# Research areas

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.

Current areas of research in **Algebra/Number Theory/Topology** include:

- Vertex operator algebras, conformal field theory, modular forms, Hopf algebras, category theory, 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
- Finite groups, their representations and representation rings. Conjectures of Alperin, Broué and Dade
- Mackey functors, biset functors, Burnside rings, fusion systems, blocks of group algebras, ring theory, module theory, category theory, homological algebra
- Algebraic topology, cobordism cohomology theories, elliptic genera and modular forms, string topology of loop spaces, topological quantum field theories
- Triangulated categories, tensor triangular geometry, stable homotopy theory, and higher category theory
- Galois and incidence geometry
- Arithmetic algebraic geometry of moduli spaces and locally symmetric varieties

Current areas of research in **Analysis, Geometry and Dynamics** include:

- Symplectic and contact geometry and topology, Floer homology, periodic orbits and dynamics of Hamiltonian systems and Reeb flows
- Dynamical systems, celestial mechanics, geometric mechanics, bifurcation theory, control theory
- Geometric integration schemes, numerical methods on manifolds
- Differential geometry, nonlinear analysis, the Ginzburg-Landau problem
- General relativity, Einstein's equations, positive mass conjecture, Teichmuller theory
- Geometric Analysis: geometric flows, harmonic maps, minimal surfaces, surfaces of constant mean curvature, min-max theory
- Functional analysis, spectral theory, and Banach algebras; Toeplitz and Hankel operators, matrices, and determinants; Wiener-Hopf factorization and Riemann-Hilbert problems; applications to random matrix theory and statistical physics
- Inverse problems in PDEs and integral geometry, X-ray/Radon transforms, non-abelian X-ray transforms, with applications to imaging sciences
- Spectral Zeta Functions and applications in Quantum Field Theory and Number Theory
- Geometric group theory and low-dimensional topology
- Global Riemannian geometry; Ricci curvature and topology

General fields of mathematical research pursued by our faculty can be explored using the links above or the navigation menus. Also, you may see individual faculty websites for more information.