Department of Mathematics
 


 
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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 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.






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