The aim of this course will be to learn algebraic geometry through the study of key examples. Topics will include homogeneous spaces (projective space, Grassmannian), curves (elliptic, hyperellipitic, plane curves, genus), and surfaces (rational, ruled, K3,Enriques, blowup of a point). Prerequisites: Basic commutative algebra and complex analysis (as coverin in Math 611-612, Math 621), notion of sooth manifold, differential forms (as covered in Math 703-704)

Symplectic geometry is a central topic in mathematics with connections to algebraic geometry, differential geometry, complex geometry and topology. A major tool which has generated much recent research interest and has many applications in a diverse set of fields, is Floer theory. This course will denote the beginning portion of the semester on a general introduction to symplectic geometry. Once the necessary background is complete, the course will introduce Floer theory, defined using holomorphic curves.

Maximum credit, 6.

Introduction to the modern methods in partial differential equations. Calculus of distributions: weak derivatives, mollifiers, convolutions and Fourier transform. Prototype linear equations of hyperbolic, parabolic and elliptic type, and their fundamental solutions. Initial value problems: Cauchy problem for wave and diffusion equations; well-posedness in the Hilbert-Sobolev setting.

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 presents some of the fundamental as well as state-of-the-art methods for the simulation of stochastic processes, with particular emphasis in high dimensional systems. Such stochastic models are ubiquitous in the applied sciences and engineering, arising in applications ranging from materials to biology to geophysics, economics and finance to name a few. The course will include a project component that will be selected and carried out in coordination between the instructor and the project participants. Material: 1. Introduction to Monte Carlo methods, 2.

Topological spaces. Metric spaces. Compactness, local compactness. Product and quotient topology. Separation axioms. Connectedness. Function spaces. Fundamental group and covering spaces.

The analysis and application of the fundamental numerical methods used to solve a common body of problems in science. Linear system solving: direct and iterative methods. Interpolation of data by function. Solution of nonlinear equations and systems of equations. Numerical integration techniques. Solution methods for ordinary differential equations. Emphasis on computer representation of numbers and its consequent effect on error propagation. Prerequisites: advanced calculus, knowledge of a scientific programming language.