Departmental Colloquium

I will discuss some combinatorial methods used to study the geometry of moduli spaces: spaces which parametrize geometric objects. These spaces play a central role in the field of algebraic geometry, the study of spaces that are patched together from zero sets of systems of polynomial equations. This talk will be accessible to all, including undergraduate math students and graduate students working in other fields.

We present new methods for solving a broad class of bound-constrained nonsmooth composite minimization problems. These methods are specially designed for objectives that are some algebraically specified (nonsmooth) mapping of a vector of outputs from a computationally expensive (black-box) function. We provide rigorous convergence analysis and guarantees, and test the implementations on synthetic problems, and on motivating problems from a wide range of applications relevant to the Department of Energy Office of Science. For this particular presentation, I will also provide introduction to the larger field of (model-based) black-box optimization.

S. Schwartzman in his Ph.D. thesis in the 1950s observed a deep result in dynamical systems. It states that a homeomorphism of a compact metric space always has two points whose iteration into the future remain closer than an arbitrary small distance. The same occurs when we consider the past. A notable extension to multidimensional dynamics is due to M. Boyle and D. Lind in 1997. It claims that any $\mathbb{Z}^d$-action (that is, $d$-commuting homeomorphisms) admits a half-space and two different points whose iterations under elements of the half-space remain arbitrarly close. While Schwartzman's result ensures this asymptotic property for both possible half-spaces of the integers, Boyle and Lind's result guarantees this property for only one of these half-spaces. In this talk we develop a geometric framework to address asymptoticity and the related property of nonexpansivity in topological dynamics when the acting group is second countable and locally compact. As an application, we show extensions of Schwartzman's theorem in this context. Also, we get new results when the acting groups is $\mathbb{Z}^d$: any half-space of Rd contains a vector defining a (oriented) nonexpansive direction in the sense of Boyle and Lind.

Kinetic models are used to simulate the collective behavior of particle systems. They provide a mesoscopic description that forms a link between continuum fluid models, which are not valid in non-equilibrium settings, and molecular dynamics models, which are often too expensive for practical purposes. In this talk, I will introduce the basic formalism of kinetic theory and present some relevant applications. I will then discuss some of the challenges of solving kinetic equations numerically, and present some of the tools being developed to address these challenges.