Graduate Program in Mathematics
Caltech has one of the best mathematics departments in the country, and offers a very strong graduate program.
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. 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 in research.
Caltech also 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.
For more information on the graduate program in mathematics, please see the sections on
- Financial Support
- Graduate Courses
- PhD Requirements
- Teaching Assistantships
- Living Arrangements
Mission of the Caltech Mathematics Graduate Program
An advanced degree in mathematics at Caltech is contingent upon an extensive research achievement. Students in the program are expected to join a research program, and carry independent research that will result in publications in peer-reviewed journals, as well as a thesis. The thesis work is then presented and evaluated by a Caltech thesis committee in a public defense. When new graduate students come to our program, they are required to solidify their basic knowledge by taking advanced courses in the three fundamental subfields of mathematics, namely analysis, algebra and geometry/ topology, and at the end of the first year they are asked in addition to pass qualifying exams in at least two of those subjects. Before the end of their third year, the students are required to pass a candidacy exam in the specific area of their proposed research in mathematics, in order to progress into the research phase of the degree.
Graduates of our program are expected to have extensive experience with modern research methods, a broad knowledge of contemporary math, and the ability to perform as independent researchers at the highest intellectual and technical levels.
Students are admitted only at the beginning of fall quarter. Applications are due by December 15 for the upcoming academic year, 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 writing to the Office of the Dean of Graduate Studies:
Office of the Dean of Graduate Studies
Mail Code 230-87
California Institute of Technology
Pasadena, CA 91125
Please do not contact the Mathematics Department for an application.
For additional information, please contact the Mathematics Graduate Admissions Committee:
Mathematics Graduate Admissions Committee
Mathematics Option 253-37
California Institute of Technology
Pasadena, CA 91125-3700
There are some fellowships available, usually for the first year of study, 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 are special fellowship for students who are US citizens, female, under represented minorities, and/or Southern California residents. Please contact the math graduate office for specific eligibility requirements or to inquire about applying.
Three basic courses in Analysis, Algebra, and Topology prepare students for the qualifying exams. 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):
Ma 105 Elliptic Curves
Ma 112 Statistics
Ma 116 Mathematical Logic and Axiomatic Set Theory
Ma 117 Computability Theory
Ma 121 Combinatorial Analysis
Ma 122 Group Theory
Ma 126 Information Theory
Ma 130 Algebraic Geometry
Ma 135 Arithmetical Geometry: Etale Cohomology
Ma 140 Functional Analysis
Ma 142 Ordinary and Partial Differential Equations
Ma 144 Probability
Ma 145 Group Representations
Ma 147 Dynamical Systems
Ma 148 Mathematical Physics
Ma 157 Manifolds
Ma 160 Number Theory
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.
The major requirement for the PhD in mathematics at Caltech is the presentation and acceptance by the faculty of a thesis containing results of original research.
For admission to candidacy for the PhD, students are required to:
- Demonstrate a good working knowledge in the three core areas: Algebra, Analysis, and Topology/Geometry by
- earning grades of B or better in the core courses, Ma 110, 120, 151 (unless excused)
- taking and passing a written qualifying exam in two of the three areas.
- Complete at least 9 quarter-courses in advanced mathematics in addition to the core courses. At least two of these must be in discrete mathematics (combinatorics, logic, complexity and computability).
- Pass a candidacy examination consisting of an oral presentation to a committee of mathematics faculty describing the proposed thesis research and the area of research it belongs to. At least a week prior to the examination, candidates submit a written summary of their presentation.
Qualifying examinations are usually taken in June. Some entering students take a qualifying examination in October in order to demonstrate knowledge of one of the core areas and be excused from taking the corresponding course. It is expected that the core courses will be completed in the first year, unless the student needs to take a preparatory course, such as Ma 109 (Introduction to Geometry and Topology). The candidacy examination is expected to be completed before the end of the student's third year.
In addition to the core courses, students are required to complete nine quarters of other advanced mathematics courses, of which at least two quarters must be in the area of discrete mathematics: combinatorics, logic, complexity, and computability. Under special circumstances (e.g., finishing the degree in three years), exceptions to these requirements may be granted by the graduate option representative.
Details of the candidacy and course requirements are given in the catalog.
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 22 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 assistantship provides a full tuition 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.