Schedule for: 21w5184 - Stochastics and Geometry (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In the fifties, Chen Ning Yang and Robert Mills made a major breakthrough in quantum field theory by extending the concept of gauge theory to non-abelian groups. The mathematical and physical consequences of going from a commutative to a non-commutative theory are major and since then, the mathematics of Yang-Mills theory has been a very active field of research. In this talk, meant to be an introductory as possible, I will focus on the last two-decade developments of Yang-Mills theory on two-dimensional manifolds with gauge group U(N) or SU(N). I will rely on the fascinating properties of the heat kernel (ie the Brownian motion and bridge) on these groups. I will in particular explain the construction of the master field, arising in the large-N limit for these models, in relation with free probability theory.

Please turn on your cameras for the "group photo" -- a screenshot in Zoom's Gallery view.

Abstract: In this talk, after a brief historical perspective, we will review some applications of the family log-Sobolev inequalities to partial differential equations, differential geometry and stochastic analysis in infinite-dimensional path spaces. A particular emphasis will be put on the fundamental contributions by Bruce Driver.

We develop a stochastic Clebsch action principle and derive the corresponding stochastic differential equations. The configuration space is a Riemannian manifold on which a Lie group acts transitively. This is joint work with D.D. Holm and T.S. Ratiu.

I will explain how tools from the theory of partial differential equations can be used to compute the eigenvalue distribution of large random matrices. I will discuss several examples where this method can be used and show lots of pictures illustrating the results. I will then explain how the method works in the simplest interesting example, for random matrices of the form $X+iY$, where $X$ is drawn from the Gaussian Unitary Ensemble and $Y$ is an arbitrary Hermitian random matrix independent of $X$. The talk should be accessible to a wide audience. The PDE approach to the subject was introduced in a work of mine with Bruce Driver and Todd Kemp and the specific example I will discuss is joint work of mine with Ching Wei Ho.

We will discuss several problems related to stochastic analysis on manifolds, especially analysis on the path space over a Riemannian manifold based on the Wiener measure (Riemannian Brownian motion), an area of stochastic analysis that Bruce Driver made groundbreaking contribution. These include the quasi-invariance of the Wiener measure under the Cameron-Martin flow, integration by parts formula and the logarithmic Sobolev inequality as well as the more general Beckner’s inequality on the path space. We survey the history of path space analysis and highlight some of its most recent developments such as sharp constants for functional inequalities and time-dependent Riemannian metrics.

We will take the time of this talk to look for some of the roots of stochastic differential geometry inside and outside of the field of probability theory, to emphasize some of its noticeable achievements, and to pay attention to which directions the leaves are growing or may be growing.

I will survey how analysis on path space can be used in the study Ricci curvature. As a motivation, I will start by discussing Driver’s foundational work on quasi-invariance and integration by parts on path space. Next, I will discuss joint work with Aaron Naber, which characterizes solutions of the Einstein equations and the Ricci flow in terms of certain sharp estimates on path space. In particular, this motivates a notion of weak solutions. Finally, I will mention joint work with Beomjun Choi, where we prove that noncollapsed limits are indeed weak solutions.

Why is the heat kernel useful? How does it help us understand other problems? This talk will be a leisure walk driven by these questions.

Presentations by Karen Habermann, David Herzog and Pierre Perruchaud

When considering a reasonable Brownian motion in a subriemannian manifold, we have a good idea of what to expect its kernel to look like for small times. It should be more or less a time singularity $t^{-Q/2}$ for some $Q$, multiplied by a Gaussian function where the subriemannian distance $d$ replaces what would be the norm in Euclidean space. Along a suitable decomposition of the tangent space, the distance $d$ behaves more or less like the $k$th power of a given smooth distance, where $k$ depends on the chosen factor in the decomposition. In many strictly hypoelliptic settings, we do not expect the kernel to be well approximated by a Gaussian. However, the grading phenomenon still occurs. In this talk, I will suggest a way to encode this geometric information in appropriate function spaces, so that we can consider non-Gaussian models, and apply some Duhamel formula to show the exact kernel vanishes at a rate prescribed by the grading.

To motivate of why it could be interesting to study multidimensional Brownian motion conditioned to have trivial signature, we discuss results on one-dimensional Brownian motion on the time interval $ [0, 1] $ conditioned to have vanishing iterated time integrals up to order $N$. We show that, in the large $N$ limit, these processes converge weakly to the zero process, which gives rise to a polynomial decomposition for Brownian motion, and we show that the associated fluctuation processes converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle.

We discuss the problem of convergence to equilibrium in two SDEs: (1) underdamped Langevin dynamics and (2) the Nos\'{e}-Hoover equation under Brownian heating. In each system, the invariant probability distribution has an explicit density which is known up to a normalization constant. Moreover, each density is of the Boltzmann-Gibbs form. In the context of statistical sampling, this form is exploited in order to take samples from a wide array of probability distributions by running the stochastic dynamics "long enough" when started from conveniently chosen prior distributions. However, outside of a particular class of target distributions, comparably little is known about how fast the stochastic dynamics converges to this equilibrium. This talk will cover joint work with my collaborators to bridge this gap, especially in the context of the singular, Lennard-Jones interaction potential.

An opportunity to present 10-15 minute talks followed by 5-10 minutes of questions; discuss open questions and ongoing work. Whiteboard will be available. Please contact Gianmarco Molino ([email protected]) with your title

An opportunity to present 10-15 minute talks followed by 5-10 minutes of questions; discuss open questions and ongoing work. Whiteboard will be available. Please contact Gianmarco Molino ([email protected]) with your title