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2020-21 UCSC General Catalog
2019-20 UCSC General Catalog
2021-22 UCSC General Catalog
2022-23 UCSC General Catalog

The Mathematics Department offers programs leading to the Master of Arts (M.A.) and Doctor of Philosophy (Ph.D.) degrees. Students admitted to the Ph.D. program may receive a master's degree en route to the Ph.D.; students admitted to the M.A. program may apply to the department to transfer to the Ph.D. program upon passing the required preliminary examinations at the Ph.D. level.

The Mathematics Department at UC Santa Cruz is small but dynamic, with an ongoing commitment to both research and teaching. The department has leading research programs in several actively developing areas on the frontiers of pure and applied mathematics, interacting strongly with theoretical physics and mechanics. 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, bisets, biset 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

In order to be prepared for the master’s or Ph.D. program, it is recommended to have a B.A. or B.S. in mathematics. Having taken more than the bare minimum of required upper-division classes in the mathematics major will be most helpful.

Applications to the graduate program can be submitted through the Graduate Division. The deadline is usually during the first half of January. Admission is decided by a faculty committee and is based on a combination of factors including: GRE scores (in particular the GRE Math Subject Score), letters of recommendation, GPA, and classes taken.

The Mathematics Department is strongly committed to the financial support of graduate students who are making good progress toward either the master's or the Ph.D. degree. For the purpose of financial support, a student’s progress is measured against the degree programs and timetables.

A teaching assistantship (TA) is the most common form of financial support for graduate students in good academic standing. TA appointments are usually made at 50 percent time (an assigned workload of approximately 220 hours for the quarter). Teaching assistants are under the supervision of the faculty member responsible for the course.

All students are strongly urged to complete a Free Application for Financial Student Aid (FAFSA) each year by the start of fall quarter to determine eligibility for need-based awards. Students are also encouraged to apply for support from the Financial Aid Office as well as from the Mathematics Department.

No need-based fellowship can be awarded to a student who does not have a current FAFSA on file. Students facing special financial hardship are urged to make this known to the department in a timely manner.

The Mathematics Department will do everything in its power to ensure that all students in good standing are granted sufficient financial aid to continue their study of mathematics.

Students in the master’s and doctoral program take the same classes in the core sequences and the same preliminary examinations. Ph.D. and master’s students have the same passing requirements in the core classes. However, the preliminary examination requirements for Ph.D. and master’s students are different and are outlined below.