embedding. We then analyze KG inference into several mechanisms: motif learning, network learning, and unstructured statistical inference, and describe experiments to measure the contributions of each mechanism.
Joint work with M. Simkin, O. Ben-Eliezer, T. Wu, S. P. Chin, T. V. Dang and A. Wood.

Knot homology and sheaves on the Hilbert scheme of points on the plane
2021-10-13 09:30 - 10:30 | MA
Speaker: Alexei Oblomkov (University of Massachusetts)
Title: Knot homology and sheaves on the Hilbert scheme of points on the plane
Abstract: The knot homology (defined by Khovavov, Rozansky) provide us with a refinement of the knot polynomial knot invariant defined by Jones. However, the knot homology are much harder to compute compared to the polynomial invariant of Jones. In my talk I present recent developments that allow us to use tools of algebraic geometry to compute the homology of torus knots and prove long-standing conjecture on the Poincare duality the knot homology. In more details, using physics ideas of Kapustin-Rozansky-Saulina, in the joint work with Rozansky, we provide a mathematical construction that associates to a braid on n strands a complex of sheaves on the Hilbert scheme of n points on the plane. The knot homology of the closure of the braid is a space of sections of this sheaf. The sheaf is also invariant with respect to the natural symmetry of the plane, the symmetry is the geometric counter-part of the mentioned Poincare duality.

Computer-Aided Mathematics and Satisfiability
2021-10-13 14:00 - 15:00 | MA
Speaker: Marijn Heule, Carnegie Mellon University
Title: Computer-Aided Mathematics and Satisfiability
Abstract: Progress in satisfiability (SAT) solving has made it possible to determine the correctness of complex systems and answer long-standing open questions in mathematics. The SAT solving approach is completely automatic and can produce clever though potentially gigantic proofs. We can have confidence in the correctness of the answers because highly trustworthy systems can validate the underlying proofs regardless of their size. We demonstrate the effectiveness of the SAT approach by presenting some recent successes, including the solution of the Boolean Pythagorean Triples problem, computing the fifth Schur number, and resolving the remaining case of Keller’s conjecture. Moreover, we constructed and validated a proof for each of these results. The second part of the talk focuses on notorious math challenges for which automated reasoning may well be suitable. In particular, we discuss our progress on applying SAT solving techniques to the chromatic number of the plane (Hadwiger-Nelson problem), optimal schemes for matrix multiplication, and the Collatz conjecture.

C-P-T Fractionalization, and Quantum Criticality Beyond the Standard Model
2021-10-15 09:30 - 10:30 | MA
Member Seminar
Speaker: Juven Wang
Title: C-P-T Fractionalization, and Quantum Criticality Beyond the Standard Model
Abstract: Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as a Z2-involution on the spacetime coordinates; but together with a charge conjugation C and the fermion parity (−1)^F, these symmetries can be further fractionalized forming nonabelian C-P-R-T-(−1)^F group structures, in various examples such as relativistic Lorentz invariant Dirac spinor quantum field theories (QFT), or nonrelativistic quantum many-body systems (involving Majorana zero modes). This result answers Prof. Shing-Tung Yau’s question on “Can C-P-T symmetries be fractionalized more than involutions?” based on arxiv:2109.15320.
In the second part of my talk, I will sketch to explain how can we modify the so(10) Grand Unified Theory (GUT) by adding a new topological term such that two GUTs of Georgi-Glashow and Pati-Salam can smoother into each other in a quantum phase transition, where the Standard Model and new dark sector physics can occur naturally near the critical region. The new modified so(10) GUT requires a double Spin structure that we name DSpin. This phenomenon is inspired by the “deconfined quantum criticality” in condensed matter. Based on arxiv:2106.16248.

Categorification and applications
2021-10-20 09:30 - 10:30 | Virtual
Speaker: Peng Shan (Tsinghua University)

Title: Categorification and applications
Abstract: I will give a survey of the program of categorification for quantum groups, some of its recent development and applications to representation theory.

Wall-crossing from Higgs bundles to vortices
2021-10-22 09:30 - 10:30 | MA
Speaker: Du Pei
Title: Wall-crossing from Higgs bundles to vortices
Abstract: Quantum field theories can often be used to uncover hidden algebraic structures in geometry and hidden geometric structures in algebra. In this talk, I will demonstrate how such “wall-crossing” can relate the moduli space of Higgs bundles with the moduli space of vortices.

Anisotropy, biased pairing theory and applications
2021-10-27 09:30 - 10:30 | MA
Speaker: Karim Adiprasito, Hebrew University and University of Copenhagen
Title: Anisotropy, biased pairing theory and applications
Abstract: Not so long ago, the relations between algebraic geometry and combinatorics were strictly governed by the former party, with results like log-concavity of the coefficients of the characteristic polynomial of matroids shackled by intuitions and techniques from projective algebraic geometry, specifically Hodge Theory. And so, while we proved analogues for these results, combinatorics felt subjugated to inspirations from outside of it.
In recent years, a new powerful technique has emerged: Instead of following the geometric statements of Hodge theory about signature, we use intuitions from the Hall marriage theorem, translated to algebra: once there, they are statements about self-pairings, the non-degeneracy of pairings on subspaces to understand the global geometry of the pairing. This was used to establish Lefschetz type theorems far beyond the scope of algebraic geometry, which in turn established solutions to long-standing conjectures in combinatorics.
I will survey this theory, called biased pairing theory, and new developments within it, as well as new applications to combinatorial problems. Reporting on joint work with Stavros Papadaki, Vasiliki Petrotou and Johanna Steinmeyer.

Why explain mathematics to computers?
2021-10-27 14:00 - 15:00 | MA
Speaker: Patrick Massot, Laboratoire de Mathématiques d’Orsay and CNRS
Title: Why explain mathematics to computers?
Abstract: A growing number of mathematicians are having fun explaining mathematics to computers using proof assistant softwares. This process is called formalization. In this talk I’ll describe what formalization looks like, what kind of things it teaches us, and how it could even turn out to be useful (in our usual sense of “useful”). This will not be a talk about foundations of mathematics, and I won’t assume any prior knowledge about formalization.
