# Prob/Stat Seminar

The Prob/Stat Seminar will meet Fridays from 11am to noon
in CU 239.

## Scaling limit of the KPZ equation

The KPZ equation in 1+1 dimensions was introduced to describe random growing interfaces by Kardar, Parisi, and Zhang in 1986. Since its introduction, the KPZ equation and its large-time asymptotics have been a major research subject in mathematics and physics. The convergence of its fundamental solutions has been a long-standing open problem. In this talk, I will review results in this direction and present my recent work that resolves this problem.

## Geometry, Topology, and the Parabolic Anderson Model

The Parabolic Anderson Model is a stochastic PDE subject to much active research. One new direction is seeing the effects of geometry on interesting properties of the solution. I will present two works on which the equation is posed over Riemannian manifolds. In the first, the equation is posed over a compact manifold and starts from a measure. The main insight is that geodesics being finite is needed for well-posedness results similar to those already known for Euclidean space. In the second, the equation is posed over a Hadamard manifold and starts from a bounded function. Here, it is revealed that having a negative curvature upper bound uniformly bounds the moments of the solution if the noise is weak. Based on ongoing work with Fabrice Baudoin and Cheng Ouyang.

## Geometry of Grushin Spaces

Much work has been done recently to try to understand broadly how to do analysis on non-Euclidean geometries. In particular, this focus has yielded many results in the realm of stochastic processes. One interesting class of non-Euclidean spaces are the almost-Riemannian manifolds, which come from a Riemannian structure that degenerates along a "singular set". The canonical example of almost-Riemannian manifold is the Grushin plane, whose geometry arises from requiring that traversal across the y axis occurs with a horizontal tangent vector. We consider a higher dimensional generalization of the Grushin plane and explore the optimality of geodesics. These spaces are modeled on R^n, so no extensive background in differential geometry will be required for this talk.

## CLTs for Poisson Functionals - The Malliavin-Stein Approach

In this talk, we will study examples of random graphs which do not behave quite as nicely as you would want them to. When those graphs are large, quantities like the sum of all edge lengths will behave almost like a Gaussian, but a fair bit of work is needed to prove this. We will go over the basics of the Malliavin-Stein method, a superbly useful tool for the derivation of bounds on the distance between a function of a Poisson process (not necessarily related to graphs - this can be much more general) and a standard Gaussian. We will see why some quantities and some graphs present difficulties and how to tweak the method to allow for those misbehaving cases. This is based on the preprint (Trauthwein 2022).

## Towards non-Euclidean space: an example of median

A major direction in the evolution of statistical learning has been expanding the scope of mathematical spaces underlying observed data, progressing from numbers to vectors, functions, and beyond. This expansion has fueled advances in both theoretical and computational breakthroughs. In this talk, I revisit the median, a robust alternative to the mean, as an example in line with this trajectory and introduce a novel generalization of this concept in the space of probability measures using the framework of optimal transport.