I will report on recent progress on influential conjectures from the 1970s and 1980s (Berry-Tabor, Bohigas-Giannoni-Schmit), which suggest that the spectral statistics of the Laplace-Beltrami operator on a given compact Riemannian manifold should be described either by a Poisson point process or by a random matrix ensemble, depending on whether the  geodesic flow is integrable or “chaotic”. This talk will straddle aspects of analysis, geometry, probability, number theory and ergodic theory, and should be accessible to a broad audience. The two most recent results presented in this lecture were obtained in collaboration with Laura Monk and with Wooyeon Kim and Matthew Welsh. 

I will present a parametric finite element formulation for structure-preserving numerical methods. The approach introduces two scalar Lagrange multipliers and evolution equations for surface energy and volume, ensuring that the resulting schemes maintain the underlying geometric and physical structures. To illustrate the method, I will discuss two applications: surface diffusion and two-phase Stokes flow. By combining piecewise linear finite elements in space with structure-preserving second-order time discretizations, we obtain fully discrete schemes of high temporal accuracy. Numerical experiments confirm that the proposed methods achieve the expected accuracy while preserving surface energy and volume.

Maximum likelihood estimation is a ubiquitous task in statistics and its applications. The task is: given some observations of a random variable, find the distribution(s) in your statistical model which best explains these observations. A modern perspective on this classical problem is to study the "likelihood geometry" of a statistical model. By focusing on models which have a polynomial parametrization, i.e., lie on an algebraic variety, this perspective brings in tools, algorithms and invariants from algebraic geometry and combinatorics.

In this talk, I will explain some of the key ideas in likelihood geometry and discuss its recent application to the study of likelihood asymptotics, i.e., understanding likelihood estimation for very large or very small observation counts. Agostini et al. showed that these asymptotics can be modeled and understood using tools from tropical geometry, and they used this to completely describe the asymptotics for linear models. In our work, we use the same approach to treat the class of log-linear models (also known as Gibbs distributions or maximum entropy models) and give a complete and combinatorial description of the likelihood asymptotics under some conditions.

This talk is based on joint work with Emma Boniface (UC Berkeley) and Serkan Hoşten (San Francisco SU), available at: https://epubs.siam.org/doi/full/10.1137/24M1656839

An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the pth-order tensor, based on lower-order derivatives. Between each recalculation of the pth-order derivative approximation, a high-order secant equation can be used to update the pth-order tensor as proposed in (Welzel 2022) or the approximation can be kept constant in a lazy manner. When refreshing the pth-order tensor approximation after m steps, an exact evaluation of the tensor or a finite difference approximation can be used with an explicit discretization stepsize. For all the newly adaptive regularization variants, we retrieve standard complexity bound to reach a second-order stationary point.  Discussions on the number of oracle calls for each introduced variant are also provided. When p = 2, we obtain a second-order method that uses quasi-Newton approximations with optimal number of iterations bound.