The study of randomness in partial differential equations (PDEs) goes back more than seventy years and include examples such as the modeling of random vibrations of strings, or the scattering of waves by objects that are imbedded in random media. Nonlinear dispersive PDEs naturally appear as models describing wave phenomena in quantum mechanics, nonlinear optics, plasma physics, water waves, and atmospheric sciences. Due to their ubiquitousness they have been at the center of profound research both from the applied community as well as from the theoretical one.

Banach and Hilbert spaces, continuous linear operators, spectral theory, Banach algebras. Prerequisite: Math 623 or 705.

Lie algebras are linear algebra devices of great usefulness in mathematics and physics as an efficient tool for the study of symmetries of objects. This course will cover the fundamentals of the subject, including nilpotent and solvable Lie algebras, as well as semisimple Lie algebras and their representations.

An introductory course in complex algebraic geometry. The basic techniques of Kahler geometry, Hodge theory, line and vector bundles which are needed for the study of the geometry and topology of complex projective algebraic varieties will be introduced and illustrated in basic examples.

This course provides an introduction to the theory of stochastic differential equations oriented towards topics useful in applications (Brownian motion, stochastic integrals, and diffusion as solutions of stochastic differential equations), and the study of diffusion in general (forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus, as well as Girsanov's theorem, Feynman-Kac formula, Martingale representation theorem). Applications to mathematical finance will be included as time permits.

This is an introductory course on symplectic topology along with its connections to differential, algebraic, complex and contact geometry and topology.

An introduction to the basic tools of algebraic topology, which studies topological spaces and continuous maps by producing associated algebraic structures (groups, vector spaces, rings, and homomorphism between them). Emphasis will be placed on being able to compute these invariants, not just on their definitions and associated theorems.

Presentation of the classical finite difference methods for the solution of the prototype linear partial differential equations of elliptic, hyperbolic, and parabolic type in one and two dimensions. Finite element methods developed for two dimensional elliptic equations. Major topics include consistency, convergence and stability, error bounds, and efficiency of algorithm. Prerequisites: Math 651, familiarity with partial differential equations.

This course covers classical methods in applied mathematics and math modeling, including dimensional analysis, asymptotics, regular and singular perturbation theory for ordinary differential equations, random walks and the diffusion limit, and classical solution techniques for PDE. The techniques will be applied to models arising throughout the natural sciences.