People

My recent has been on classifying Ancient Solutions and finding soliton solutions to Ricci flow and Mean Curvature flow. I am also interested in mathematical modeling of systems in cell biology.

I work on problems in approximation theory,complex and harmonic analysis, and spectral theory. I also study mathematical theory of fluidsand wave propagation.

I am interested in nonlinear PDE and free boundary problems. My recent research include study of shock reflection in gas dynamics, which involves study of free boundary problems for nonlinear degenerate-elliptic equations. I also work on semigestrophic system of PDE, which models atmospheric flows, and this work involves methods related to Monge-Kantorovich mass transport theory.

My research is in several complex variables and dynamical systems. I have worked on normal formsof real submanifolds in C^n. I am interested in derivative estimates for solutions of the d-bar equationon domains whose boundaries have minimum smoothness. My recent research includes using d-bar solution operatorsto study the stability of deformations of complex structures on a domain.

My research is in harmonic analysis and its connections to analytic number theory and geometric measure theory.

I am interested in applied PDEs in the fields of kinetic theory and related areas.My recent research include study of hydrodynamic limit from the Boltzmann equation toward various fluid models. I also work on long time behavior of the Vlasov equation, which models plasma and galaxies.

I study problems in the analytic theory of automorphic forms, and harmonic analysis on symmetric spaces. These include questions about counting the number of automorphicforms in families, and understanding the asymptotic behaviour of eigenfunctions with large eigenvalue.

Andreas' research interests include singular integrals, pointwise convergence of Fourier series, Fourier and spectral multiplier transformations, function spaces, wave propagation, oscillatory and Fourier integral operators, maximal functions related to geometric questions, and other applications of harmonic analysis.

Betsy Stovall's main research focus is on harmonic analysis.

I work in problems related to sub-Riemannian geometry, especially linear and multi-linear singular integrals, and partial differential equations.

Albert Ai
Van Vleck Assistant Professor
UC Berkeley, 2019
aai {at} math {dot} wisc {dot} edu
Research Interests: Nonlinear dispersive PDEs, fluid dynamics, harmonic analysis
Research Description:
My research is on quasilinear dispersive PDEs with a focus on models from fluids and relativity. My work focuses on well-posedness in low regularity Sobolev spaces and uses a variety of tools from microlocal and harmonic analysis, including paradifferential normal forms, balanced energy estimates, and Strichartz estimates.

I am interested in topics surrounding complex analysis and fractal geometry,including complex dynamics and analysis on metric spaces in particular. Much of my work in complexanalysis surrounds the study of examples to examine possible topology and geometry of multiply connectedwandering Fatou components, and I am interested in discovering more general ways to categorize the phenomena that occurin these examples.

My research focuses on Euclidean harmonic analysis and related areas. Particular examples are questions related to the Fourier restriction conjecture, decoupling inequalities, local smoothing estimates, averages along manifolds, maximal and variation norm Radon transforms, the Kakeya conjecture, extremisers for Strichartz estimates, regularity of maximal functions, pseudodifferential operators, weighted inequalities and sparse domination.

Dominique Kemp
NSF Postdoctoral Fellow (2021-2022)
Indiana University-Bloomington, 2021
ddkemp at math dot wisc dot edu
Research Interests:
I'm interested in problems that connect harmonic analysis with geometry. In particular, decoupling, restriction and Kakeya theory, and Böchner-Riesz are my current focus. I also enjoyexploring applications of decoupling inequalities to problems in number theory and geometry.

I am working on partial differential equations, in connection with gradient flows,optimal transport, and free boundary problems. In particular, I am interested in mean curvature flows,degenerate diffusion, and optimal control.

I am interested in harmonic analysis and its connectionsto number theory, combinatorics, and PDE. My recent research deals with decoupling theoryand connections between decoupling and efficient congruencing.

Trinh T. Nguyen
Van Vleck Assistant Professor
Penn State 2020
Research Interests: Analysis of PDEs, Fluid Dynamics, Kinetic Theory
Research Description:
Boundary layers and the inviscid limit of the Navier-Stokes equations, point-vortex dynamics, hydrodynamic limits of the Boltzmann equations

Simon Schulz
Van Vleck Assistant Professor
University of Oxford 2020

Research Interests: Analysis of PDEs

Research Description: I work on both hyperbolic conservation laws and parabolicPDEs- with a specific focus on the compressible Euler equations of gas dynamics, and degenerate parabolic systems comprising cross-diffusion. I have also worked on Liouville type theorems for the MHD system, and on the Morawetz problem in transonic flow.

Research Description:

I study nonlinear partial differential equations arising from mathematical physics and fluid dynamics. I am particularly interested in free boundary problems, geophysical flows, and complex fluids.

Tongou Yang
Van Vleck Assistant Professor
University of British Columbia, 2021
tyang at math dot wisc dot edu

Research Description:

I study Euclidean harmonic analysis and geometric measure theory. In particular, I am interested in Fourier decoupling theory and its applications.