Research

Algebra and Number Theory

The algebra/number theory group (Professors Boltje, Cooperstein, Dong, Mason and Weissman) is an active research unit with research interests that touch a number of current topics anchored by two main themes: group theory and representation theory, and conformal field theory (vertex operator algebras). Boltje and his students are involved with the conjectures of Alperin, Broue and Dade in the theory of modular representations of finite groups, and Cooperstein studies finite groups, in particular groups of Lie type, in the context of finite geometry and combinatorics. Dong and Mason have a long history of collaboration in the theory of vertex operator algebras, including representation theory and orbifold theory. There are extensive connections with topics such as elliptic cohomology, Hopf algebras and tensor categories, Kac-Moody Lie algebras, theoretical physics. Boltje also works in the area of algebraic number theory, where he has developed functorial methods to understand Galois actions on rings of algebraic integers in number fields. Vertex operator algebra theory has deep connections to parts of analytic number theory, in particular the theory of elliptic modular forms, Siegel modular forms and vector-valued modular forms. These topics are currently being pursued by Dong and Mason, both for applications to conformal field theory and for their own sake.

Mechanics, Symplectic Geometry, and Dynamic Systems

Symplectic geometry is the geometry underlying classical mechanics and is important to quantum mechanics and low-dimensional topology, and is an active area of research.

In the department Professors Ginzburg, Lewis, Montgomery, and Weitsman represent symplectic geometry and its links to mechanics and dynamics. Ginzburg is internationally known for achievements in the Hamiltonian Seifert conjecture and for work in Poisson topology. Hamiltonian systems are the natural dynamical systems for a symplectic geometry. Each such system has a `Hamiltonian', or energy, which is constant along solutions to the system. The Hamiltonian Seifert conjecture proposes that if the Hamiltonian has a sphere as one of its level sets, then on that sphere there is a periodic orbit. Ginzburg and his graduate students have constructed lowest dimensional counterexamples to this conjecture. Their work has provided strong constraints to theorems in symplectic topology. Lewis is an expert in relative equilibria for Hamiltonian systems and their stability and bifurcations relative equilibria. A relative equilibrium for a dynamical system is a solution which is an equilibrium modulo a symmetry group action. Her work is an interplay between group theory, symplectic geometry, and uses a good deal of symbolic manipulation. Montgomery is known for his work on subRiemannian geodesics, and the N-body problem. He and Alain Chenciner (Paris 7 and the Bureau des Longitudes, Paris) proved the existence of a figure eight shaped orbit for the three-body problem. Their result, and the methods they pioneered generated much recent work in mathematical celestial mechanics. Weitsman's work centers around the interface between symplectic geometry, algebraic topology, and mathematical physics. He has results on the cohomology ring of various symplectic reduced spaces, and on a nonstandard quantization scheme.

Analysis: Geometry, Fluids, and Random Matrices

Geometric Analysis

Professor Qing works on conformal geometry, the AdS-CFT correspondence, and 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 which bounds it. Qing has some of the strongest uniqueness results available in AdS/CFT for conformal spheres. Professor Tromba's work is in minimal surfaces. These are surfaces which 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.

Analysis: Non-linear Partial Differential Equations and fluids

Professor 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. Questions of existence and regularity of the solutions are also considered in her work.

Random Matrices

This theory grew out of speculations in the 1950s by physicist Eugene Wigner that the spectrum of atoms with large numbers of nuclei, such as Uranium, behaves like the spectrum of a large "random'' hermitian matrix. It has blossomed in the last 10 years, in part due to deep yet poorly understood relations which have been uncovered between random matrix theory and the distribution of zeros on the Riemann zeta function.

Emeritus Professor Widom with Professor Tracy of UC Davis had 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 remains actively involved in the department. Two new professors whose work involves random matrices have joined the department. They are Professors Ehrhardt and Gamburd.  Ehrhardt's work involves the asymptotics of the determinants of Toeplitz, Wiener-Hopf and Hankel operators. Gamburd has found surprising links between expander graphs and random matrices. 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. [Read Focus Article]

Algebraic Geometry

Professor Todorov works on algebraic geometrical problems inspired by string theory. He has some fundamental results on the moduli space of K3 surfaces and of Calabi-Yau manifolds.

Algebraic Topology

Professor Tamanoi works in algebraic topology, particularly elliptic cohomology theory and relations between algebraic topology and conformal field theory. He also has work on generalized Schwarzian derivatives in several variables, Moebius invariant differential operators and generalized hypergeometric functions.

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