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.

All mathematical objects of a given type can often be organized into a category. Category theory offers a framework to examine how a given category of mathematical objects interacts with those of a different nature (like topological spaces and groups), and to formalize constructions that map one mathematical object to another. Algebraic topology, for example, illustrates numerous instances of this concept, where algebraic invariants are derived from topological spaces using appropriate functorial constructions.

In the last decade, generative models and generative artificial intelligence have produced breakthrough results in image generation, text and speech synthesis, and more recently in scientific research itself, in fields such as aerospace, astronomy, biology, materials science, and medicine. Not unexpectedly, mathematics plays a foundational role in generative AI. It is essential for gaining a deeper understanding of existing methods, establishing limitations, quantifying trustworthiness, and developing new, provably more robust, or more energy-efficient methods.

Generating functions are a basic counting technique in combinatorics with many applications to different fields, including geometry, number theory, and representation theory. This course will be a fast-paced introduction to the subject with an emphasis on examples. Topics will include ordinary and exponential generating functions, rational generating functions, algebraic generating functions, combinatorics of gaussian integrals and matrix models, and applications to counting graphs, hypergraphs, and maps on surfaces.

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.