Department of Mathematics  
Address: Mathematics
25337

Caltech  Pasadena, CA 91125
Telephone: (626) 3954335  Fax: (626) 5851728 
Project MATHEMATICS!  Caltech Home Physics, Mathematics, and Astronomy @ Caltech 

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. 
