Department of Mathematics 
Address:
Mathematics 25337  Caltech  Pasadena, CA 91125 Telephone: (626) 3954335  Fax: (626) 5851728 
Project MATHEMATICS!  Caltech Home  Division of Physics, Mathematics and Astronomy 
Introduction 
Caltech
is
a
small,
highly
selective
institution,
emphasizing
research
in
the
sciences
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 facultystudent interaction. Caltech has
one of the best mathematics departments in the country, and offers a
very strong graduate program. In the 2010 U.S.
News
and
World
Report
survey, the graduate program was ranked 7th. In 2011 and 2012,
the
Times
Higher
Education
ranked
Caltech
the
world's
number
one university. in 2014, US
News ranked the Caltech Mathematics Department 7th in the United
states. 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 20132014 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
office website: Please
do
not
contact
the
Mathematics Department for an application as they are all
done online 
The
Faculty 
Faculty 
Fields
of
Interest 
Michael Aschbacher  Group Theory, Combinatorics 
Matthias
Flach 
Number Theory, Algebraic
Geometry 
Rupert
Frank 
Mathematical Physics 
Tom Graber 
Algebraic Geometry 
Sergei Gukov  String Theory, Geometry and Topology 
Anton
N. Kapustin 
Theoretical Physics and
Mathematics 
Nets Katz 
Analysis 
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 
Matilde
Marcolli 
Noncommutative Geometry 
Vladimir Markovic  Low Dimensional Geometry, Teichmüller theory 
Yi Ni  Geometry and Topology 
Hiroshi Ooguri 
Theoretical Physics and Mathematics 
Eric Rains  
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 firstyear 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 RadonNikodyn 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
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 Institutewide 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. 
Ph.D.
Requirements 

Teaching
Assistantships 
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 problemsolving 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. 
Living
Arrangements 
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 privatelyowned apartment buildings and houses near campus. 
Admission
Requirements 
Students are admitted only at the beginning of the Fall quarter. Applications are due by January 15, 2014 (for admission in the academic year 201415), 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: Office
of
the
Dean
of
Graduate Studies Please
do
not
contact
the
Mathematics Department for an application.
Mathematics
Graduate
Admissions
Committee or email questions to: Kristy
Aubry
(Graduate
Program
Administrator) or Professor
Nikolai
Makarov
(Chair
of
the
Mathematics
Graduate
Admissions
Committee) 
California
Institute
of
Technology
is
an
Equal
Opportunity
Employer. 