Analysis: Geometry, Fluids, and Random Matrices

There are six faculty active in mathematical analysis in the department.

Geometric Analysis

Jie Qing works on conformal geometry, the AdS-CFT correspondence, and most recently in general relativity. In conformal geometry, only the angle between two vectors can be measured but not the vector's lengths. The AdS-CFT correspondence relates the Riemannian or pseudo-Riemannian (general relativistic) geometry on one manifold to the conformal geometry of a manifold that bounds it. AdS-CFT has roots going back about 30 years but became very popular in the last decade due to its relevance to string theory. Qing has some of the strongest uniqueness results available in AdS/CFT for conformal spheres.

Tony Tromba's work is in minimal surfaces. These are surfaces that minimize area among all surfaces bounding a given curve. He has developed an index theory for minimal surfaces paralleling Morse's index theory for geodesics. He has written several books on the subject.

Fluids and Partial Differential Equations

Maria Schonbek's research centers on the study of non-linear diffusive partial equations such as the Navier-Stokes and the Quasi-geostrophic equations arising in fluid dynamics. These equations can be used to describe ocean currents, water flow in a pipe and other fluid behavior. The main direction of her research is towards understanding the qualitative behavior of solutions, in particular their energy asymptotics and decay rates. Questions of existence and regularity of the solutions are also considered in her work.

Random Matrices

Torsten Ehrhardt's research interests range from functional analysis (in particular, operator theory and the theory of Banach algebras), over harmonic analysis and Wiener-Hopf factorization theory to complex analysis. His research is both motivated by applications in such areas as statistical physics and random matrix theory, as well as by questions arising in different areas of "pure'' mathematics.

Alexander Gamburd has found surprising links between expander graphs and random matrices. Expander graphs are families of sparse (few vertices) but highly connected graphs. He has investigated the asymptotics of the quantized cat map. He is actively involved in uncovering relations between combinatorics, number theory, discrete group theory, and random matrices. In collaboration with Philip Kuekes (Computer Architect at Hewlett-Packard Laboratories in Palo Alto), Fraser Stoddart (Director of California NanoSystems Institute), and Omar Yaghi (Professor of Chemistry, UCLA), Gamburd is working on constructing expander-based computer architectures via the process of nano-scale self-assembly.

Emeritus Professor Harold Widom, with Craig Tracy of UC Davis, has established some of the fundamental results and methods in the field. The Tracy-Widom distribution is named after them. They have developed methods for obtaining explicit asymptotic formulae for relevant distributions via infinite determinants. Widom is the winner of numerous awards, including most recently being elected to the American Academy of Arts and Sciences. He remains actively involved in the department.