(Analysis I) The important topics to be addressed will be: construction of real numbers, basic topology, numerical sequences and series, continuity and differentiation, sequences and series of functions, and uniform convergence.
(Analysis II) Continuation of AS.110.415, introduction to real analysis. Topics include Lebesgue integration and differentiation. Elementary Hilbert and Banach space theory. Baire category theorem.
First exposure to the theory of Partial Differential Equations by examples. Basic examples of PDEs (Boundary value problems and initial value problems): Laplace equation, heat equation and wave equation. Method of separation of variables. Fourier series. Examples of wave equations in one and two dimensions. Sturm-Liouville eigenvalue problems and generalized Fourier series. Self-adjoint operators and applications to problems in higher dimensions. Nonhomogeneous PDEs. Green's functions and fundamental solution for the heat equation.
This course is the sequence about the general theory of PDEs. The beginning of the course will describe several important results of functional analysis which are instrumental for the study of PDEs: Hahn-Banach theorem, Uniform boundedness and closed graph theorems, reflexive spaces and weak topologies. The topics covered include: theory of Sobolev spaces. Harmonic functions and their properties. Weyl theorem. General Elliptic operators. Existence theory for elliptic boundary value problems. Lax-Milgram theorem. Dirichlet principle. Fine properties of solutions of elliptic equations such as maximum principles, Harnack principles, Sobolev and Hölder regularity.
Introduction to foundational material in Riemannian Geometry. Riemannian manifolds, a smooth manifold equipped with a Riemannian metric. Topics include connections, geodesics, Jacobi fields, submanifold theory including the second fundamental form and Gauss equations, manifolds of constant curvature, comparison theorems, Morse index theorem, Hadamard theorem and Bonnet-Myers theorem
This course is an introduction to stochastic differential equations and applications. Basic topics to be reviewed include Ito and Stratonovich integrals, Ito formula, SDEs and their integration. The course will focus on diffusion processes and diffusion theory, with topics include Markov properties, generator, Kolmogrov's equations (Fokker-Planck equation), Feynman-Kac formula, the martingale problem, Girsanov theorem, stability and ergodicity. The course will briefly introduce applications, with topics include statistical inference of SDEs, filtering and control.