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Department of Mathematics
Address: Mathematics 253-37 | Caltech | Pasadena, CA 91125 
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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 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. 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 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 office website:

Caltech Graduate Office



Please do not contact the Mathematics Department for an application as they are all done online
Apply 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 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 104 Elliptic Curves
Ma 111 Analysis II
Ma 112 Statistics
Ma 116 Mathematical Logic and Axiomatic Set Theory
Ma 117 Computability Theory
Ma 121 Combinatorial Analysis
Ma 122 Group Theory
Ma 130 Algebraic Geometry
Ma 135 Arithmetic Geometry
Ma 142 Ordinary and Partial Differential Equations
Ma 144 Probability
Ma 145 Group Representations
Ma 147 Dynamical Systems
Ma 148 Mathematical Physics
Ma 157 Riemannian Geometry
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.


Ph.D. Requirements

The major requirement for the Ph.D. in Mathematics at Caltech is the presentation and acceptance by the faculty of a thesis containing results of original research.

To be recommended for candidacy for the Ph.D., students are required to:

(a) Demonstrate a good working knowledge in the three core areas: Algebra, Analysis, and Topology/Geometry by

  • (i) earning grades of B or better in the core courses, Ma 110, 120, 151 (unless excused),
  • (ii) taking and passing a written qualifying exam in two of the three areas.

(b) Complete at least 9 quarter-courses in advanced mathematics Ma 111a and above in addition to the core courses. At least two of these must be in 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.

  • (i) These 9 courses MUST be taken for grades except for courses that are given Pass/Fail only.
  • (ii) Two of the 9 courses must be in discrete math.  Any quarters of 116, 117 or 121 count automatically.  Other courses can be used with approval by Dr. Wilson or Dr. Kechris.  Preferably you should get the approval in advance and ask the approver to send an email to the Graduate Program Administrator (Kristy Aubry)
  • (iii) Normally the 9 courses should be in Ma (or joint listed with Ma first). Other courses can be used with approval of the Graduate Option Representative for the Mathematics Department (Dr. Graber). Preferably you should get the approval in advance and ask the approver to send an email to Kristy for your file.

(c) Before the end of their third year, students are expected to finish the process of applying for admission to candidacy for the Ph.D. degree. This formal step requires completion of the requirements for qualifying examinations and core courses, as well as a satisfactory oral presentation to a committee of faculty members. The presentation will describe both
the general area of the student’s proposed thesis research and the specific problem or problems to be addressed. A written summary of the presentation, typically 3–10 pages, must be given to the committee members at least one week before the presentation. The student and his or her adviser will arrange the formation of this committee, which will have four members and meet the requirements listed in the subsection Graduate Policies and Proceedure entitled ‘Degree of Doctor of Philosophy’.

Qualifying examinations in the three core areas—analysis, algebra, and geometry/topology—are offered in October and June. These examinations emphasize mastery of the basic concepts and theorems and the ability to apply them to
specific cases. Students are required to take and pass two of the three examinations, and for the one not taken, to complete the corresponding core course with a grade of B or better. Normally, the examination requirements are completed at the end of the first year or the beginning of the second

Details of the candidacy and course requirements are given in the catalog.


Thesis and Final Examination
.

On or before the first Monday in May of the year in which the degree is to be conferred, candidates for the degree of Doctor of Philosophy must deliver copies of their theses to their advisers, to the Graduate Office, and to the members of the committee (of four members that meets the requirements listed in the subsection Graduate Policies and Procedure entitled ‘Degree of Doctor of Philosophy’) that will conduct the final oral examination on the thesis. The examination
must be held at least three weeks before the date on which the degree will be conferred and at least two weeks after the delivery of the copies of the thesis.

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 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.


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 privately-owned 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 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:

Office of the Dean of Graduate Studies
http://www.gradoffice.caltech.edu/

Please do not contact the Mathematics Department for an application.
Apply online!


Additional information
:

Mathematics Graduate Admissions Committee
Mathematics Option 253-37
California Institute of Technology

Pasadena, CA 91125-3700
mathgrad@caltech.edu

or e-mail questions to:

Kristy Aubry (Graduate Program Administrator)
kaubry@caltech.edu

or

Professor Nikolai Makarov (Chair of the Mathematics Graduate Admissions Committee)
makarov@caltech.edu



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Women, minorities, veterans, disabled persons are encouraged to apply.




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