M 383C (57585) Methods of Applied Mathematics

Time: TTH 3:30 – 5:00 pm

Room: RLM 12.166

Instructor: Luis Caffarelli

Course Description:

This is the first semester of a course on methods of applied mathematics. It is open to mathematics, science, engineering, and finance students.

The first semester is an introduction to the main concepts of functional

analysis:

Preliminaries: (the student is supposed to have some familiarity)

·       Basic finite dimensional linear algebra, matrices ( orthonormal, symmetric, diagonalization, eigenvalues, Jordan canonical form)

·       Topology of Euclidean space: open, closed, compact sets, uniform continuity Lebesgue measure and integration, at least on the line.

Topics to be covered:

·       Basic theory of Banach spaces:

·       Hahn -Banach, Open Mapping, Uniform boundness theorems

·       Duality, compactness and weak convergence

·       Hilbert spaces: inner products, orthogonal projections, orthonormal bases.

·       Compact operators and spectral theory

·       Generalized functions: basic spaces, tests functions, distributions