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.

Algebraic geometry is the study of geometric spaces locally defined by polynomial equations. It is a central subject in mathematics with strong connections to differential geometry, number theory, and representation theory. This course will be a fast-paced introduction to the subject with an emphasis on examples. In the algebraic approach to the subject, local data is studied via the commutative algebra of quotients of polynomial rings in several variables. Topics will include projective varieties, resolution of singularities, divisors and differential forms.

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.

Inverse and implicit functions theorems, rank of a map. Regular and critical values. Sard's theorem. Differentiable manifolds, submanifolds, embeddings, immersions and diffeomorphisms. Tangent space and bundle, differential of a map. Partitions of unity, orientation, transversality embed-dings in Rn. Vector fields, local flows. Lie bracket, Frobenius theorem. Lie groups, matrix Lie groups, left invariant vector fields, tensor fields, differential form, integration, closed and exact forms. DeRham's cohomology, vector bundles, connections, curvature.

This course is an introduction to stochastic control theory and reinforcement learning. The goal is to explore the theory of reinforcement learning through control theory and game theory, enabling the students to design novel algorithms in single agent and multi-agent settings, and evaluate the accuracy, efficiency and robustness of the designed algorithms theoretically as well as empirically.

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.

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.