Posted January 12, 2022

Last modified January 25, 2022

Colloquium Questions or comments?

11:30 am - 12:20 pm Zoom
Guangqu Zheng, University of Edinburgh

Wiener chaos, Gaussian analysis and Stochastic partial differential equations

Abstract: This talk goes around the concept of Wiener chaos, which was first introduced by N. Wiener (1938) and later modified by K. Ito (1951, 56). It has been recurrently brought up in recent years, as it arises naturally in the study of stochastic partial differential equations, parameter estimation, nodal statistics of a Gaussian random fields and stochastic geometry, to name a few. Notably, Nualart-Peccati’s fourth moment theorem (2004) and Nourdin-Peccati’s Malliavin-Stein approach (2008) further push Wiener chaos to the center of Gaussian analysis, and it has turned out to be very effective in obtaining quantitative limit theorems in practice. In this talk, we will focus on the central limit theorem for the stochastic wave equation driven by Gaussian noise. We will present how the Wiener chaos enters the picture, and then highlight the key ideas and sketch main steps for obtaining relevant limit theorems. If time permits, we will talk about how this line of research (ideas, techniques) may lead to some other interesting results, for example: (i) extending random field solution theory for nonlinear SPDEs driven by colored noise, (ii) obtaining Gaussian fluctuations for (renormalized) singular stochastic dispersive/parabolic PDEs.

Posted January 25, 2022

Colloquium Questions or comments?

3:30 pm - 4:20 pm Zoom
Ana Balibanu, Harvard University

Poisson transversals in representation theory

Abstract: Geometric representation theory studies groups and algebras by realizing their representations geometrically, through actions on associated algebraic varieties. Symplectic and Poisson structures appear naturally in this setting, and give key insights into the geometry of the spaces that carry them. In turn, these spaces provide foundational examples for new research directions in Poisson geometry. The purpose of this talk is to illustrate this interplay in the framework of transversal structures. We will begin by introducing the notion of Poisson transversality, and by giving examples of several well-known representation-theoretic algebraic varieties that arise as Poisson transversals. Motivated by multiplicative analogues of these varieties, we will then define a general class of transversal slices for quasi-Poisson structures. This construction is based on the algebraic data that comes from an associated complex semisimple group, and can be used to produce canonical compactifications of these spaces which have interpretations in the setting of mathematical physics.

Posted January 26, 2022

Colloquium Questions or comments?

3:30 pm - 4:20 pm Zoom
Michael Lindsey, Courant Institute, NYU

A sampling of Monte Carlo methods

Abstract: In this talk I discuss recent work in several different areas involving Monte Carlo sampling. In the first part of the talk, I consider generic sampling problems, especially in low to moderate dimension, for which I introduce an ensemble Markov chain Monte Carlo (MCMC) method that overcomes the difficulty of slow transitions between isolated modes. In the second part, I consider the problems of computing ground states and excited states of quantum many-body systems, which are eigenpairs of exponentially high-dimensional Hermitian operators. I present a new optimization method within the framework of variational Monte Carlo (VMC). The VMC framework approaches these problems by stochastic optimization over a parametric class of wavefunctions. Of particular interest are recently introduced neural-network-based parametrizations for which this approach yields state-of-the-art results. In the last part, I consider a lattice model of quantum critical metals that captures a plausible mechanism for high-temperature superconductivity. This model can be studied numerically by Monte Carlo methods, but previous approaches cannot reach the large-volume limit needed to reveal critical scaling properties due to cubic computational cost in the lattice volume. I present recent work toward a linear-scaling approach.

Posted January 25, 2022

Colloquium Questions or comments?

3:30 pm - 4:20 pm Zoom
Spencer Leslie, Duke University

Periods, L-values, and stabilization

Abstract: The study of period integrals of automorphic forms originates in deep questions about cohomology of locally symmetric spaces. A particularly powerful tool for studying periods is a relative trace formula, which often allows one to relate these integrals to other arithmetic objects like L-functions. In this talk, I review some of this story, discuss this modern approach to relating period integrals to L-functions, and introduce an important case of active research: unitary Friedberg-Jacquet periods. These periods are conjecturally related to central values of certain L-functions and are thus connected to deep conjectures on the cohomology of the associated locally symmetric spaces. To prove these conjectural relationships, a promising approach is to use a relative trace formula. However, new problems (known as instability) arise in this setting that must be overcome if one is to prove this relation. I will discuss my work on a theory of endoscopy and a stable relative trace formula to overcome these problems. This gives a refinement of the relative trace formula amenable to proving this conjecture.