|Department of Mathematics|
Mathematics 253-37 | Caltech | Pasadena, CA 91125
Telephone: (626) 395-4335 | Fax: (626) 585-1728
Project MATHEMATICS! | Caltech Home | Division of Physics, Mathematics and Astronomy
and engineering. With undergraduate and graduate students each
numbering less than 1,000, and over 280 faculty members, Caltech offers
an exceptionally high level of faculty-student interaction. Caltech has
one of the best mathematics departments in the country, and offers a
very strong graduate program. In the 2010 U.S.
survey, the graduate program was ranked 7th. In 2011 and 2012,
one university. in 2014, US
News ranked the Caltech Mathematics Department 7th in the United
The mathematics department has 21 regular faculty members, 19 postdoctoral fellows and instructors, and 36 graduate students. The graduate program in mathematics is designed to prepare students for research careers in universities, industry or government. Accordingly, only those students seeking the Ph.D. degree are admitted. In the past fifteen years, Caltech has awarded doctorates in mathematics to about 90 students, many of whom have moved into attractive positions as members of university faculty and in industry and government research.
Graduate students are encouraged to engage in creative research work after passing their qualifying exams (see below). As a result of the informal atmosphere, small size of the department, and the large faculty to student ratio, graduate students have ample opportunity to interact closely with the faculty. The research interests of the faculty as well as the courses and seminars offered are listed below. Further details appear in the 2013-2014 catalog and online at caltech.edu/catalog/ .
Caltech has an excellent mathematics library of over 20,000 volumes and 250 mathematical journals. Students have access to a wide variety of computing equipment in the computing center and to personal computers in the Mathematics building and other campus locations.
Financial Support. There are some fellowships available, usually for the first year, but the predominant mode of financial support is a teaching assistantship, which includes a full tuition grant, as well as a stipend to cover living expenses. The teaching load is very light and typically consists of assisting with a freshman or sophomore level course. All offers of admission in recent years have been accompanied by a fellowship or assistantship offers, and such support is independent of whether the student is domestic or international. In addition there is a special fellowship for which minority citizens of the U.S. can apply. For more detailed information, see the section on Teaching Assistantships.
Further information can
be obtained by visiting the Graduate
Mathematics Department for an application as they are all
|Michael Aschbacher||Group Theory, Combinatorics|
||Number Theory, Algebraic
|Sergei Gukov||String Theory, Geometry and Topology|
||Theoretical Physics and
|Alexander S. Kechris||Mathematical Logic, Set Theory|
|Alexei Kitaev||Theoretical Physics, Computer Science, and Mathematics|
|Nikolai Makarov||Harmonic Analysis, Complex Dynamics|
|Elena Mantovan||Number Theory, Arithmetic Geometry|
|Vladimir Markovic||Low Dimensional Geometry, Teichmüller theory|
|Yi Ni||Geometry and Topology|
||Theoretical Physics and Mathematics|
|Dinakar Ramakrishnan||Number Theory, Automorphic Forms, Algebraic Geometry|
|Christina M. (Chris) Shannon||Economic theory, Mathematical Economics|
|Barry Simon||Mathematical Physics, Functional Analysis|
|Richard Wilson||Combinatorics, Computing|
Graduate Courses and Seminars
The three basic courses in Analysis, Algebra, and Topology (see below) prepare students for the qualifying exams described in Section 4. Students who have not already completed equivalent courses are expected to take these during the first year. In some cases, a first-year student will be allowed to postpone one of the basic courses to the second year.
Ma 110 abc Real and Complex Analysis
First, second, third terms. Analytic functions, conformal mappings, Riemann surfaces, abstract measure theory, Fubini and Radon-Nikodyn theorems, Riesz representation theorem. Banach spaces, duality, L p spaces, Hilbert spaces. Application to Fourier series and integrals, elements of spectral theory.
Ma 120 abc Abstract Algebra
First, second, third terms. Abstract development of the basic structure theorems for groups, commutative and noncommutative rings, modules, algebras, fields (including Galois theory), and group representations.
Ma 151 abc Topology and Geometry
First, second, third terms. Fundamental groups and covering spaces, homology, cohomology and calculation of homology groups, exact sequences. Fibrations, higher homotopy groups and exact sequences of fibrations, structure of differentiable manifolds, degree theory, de Rham cohomology, elements of Morse theory. Geometry of Riemannian manifolds, covariant derivatives, geodesics, curvature, relations between curvature and topology.
Additional courses in a variety of areas are offered on a regular basis (see the catalog for descriptions):
104 Elliptic Curves
Advanced courses in special topics of interest to the faculty are given periodically. There are regular seminars in analysis, mathematical physics, group theory, number theory/algebraic geometry, combinatorics, logic, and geometry/topology. The mathematical physics seminar has active participation of mathematical physicists from various universities in the Los Angeles area. Additionally, there are seminars, held jointly with UCLA, in Number Theory and in Logic. Also of interest is the string theory/conformal field theory seminar run in the Physics department and the seminars in the Control Dynamical Systems and Applied Mathematics at Caltech. Options, as well as the Institute-wide theory seminar, workshops, research conferences, and regional seminars are held periodically in many of the above areas.
The Department colloquium takes place on Tuesday afternoon.
A teaching assistant is normally assigned to a freshman or sophomore calculus course taught by a faculty member and is responsible for one recitation section of approximately 30 students. Teaching assistants meet with their section one hour per week for a problem-solving session. They also have office hours (at most three hours per week) and grade exams and their students' homework. In return, the teaching assitantship provides a full tutition waver, as well as a stipend to cover living expenses. Although there are no classes during the summer, graduate students normally remain in residence (except for vacation) to study and do research. Summer stipends to support this are generally available.
Caltech housing is available in graduate student dormitories, apartments, and Avery House. Meals are available at the campus dining halls. Some students rent accommodations in privately-owned apartment buildings and houses near campus.
Students are admitted only at the beginning of the Fall quarter. Applications are due by January 15, 2014 (for admission in the academic year 2014-15), though later applications may be considered. Applicants are required to take the entire Graduate Record Examination (GRE) including the section in Advanced Mathematics. Foreign students must take the Test of English as a Foreign Language (TOEFL).
Application forms for admission and financial support may be obtained by visiting:
Mathematics Department for an application.
or e-mail questions to: