Georgia Institute of Technology College of Sciences

Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic continuum models. Numerically approximating kinetic equations presents several difficulties: 1) high dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting fluid models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties. This is joint work with Ruiwen Shu (University of Maryland).

Given an action of a finite group $G$ on a complex vector space $V$, the Chevalley-Shephard-Todd Theorem gives a beautiful characterization for when the quotient variety $V/G$ is smooth. In his 1986 ICM address, Popov asked whether this criterion could be extended to the case of Lie groups. I will discuss my contributions to this problem and some intriguing questions in combinatorics that this raises. This is based on joint work with Dan Edidin.

Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This work studies regression of a $s$-Hölder function $f$ in $\mathbb{R}^D$ which varies along an active subspace of dimension $d$ while $d\ll D$. A direct approximation of $f$ in $\mathbb{R}^D$ with an $\varepsilon$ accuracy requires the number of samples $n$ in the order of $\varepsilon^{-(2s+D)/s}$. In this work, we modify the Generalized Contour Regression (GCR) algorithm to estimate the active subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the active subspace, but its sample complexity is an open question. Our modified GCR improves the efficiency over the original GCR and leads to a mean squared estimation error of $O(n^{-1})$ for the active subspace, when $n$ is sufficiently large. The mean squared regression error of $f$ is proved to be in the order of $\left(n/\log n\right)^{-\frac{2s}{2s+d}}$, where the exponent depends on the dimension of the active subspace $d$ instead of the ambient space $D$. This result demonstrates that GCR is effective in learning low-dimensional active subspaces. The convergence rate is validated through several numerical experiments.

This is a joint work with Wenjing Liao.

Decoupling is a Fourier analytic tool that  has repeatedly proved its extraordinary potential for a broad range of applications to number theory (counting solutions to Diophantine systems, estimates for the growth of the Riemann zeta), PDEs (Strichartz estimates, local smoothing for the wave equation, convergence of solutions to the initial data), geometric measure theory (the Falconer distance conjecture)  and harmonic analysis (the Restriction Conjecture). The abstract theorems are formulated and proved in a continuous framework, for arbitrary functions with spectrum supported near curved manifolds. At this level of generality, the proofs involve no number theory, but rely instead on  wave packet analysis and incidence geometry related to the Kakeya phenomenon.   The special case when the spectrum is localized near lattice points leads to unexpected  solutions of conjectures once thought to pertain to the realm of number theory.