• November 16, 2022 12:30 PM
Speaker: Hidenori Tanaka
Title: Noether’s Learning Dynamics: Role of Symmetry Breaking in Neural Networks
Venue: CMSA Room G10

Colloquium Speaker: Hidenori Tanaka (NTT Research at Harvard) Title: Noether’s Learning Dynamics: Role of Symmetry Breaking in Neural Networks Abstract: In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle, but the role of symmetry breaking is not well understood. Here, we develop a Lagrangian formulation to study the geometry of learning dynamics in neural networks and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. Then, we generalize Noether’s theorem known in physics to describe a unique symmetry breaking mechanism in learning and derive the resulting motion of the Noether charge: Noether’s Learning Dynamics (NLD). Finally, we apply…

  • November 02, 2022 12:45 PM
Speaker: Liang Fu
Title: Doping and inverting Mott insulators on semiconductor moire superlattices
Venue: CMSA Room G10

Speaker: Liang Fu (MIT) Title: Doping and inverting Mott insulators on semiconductor moire superlattices Abstract: Semiconductor bilayer heterostructures provide a remarkable platform for simulating Hubbard models on an emergent lattice defined by moire potential minima. As a hallmark of Hubbard model physics, the Mott insulator state with local magnetic moments has been observed at half filling of moire band. In this talk, I will describe new phases of matter that grow out of the canonical 120-degree antiferromagnetic Mott insulator on the triangular lattice. First, in an intermediate range of magnetic fields, doping this Mott insulator gives rise to a dilute gas of spin polarons, which form a pseudogap metal. Second, the application of an electric field between…

  • October 26, 2022 12:30 PM
Speaker: Virginia Vassilevska Williams
Title: Clique listing algorithms
Venue: CMSA Room G10

Speaker: Virginia Vassilevska Williams (MIT) Title: Clique listing algorithms Abstract: A k-clique in a graph G is a subgraph of G on k vertices in which every pair of vertices is linked by an edge. Cliques are a natural notion of social network cohesiveness with a long history. A fundamental question, with many applications, is “How fast can one list all k-cliques in a given graph?”. Even just detecting whether an n-vertex graph contains a k-Clique has long been known to be NP-complete when k can depend on n (and hence no efficient algorithm is likely to exist for it). If k is a small constant, such as 3 or 4 (independent of n), even the brute-force algorithm…

  • October 19, 2022 12:30 PM
Speaker: Patrick Lopatto
Title: The Mobility Edge of Lévy Matrices
Venue: CMSA Room G10

Colloquium Speaker: Patrick Lopatto (Brown) Title: The Mobility Edge of Lévy Matrices Abstract: Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an alpha-stable law; such distributions have infinite variance when alpha is less than 2. Due to the ubiquity of heavy-tailed randomness, these models have been broadly applied in physics, finance, and statistics. When the entries have infinite mean, Lévy matrices are predicted to exhibit a phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the physical context for this conjecture, and describe a result establishing it for values of alpha close to zero and one. This is joint work with Amol Aggarwal…

  • October 12, 2022 12:30 PM
Speaker: James Cummings
Title: Complete disorder is impossible: Some topics in Ramsey theory
Venue: CMSA Room G10

Colloquium Title: Complete disorder is impossible: Some topics in Ramsey theory Speaker: James Cummings, Carnegie Mellon University Abstract: The classical infinite Ramsey theorem states that if we colour pairs of natural numbers using two colours, there is an infinite set all of whose pairs get the same colour. This is the beginning of a rich theory, which touches on many areas of mathematics including graph theory, set theory and dynamics. I will give an overview of Ramsey theory, emphasizing the diverse ideas which are at play in this area.

  • October 05, 2022 04:00 PM
Speaker: Subir Sachdev
Title: Quantum statistical mechanics of charged black holes and strange metals
Venue: CMSA Room G10

Colloquium Please note this colloquium will be held at a special time:  4:00-5:00 pm. Speaker: Subir Sachdev (Harvard) Title: Quantum statistical mechanics of charged black holes and strange metals Abstract: The Sachdev-Ye-Kitaev model was introduced as a toy model of interacting fermions without any particle-like excitations. I will describe how this toy model yields the universal low energy quantum theory of generic charged black holes in asymptotically 3+1 dimensional Minkowski space. I will also discuss how extensions of the SYK model yield a realistic theory of the strange metal phase of correlated electron systems. Slides: cmsa22

  • September 28, 2022 12:30 PM
Speaker: Dima Sinapova
Title: The Tree Property and uncountable cardinals
Venue: CMSA Room G10

Colloquium Speaker: Dima Sinapova (Rutgers University) Title: The Tree Property and uncountable cardinals Abstract: In the late 19th century Cantor discovered that there are different levels of infinity. More precisely he showed that there is no bijection between the natural numbers and the real numbers, meaning that the reals are uncountable. He then went on to discover a whole hierarchy of infinite cardinal numbers. It is natural to ask if finitary and countably infinite combinatorial objects have uncountable analogues. It turns out that the answer is yes. We will focus on one such key combinatorial property, the tree property. A classical result from graph theory (König’s infinity lemma) shows the existence of this property for countable trees. We…

  • September 21, 2022 12:30 PM
Speaker: Melody Chan
Title: Moduli spaces of graphs
Venue: CMSA Room G10

Colloquium Speaker: Melody Chan Title: Moduli spaces of graphs Abstract: A metric graph is a graph—a finite network of vertices and edges—together with a prescription of a positive real length on each edge. I’ll use the term “moduli space of graphs” to refer to certain combinatorial spaces—think simplicial complexes—that furnish parameter spaces for metric graphs. There are different flavors of spaces depending on some additional choices of decorations on the graphs, but roughly, each cell parametrizes all possible metrizations of a fixed combinatorial graph. Many flavors of these moduli spaces have been in circulation for a while, starting with the work of Culler-Vogtmann in the 1980s on Outer Space. They have also recently played an important role in…

  • September 14, 2022 12:00 PM
Speaker: Yannai Gonczarowski
Title: Strategyproof-Exposing Mechanisms Descriptions
Venue: CMSA Room G10

Colloquium Speaker: Yannai Gonczarowski (Harvard) Title: Strategyproof-Exposing Mechanisms Descriptions Abstract: One of the crowning achievements of the field of Mechanism Design has been the design and usage of the so-called “Deferred Acceptance” matching algorithm. Designed in 1962 and awarded the Nobel Prize in 2012, this algorithm has been used around the world in settings ranging from matching students to schools to matching medical doctors to residencies. A hallmark of this algorithm is that unlike many other matching algorithms, it is “strategy-proof”: participants can never gain by misreporting their preferences (say, over schools) to the algorithm. Alas, this property is far from apparent from the algorithm description. Its mathematical proof is so delicate and complex, that (for example) school…

  • May 18, 2022 09:30 AM
Speaker: David Nelson
Title: Statistical Mechanics of Mutilated Sheets and Shells
Venue: Virtual

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…

  • April 27, 2022 09:30 AM
Speaker: Venkatesan Guruswami
Title: Long common subsequences between bit-strings and the zero-rate threshold of deletion-correcting codes
Venue: Virtual

Speaker: 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…

  • April 13, 2022 09:30 AM
Speaker: Yuri Manin
Title: Quantisation in monoidal categories and quantum operads
Venue: Virtual

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…

  • April 06, 2022 09:30 AM
Speaker: Johannes Kleiner
Title: What is Mathematical Consciousness Science?
Venue: Virtual

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.

  • March 30, 2022 09:30 AM
Speaker: Rob Leigh
Title: Edge Modes and Gravity
Venue: Virtual

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.

  • March 23, 2022 09:30 AM
Speaker: Joel Cohen
Title: Fluctuation scaling or Taylor’s law of heavy-tailed data, illustrated by U.S. COVID- cases and deaths
Venue: Virtual

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…

  • March 09, 2022 09:30 AM
Speaker: Yen-Hsi Richard Tsai
Title: Side-effects of Learning from Low Dimensional Data Embedded in an Euclidean Space
Venue: Virtual

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…

  • March 02, 2022 09:30 AM
Speaker: Richard Kenyon
Title: Dimers and webs
Venue: Virtual

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.

  • February 23, 2022 01:22 PM
Speaker: Bartek Czech
Title: Holographic Cone of Average Entropies and Universality of Black Holes
Venue: Virtual

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…

  • February 23, 2022 09:30 AM
Speaker: Bartek Czech
Title: CMSA Colloquium
Venue: Virtual

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…

  • February 16, 2022 09:30 AM
Speaker: Takuro Mochizuki
Title: Kobayashi-Hitchin correspondences for harmonic bundles and monopoles
Venue: Virtual

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.