# Mathematics

4111 McHenry

(831) 459-2969

https://www.math.ucsc.edu

## Programs Offered

Mathematics Contiguous Bachelor's/Master's Pathway

## Other Programs of Interest

Economics and Mathematics Combined B.A.

## Undergraduate Program

Mathematics is both a fundamental discipline and an essential tool for students of biology, chemistry, computer engineering, computer science, Earth sciences, economics, electrical engineering, information systems management, physics, and psychology. Researchers in all these areas are constantly developing new ways of applying mathematics to their fields. A strong mathematics background is vital to the advanced study of many disciplines, including the physical and biological sciences, engineering, and the social sciences.

The UC Santa Cruz mathematics program offers a wide variety of undergraduate courses:

- Students interested in studying mathematics are strongly encouraged to take algebra, geometry, and trigonometry before entering the university. Students needing mathematics courses for their intended major are strongly encouraged to consider their options, and take the necessary steps for assessment and placement as early as possible. Progress in some majors could be delayed if the calculus series is not begun upon arrival at UCSC. Students concerned about their ability to place into courses above MATH 3 should consider taking MATH 2 or its equivalent prior to entering UCSC.
- Lower-division courses with numbers in the range MATH 11A through MATH 24 (calculus, linear algebra, vector calculus, and differential equations) prepare students for further study in mathematics, the physical and biological sciences, engineering, or quantitative areas of the social sciences. Science and engineering majors take some or all of these courses as part of their undergraduate studies.
- Upper-division courses, with numbers in the range MATH 100-MATH 199, are intended for majors in mathematics and closely related disciplines. Some of these courses provide students with a solid foundation in key areas of mathematics such as algebra, analysis, geometry, and number theory, whereas others introduce students to more specialized areas of mathematics. Calculus, linear algebra, vector calculus, and proof and problem solving are prerequisite to most of these advanced courses.

All students should review the catalog's requirements for their major or intended major, and possibly consult with the department sponsoring their major (or expected major) before deciding which math courses to take. More information on which courses are intended for the various types of students may be found here at the Mathematics Department website.

### Enrollment Requirements

Students who plan to take a mathematics course at UC Santa Cruz must first demonstrate sufficient preparation for that course by completing mathematics placement (see below), the College Entrance Examination Board's Advanced Placement (AP) calculus examination, the International Baccalaureate (IB) Mathematics Examination, or by passing the appropriate prerequisite course.

Through placement, some students may be required to start their UCSC mathematics classes with MATH 2 (followed by MATH 3) or MATH 3 before progressing to one of the calculus series. Students who have passed MATH 2 may enroll in MATH 3. Students who have passed MATH 3 may enroll in course MATH 11A or MATH 19A. Students who have passed a precalculus course at a college or university may enroll in course MATH 11A or MATH 19A, but they must first verify eligibility of the course (on Assist.org) and course completion with the mathematics adviser.

#### UC Santa Cruz Mathematics Placement

Mathematics placement assesses student readiness for their first UC Santa Cruz mathematics class. Students whose areas of study require precalculus or calculus courses are strongly advised to complete placement, as well as any required courses, early in their academic careers. Students intending to take one or more mathematics courses at UCSC should begin placement as early as possible to fully benefit from the process.

Students completing placement by assessment through ALEKS PPL should familiarize themselves with the instructions and guidelines, course eligibility cut-offs, and score posting schedule.

#### Advanced Placement (AP) Calculus Examinations

Students completing placement requirements by using their scores on the College Board's Advanced Placement Calculus Exam should refer to this table for course equivalencies.

#### International Baccalaureate (IB) Examination in Mathematics

Students completing placement requirements by using their scores from the International Baccalaureate Exam should refer to this table for course equivalencies.

## Graduate Program

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.

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.

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, and Vertex Operators 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, 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.

Further information about the research pursued by our faculty can be explored via the links above. Also, see individual faculty websites for more information.

### Admission to the Graduate Program

Submit your application 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.

### Financial Support

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 requirements 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.

### Relationship of Master’s and Doctoral Programs

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 within the requirements section of each program.