# Research

The Mathematics Department at UC Santa Cruz is small but dynamic, with an ongoing commitment to both research and teaching. The department has leading programs in several actively developing areas on the frontiers of pure and applied mathematics, interacting strongly with theoretical physics and mechanics. The extraordinary level of National Science Foundation support received by our faculty reflects the high caliber of the research carried out in the department.

The current areas of research include:

- Vertex operator algebras, higher genus conformal field theory, modular forms, quasi-Hopf algebras, infinite-dimensional Lie algebras, mathematical physics.
- Representations of Lie and p-adic groups, applications to number theory, Bessel functions, Rankin-Selberg integrals, Gelfand-Graev models.
- Algebra, group theory, finite groups and their representations, conjectures of Alperin, Dade and Broué, Mackey functors, modular representation theory, fusion systems, blocks of finite groups, bisects, bisect functors, Burnside rings, representations of algebras, ring theory, module theory.
- Algebraic topology, elliptic cohomology, quantum field theory, automorphic forms, string topology, topology of Lie groups, loop spaces.
- Symplectic geometry and topology, Floer homology, Poisson Lie groups.
- Dynamical systems, celestial mechanics, geometric mechanics, bifurcation theory, control theory.
- Fluid and continuum mechanics, the Navier-Stokes equation, long time behavior of solutions of PDEs.
- Geometric integration schemes, numerical methods on manifolds.

Algebraic geometry. - Differential geometry, nonlinear analysis, harmonic maps, Ginzburg-Landau problem.
- General relativity, Einstein's equations, positive mass conjecture, Teichmuller theory.
- Galois and incidence geometry.
- Algebraic number theory, elliptic curves, L-functions, p-adic L-functions, special values of L-functions, Gross-Stark conjecture, Heegner points.
- Graph theory, expander graphs, prime number distribution.
- Functional analysis, random matrix theory, spectral gap, operator theory, Banach algebras, harmonic analysis, Wiener-Hopf factorization, statistical physics.

General fields of mathematical research pursued by faculty within the department can be explored using the links to the left. Also, you may see individual faculty information for more information.