All speakers will give two talks with a 10-minute break. All times are US Eastern Standard Time (EST).

• Monday Jan 4, 2021
• Yoshiko Ogata (University of Tokyo)
8:00am   I: Classification of symmetry protected topological phases in quantum spin chains
9:00am   II: Classification of symmetry protected topological phases in quantum spin chains
• Prof. Ogata’s lecture notes linked here.

• Tuesday Jan 5, 2021
• Andras Vasy (Stanford University)
4:00pm   I: The black hole stability problem — an introduction and results
5:00pm   II: Analysis and geometry in the black hole stability problem
• Prof. Vasy’s lecture slides

• Wednesday Jan 6, 2021
• Jinxin Xue (Tsinghua University)
9:00am   I: Painleve conjecture in Newtonian N-body problem
10:00am    II: Painleve conjecture in Newtonian N-body problem
• Prof. Xue’s lecture notes

• Thursday Jan 7, 2021
• Michael Aizenman (Princeton University)
10:00am   I: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
11:00am   II: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
• Prof. Aizenman’s lecture slides

For online registration Click Here.

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).

All speakers will give two talks with a 10-minute break. All times are US Eastern Standard Time (EST).

• Monday Jan 4, 2021
• Yoshiko Ogata (University of Tokyo)
8:00am   I: Classification of symmetry protected topological phases in quantum spin chains
9:00am   II: Classification of symmetry protected topological phases in quantum spin chains
• Prof. Ogata’s lecture notes linked here.

• Tuesday Jan 5, 2021
• Andras Vasy (Stanford University)
4:00pm   I: The black hole stability problem — an introduction and results
5:00pm   II: Analysis and geometry in the black hole stability problem
• Prof. Vasy’s lecture slides

• Wednesday Jan 6, 2021
• Jinxin Xue (Tsinghua University)
9:00am   I: Painleve conjecture in Newtonian N-body problem
10:00am    II: Painleve conjecture in Newtonian N-body problem
• Prof. Xue’s lecture notes

• Thursday Jan 7, 2021
• Michael Aizenman (Princeton University)
10:00am   I: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
11:00am   II: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
• Prof. Aizenman’s lecture slides

For online registration Click Here.

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).

Does quantum theory impose any limits on how accurately we can map out spacetime and, if yes, what are they? This question has been studied already in the early days of quantum theory, but it is still a topic of current research. If one takes an operational perspective then the answer obviously depends on how accurately we can generate and measure time signals. In this talk I will present a bound on the latter. Specifically, I will show that the accuracy of a time signal generated by a quantum device is fundamentally limited by an information-theoretic quantity, which we call the “controllable dimension” of the device. (This is joint work with Yuxiang Yang, arXiv:2004.07857.)

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

All speakers will give two talks with a 10-minute break. All times are US Eastern Standard Time (EST).

• Monday Jan 4, 2021
• Yoshiko Ogata (University of Tokyo)
8:00am   I: Classification of symmetry protected topological phases in quantum spin chains
9:00am   II: Classification of symmetry protected topological phases in quantum spin chains
• Prof. Ogata’s lecture notes linked here.

• Tuesday Jan 5, 2021
• Andras Vasy (Stanford University)
4:00pm   I: The black hole stability problem — an introduction and results
5:00pm   II: Analysis and geometry in the black hole stability problem
• Prof. Vasy’s lecture slides

• Wednesday Jan 6, 2021
• Jinxin Xue (Tsinghua University)
9:00am   I: Painleve conjecture in Newtonian N-body problem
10:00am    II: Painleve conjecture in Newtonian N-body problem
• Prof. Xue’s lecture notes

• Thursday Jan 7, 2021
• Michael Aizenman (Princeton University)
10:00am   I: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
11:00am   II: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
• Prof. Aizenman’s lecture slides

For online registration Click Here.

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).

All speakers will give two talks with a 10-minute break. All times are US Eastern Standard Time (EST).

• Monday Jan 4, 2021
• Yoshiko Ogata (University of Tokyo)
8:00am   I: Classification of symmetry protected topological phases in quantum spin chains
9:00am   II: Classification of symmetry protected topological phases in quantum spin chains
• Prof. Ogata’s lecture notes linked here.

• Tuesday Jan 5, 2021
• Andras Vasy (Stanford University)
4:00pm   I: The black hole stability problem — an introduction and results
5:00pm   II: Analysis and geometry in the black hole stability problem
• Prof. Vasy’s lecture slides

• Wednesday Jan 6, 2021
• Jinxin Xue (Tsinghua University)
9:00am   I: Painleve conjecture in Newtonian N-body problem
10:00am    II: Painleve conjecture in Newtonian N-body problem
• Prof. Xue’s lecture notes

• Thursday Jan 7, 2021
• Michael Aizenman (Princeton University)
10:00am   I: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
11:00am   II: Marginal triviality of the scaling limits of 4D critical Ising and Φ₄⁴ models
• Prof. Aizenman’s lecture slides

For online registration Click Here.

Organizers: David Jerison, Paul Seidel, Nike Sun (MIT); Denis Auroux, Mark Kisin, Lauren Williams, Horng-Tzer Yau, Shing-Tung Yau (Harvard).

Schur functions serve as characters of representations of unitary (and general linear) groups; the multiplicative structure of Schur functions determines the fusion rules for representations of unitary groups. We will discuss an analogous family of symmetric polynomials for symmetric groups, and their construction in terms of Lie algebra cohomology. Although these polynomials have much in common with Schur functions, in some sense they are significantly more complicated. We will give some examples and applications.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

TITLE: Quantum topology and new types of modularity

ABSTRACT: The talk concerns two fundamental themes of modern 3-dimensional topology and their unexpected connection with a theme coming from number theory. A deep insight of William Thurston in the mid-1970s is that the vast majority of complements of knots in the 3-sphere, or more generally of 3-manifolds, have a unique metric structure as hyperbolic manifolds of constant curvature -1, so that 3-dimensional topology is in some sense not really a branch of topology at all, but of differential geometry. In a different direction, the work of Vaughan Jones and Ed Witten in the late 1980s gave rise to the field of Quantum Topology, in which new types of invariants of knot complements and 3-manifolds are introduced that have their origins in ideas coming from quantum field theory. These two themes then became linked by Kashaev’s famous Volume Conjecture, now some 25 years old, which says that the Kashaev invariant _N of a hyperbolic knot K (this is a quantum invariant defined for each positive integer N and whose values are algebraic numbers) grows exponentially as N tends to infinity with an exponent proportional to the hyperbolic volume of the knot complement. About 10 years ago, I was led by numerical experiments to the discovery that Kashaev’s invariant could be upgraded to an invariant having rational numbers as its argument (with the original invariant being the value at 1/N) and that the Volume Conjecture then became part of a bigger story saying that the new invariant has some sort of strange transformation property under the action x -> (ax+b)/(cx+d) of the modular group SL(2,Z) on the argument. This turned out to be only the beginning of a fascinating and multi-faceted story relating quantum invariants, q-series, modularity, and many other topics. In the talk, which is intended for a general mathematical audience, I would like to recount some parts of this story, which is joint work with Stavros Garoufalidis (and of course involving contributions from many other authors). The “new types of modularity” in the title refer to a specific byproduct of these investigations, namely that there is a generalization of the classical notion of holomorphic modular form – which plays an absolutely central role in modern number theory – to a new class of holomorphic functions in the upper half-plane that no longer satisfy a transformation law under the action of the modular group, but a weaker extendability property instead. This new class, called “holomorphic quantum modular forms”, turns out to contain many other functions of a more number-theoretical nature as well as the original examples coming from quantum invariants.

Talk chair: Mark Kisin

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

The talk will discuss the main approaches that combine machine learning with automated theorem proving and automated formalization. This includes learning to choose relevant facts for “hammer” systems, guiding the proof search of tableaux and superposition automated provers by interleaving learning and proving (reinforcement learning) over large ITP libraries, guiding the application of tactics in interactive tactical systems, and various forms of lemmatization and conjecturing. I will also show some demos of the systems, and discuss autoformalization approaches such as learning probabilistic grammars from aligned informal/formal corpora, combining them with semantic pruning, and using neural methods to learn direct translation from Latex to formal mathematics.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

One of the central problems in the study of deep learning theory is to understand complexity of the neural networks. In this talk, we will introduce the topological entropy from dynamic system as a measure of complexity of neural networks. We will also discuss the depth-width trade-offs of neural networks via topological entropy.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Richard Feynman once said “Anyone who wants to analyze the properties of matter in a real problem might want to start by writing down the fundamental equations and then try to solve them mathematically. Although there are people who try to use such an approach, these people are the failures in this field. . . ”

I will summarize efforts to solve microscopic models of the cuprates using quantum Monte Carlo and density matrix renormalization group computational methods, with emphasis on how far one can get before failing to describe the real materials. I will start with an overview of the quantum chemistry of the cuprates that guides our choices of models, and then I will discuss “phases” of these models, both realized and not. I will lastly discuss the transport properties of the models in the “not-so-normal” regions of the phase diagram.

Zoom: https://harvard.zoom.us/j/977347126

In this talk, I will summarize the current state of the art of transformer based language models and give examples on non-trivial reasoning task language models can solve in higher order logic reasoning. I will also discuss how to inject injective bias into transformer networks via pretraining on very simple synthetic tasks and representing graph structures for transformer networks.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

Zoom: https://harvard.zoom.us/j/977347126

Designing encoding and decoding circuits to send messages reliably over many uses of a noisy channel is a central problem in communication theory. When studying the optimal transmission rates achievable with asymptotically vanishing error, it is usually assumed that these circuits can be implemented using noise-free gates. While this assumption is satisfied for classical machines in many scenarios, it is not expected to be satisfied in the near-term future for quantum machines, where decoherence leads to faults in the quantum gates. As a result, fundamental questions regarding the practical relevance of quantum channel coding remain open. By combining techniques from fault-tolerant quantum computation with techniques from quantum communication, we initiate the study of these questions. As our main result, we prove threshold theorems for quantum communication, i.e. we show that coding near the (standard noiseless) classical or quantum capacity is possible when the gate error is below a threshold. (Joint work with Alexander Müller-Hermes, https://arxiv.org/abs/2009.07161)

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

All five-dimensional non-abelian gauge theories have a U(1) global symmetry associated with instantonic particles. I will describe a mixed ’t Hooft anomaly between this and other global symmetries such as the one-form center symmetry or the ordinary flavor symmetry for theories with fundamental matter. I will also discuss how these results can be applied to supersymmetric gauge theories in five dimensions, analyzing the symmetry enhancement patterns occurring at their conjectured RG fixed points.

Zoom: https://harvard.zoom.us/j/977347126

We begin with an introduction to arithmetic dynamics and heights attached to rational maps. We then introduce a dynamical version of Lang’s conjecture concerning the minimal canonical height of non-torsion rational points in elliptic curves (due to Silverman) as well as a conjectural analogue of Mazur/Merel’s theorem on uniform bounds of rational torsion points in elliptic curves (due to Morton-Silverman). It is likely that the two conjectures are harder in the dynamical setting due to the lack of structure coming from a group law. We describe joint work with Pierre Le Boudec in which we establish statistical versions of these conjectures for polynomial maps.

Zoom: https://harvard.zoom.us/j/99334398740

Password: The order of the permutation group on 9 elements.

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.

Zoom: https://harvard.zoom.us/j/99018808011?pwd=SjRlbWFwVms5YVcwWURVN3R3S2tCUT09

Given fixed integers a, b, c, which primes can be written as aX^2 + bXY + cY^2 for integers X, Y? This simple question in number theory has generated vast amounts of mathematics over the past 400 years. Of central importance to the answer in the general case are more abstract quantities called class numbers. These are individually very mysterious, but on average seem to be well-behaved. This talk is about an asymptotic law for truncated averages of class numbers, originally from Sarnak’s 1980 Stanford thesis. The proof will take us far away from the elementary terms in which the original question was stated: first to the Riemannian geometry of modular curves, and then (via the Selberg trace formula) to the analytic theory of automorphic forms.

Zoom: https://harvard.zoom.us/j/96759150216?pwd=Tk1kZlZ3ZGJOVWdTU3JjN2g4MjdrZz09

TITLE: Discrepancy Theory and Randomized Controlled Trials

ABSTRACT: Discrepancy theory tells us that it is possible to partition vectors into sets so that each set looks surprisingly similar to every other.  By “surprisingly similar” we mean much more similar than a random partition. I will begin by surveying fundamental results in discrepancy theory, including Spencer’s famous existence proofs and Bansal’s recent algorithmic realizations of them.

Randomized Controlled Trials are used to test the effectiveness of interventions, like medical treatments.  Randomization is used to ensure that the test and control groups are probably similar.  When we know nothing about the experimental subjects, uniform random assignment is the best we can do.

When we know information about the experimental subjects, called covariates, we can combine the strengths of randomization with the promises of discrepancy theory.  This should allow us to obtain more accurate estimates of the effectiveness of treatments, or to conduct trials with fewer experimental subjects.

I will introduce the Gram-Schmidt Walk algorithm of Bansal, Dadush, Garg, and Lovett, which produces random solutions to discrepancy problems. I will then explain how Chris Harshaw, Fredrik Sävje, Peng Zhang and I use this algorithm to improve the design of randomized controlled trials. Our Gram-Schmidt Walk Designs have increased accuracy when the experimental outcomes are correlated with linear functions of the covariates, and are comparable to uniform random assignments in the worst case.

Written articles will accompany each lecture in this series and be available as part of the publication “History and Literature of Mathematical Science.”

For more information, please visit the event page.

Register here to attend.

I describe a proposal for constructing lattice theories that target certain chiral gauge theories in the continuum limit. The models use reduced staggered fermions and employ site parity dependent Yukawa interactions of Fidkowski-Kitaev type to gap a subset of the lattice fermions without breaking symmetries. I show how the structure of these interactions is determined by a certain topological anomaly which is captured exactly by the generalizations of staggered fermions to triangulations of arbitrary topology. In the continuum limit the construction yields a set of sixteen Weyl fermions in agreement both with results from condensed matter physics and arguments rooted in the Dai-Freed theorem.  Finally, I point out the connection to the Pati-Salam GUT model.

Zoom: https://harvard.zoom.us/j/977347126