Abstract: The theory of wave turbulence, which started in the 1920s as the wave analog of Boltzmann’s kinetic theory, has been an active field of physics in the last century, with substantial scientific applications. In this talk I will review some recent works, joint with Zaher Hani, that establish the first rigorous mathematical foundation of the wave turbulence theory, by justifying the derivation of the wave kinetic equation, the fundamental equation of this subject.

]]>Abstract: A brief introduction to ChatGPT as it is commonly used by students, followed by a discussion of academic integrity concerns, building toward a framework in which AI can be used constructively in undergraduate courses.

]]>Abstract: Vacillating tableaux are sequences of integer partitions that satisfy specific conditions. The concept of vacillating tableaux stems from the representation theory of the partition algebra and the combinatorial theory of crossings and nestings of matchings and set partitions.

We further investigate the enumeration of vacillating tableaux and derive multiple combinatorial identities and integer sequences relating to the number of vacillating tableaux, simplified vacillating tableaux, and limiting vacillating tableaux. This is joint work with Z. Berikkyzy, P. E. Harris, A. Pun and C. Yan.

Abstract: We will follow Chris Wendl's introduction of Symplectic Field Theory. SFT is an algebraic formalism used to define invariants of contact manifolds. We will discuss historical background involving the Arnold and Weinstein conjectures, as well as Morse, Floer, and contact homology. Then, we will sketch the algebraic structure of SFT and give example applications to distinguishing contact structures and obstructing exact symplectic cobordisms.

]]>Abstract: Consider the cosphere bundle of a manifold with the standard contact structure and Lagrangian cobordisms in its symplectization between Legendrian submanifolds. We construct a fully faithful functor between the categories of sheaves with singular support on the corresponding Legendrians, after enhancing the category at the negative end by local systems on the cobordism. In the special case where the contact manifold is the 1-jet bundle, we give a geometric model of the enhanced category as sheaves on the original manifold times the real line with microsupport on the cobordism. This gives a sheaf theoretic description of the maps between Legendrian contact homologies with coefficients enhanced by chains of the based loop spaces, defined using pseudoholomorphic curves.

]]>Abstract: We will discuss a mathematical formulation of the open/closed correspondence originally proposed by Mayr in physics, which is a correspondence in genus zero between the open Gromov-Witten theory of toric Calabi-Yau 3-folds and the closed Gromov-Witten theory of toric Calabi-Yau 4-folds. We will discuss different aspects of the correspondence on both the A- and B-sides of mirror symmetry. We will also discuss some applications. This is based on joint works with Chiu-Chu Melissa Liu and Zhengyu Zong.

]]>Abstract: Is learning a distribution always necessary for generating new samples from the distribution? To study this, we introduce the problem of “sample amplification”: given n independent draws from an unknown distribution, D, to what extent is it possible to output a set of m > n datapoints that are indistinguishable from m i.i.d. draws from D? Curiously, we show that nontrivial amplification is often possible in the regime where the number of datapoints n is too small to learn D to any nontrivial accuracy. We prove upper bounds and matching lower bounds on sample amplification for a number of distribution families including discrete distributions, Gaussians, and any continuous exponential family.

This is based on joint work with Brian Axelrod, Yanjun Han, Shivam Garg and Greg Valiant. Most of the talk will be based on this work, but we will also touch on this one.