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. |
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. |
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 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 |
3/2/2022 | Richard Kenyon (Yale University) | Title: Dimers and webs
Abstract: We consider SL_n-local systems on graphs on surfaces and show how the associated Kasteleyn matrix can be used to compute probabilities of various topological events involving the overlay of n independent dimer covers (or “n-webs”). This is joint work with Dan Douglas and Haolin Shi. |
3/9/2022 | Yen-Hsi Richard Tsai (UT Austin) | Title: Side-effects of Learning from Low Dimensional Data Embedded in an Euclidean Space
Abstract: The low dimensional manifold hypothesis posits that the data found in many applications, such as those involving natural images, lie (approximately) on low dimensional manifolds embedded in a high dimensional Euclidean space. In this setting, a typical neural network defines a function that takes a finite number of vectors in the embedding space as input. However, one often needs to consider evaluating the optimized network at points outside the training distribution. We analyze the cases where the training data are distributed in a linear subspace of Rd. We derive estimates on the variation of the learning function, defined by a neural network, in the direction transversal to the subspace. We study the potential regularization effects associated with the network’s depth and noise in the codimension of the data manifold. |
3/23/2022 | Joel Cohen (University of Maryland) | Title: Fluctuation scaling or Taylor’s law of heavy-tailed data, illustrated by U.S. COVID-19 cases and deaths
Abstract: Over the last century, ecologists, statisticians, physicists, financial quants, and other scientists discovered that, in many examples, the sample variance approximates a power of the sample mean of each of a set of samples of nonnegative quantities. This power-law relationship of variance to mean is known as a power variance function in statistics, as Taylor’s law in ecology, and as fluctuation scaling in physics and financial mathematics. This survey talk will emphasize ideas, motivations, recent theoretical results, and applications rather than detailed proofs. Many models intended to explain Taylor’s law assume the probability distribution underlying each sample has finite mean and variance. Recently, colleagues and I generalized Taylor’s law to samples from probability distributions with infinite mean or infinite variance and higher moments. For such heavy-tailed distributions, we extended Taylor’s law to higher moments than the mean and variance and to upper and lower semivariances (measures of upside and downside portfolio risk). In unpublished work, we suggest that U.S. COVID-19 cases and deaths illustrate Taylor’s law arising from a distribution with finite mean and infinite variance. This model has practical implications. Collaborators in this work are Mark Brown, Richard A. Davis, Victor de la Peña, Gennady Samorodnitsky, Chuan-Fa Tang, and Sheung Chi Phillip Yam. |
3/30/2022 | Rob Leigh (UIUC) | Title: Edge Modes and Gravity
Abstract: In this talk I first review some of the many appearances of localized degrees of freedom — edge modes — in a variety of physical systems. Edge modes are implicated for example in quantum entanglement and in various topological and holographic dualities. I then review recent work in which it has been realized that a careful treatment of such modes, paying attention to relevant symmetries, is required in order to properly understand such basic physical quantities as Noether charges. From many points of view, it is conjectured that this physics may be pointing at basic properties of quantum spacetimes and gravity. |
4/6/2022 | Johannes Kleiner (LMU München) | Title: What is Mathematical Consciousness Science?
Abstract: In the last three decades, the problem of consciousness – how and why physical systems such as the brain have conscious experiences – has received increasing attention among neuroscientists, psychologists, and philosophers. Recently, a decidedly mathematical perspective has emerged as well, which is now called Mathematical Consciousness Science. In this talk, I will give an introduction and overview of Mathematical Consciousness Science for mathematicians, including a bottom-up introduction to the problem of consciousness and how it is amenable to mathematical tools and methods. |
4/13/2022 | Yuri Manin (Max-Planck-Institut für Mathematik) | Title: Quantisation in monoidal categories and quantum operads
Abstract: The standard definition of symmetries of a structure given on a set S (in the sense of Bourbaki) is the group of bijective maps S to S, compatible with this structure. But in fact, symmetries of various structures related to storing and transmitting information (such as information spaces) are naturally embodied in various classes of loops such as Moufang loops, – nonassociative analogs of groups. The idea of symmetry as a group is closely related to classical physics, in a very definite sense, going back at least to Archimedes. When quantum physics started to replace classical, it turned out that classical symmetries must also be replaced by their quantum versions, e.g. quantum groups. In this talk we explain how to define and study quantum versions of symmetries, relevant to information theory and other contexts |
4/27/2022 | Venkatesan Guruswami (UC Berkeley) | Title: Long common subsequences between bit-strings and the zero-rate threshold of deletion-correcting codes
Abstract: Suppose we transmit n bits on a noisy channel that deletes some fraction of the bits arbitrarily. What’s the supremum p* of deletion fractions that can be corrected with a binary code of non-vanishing rate? Evidently p* is at most 1/2 as the adversary can delete all occurrences of the minority bit. It was unknown whether this simple upper bound could be improved, or one could in fact correct deletion fractions approaching 1/2. We show that there exist absolute constants A and delta > 0 such that any subset of n-bit strings of size exp((log n)^A) must contain two strings with a common subsequence of length (1/2+delta)n. This immediately implies that the zero-rate threshold p* of worst-case bit deletions is bounded away from 1/2. Our techniques include string regularity arguments and a structural lemma that classifies bit-strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence. This is joint work with Xiaoyu He and Ray Li. |
5/18/2022 | David Nelson (Harvard) | Title: Statistical Mechanics of Mutilated Sheets and Shells
Abstract: Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, a precursor of general relativity characterized by a dimensionless coupling constant (the “Foeppl-von Karman number”) that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) A crumpling transition out of the flat phase for thermalized elastic membranes has been predicted when kT is large compared to the microscopic bending stiffness, which could have interesting consequences for Dirac cones of electrons embedded in graphene. It may be possible to lower the crumpling temperature for graphene to more readily accessible range by inserting a regular lattice of laser-cut perforations, an expectation an confirmed by extensive molecular dynamics simulations. We then move on to analyze the physics of sheets mutilated with puckers and stitches. Puckers and stitches lead to Ising-like phase transitions riding on a background of flexural phonons, as well as an anomalous coefficient of thermal expansion. Finally, we argue that thin membranes with a background curvature lead to thermalized spherical shells that must collapse beyond a critical size at room temperature, even in the absence of an external pressure. |
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. 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. |
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 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. |