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.

Representation theory studies the way groups, rings and other algebraic structures can act by linear symmetries. We will consider representations of finite groups, the general linear group, the symmetric group, and quiver algebras. If time permits, we cover the theory of Soergel bimodules and applications to knot homology. We will provide suggestions for further reading to explore the beyond the listed examples. Students will prepare a presentation on a topic related to class discussion. A list of topics will offered in class.

Elliptic curves, as the only smooth projective algebraic curves equipped with a group law, play a central role in modern arithmetic geometry. The goal of this course is to learn the tools and techniques required to study these groups over the rational numbers by first studying them over finite fields, p-adic fields and archimedean fields.

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.

Not available at this time

The course covers a varieties of biological neuronal network and artificial neural network topics. We will begin with an introduction to mathematical neuroscience, covering mathematical models for single neurons and neuronal networks. We then delve into how artificial neural networks draw inspiration from biological counterparts. Selected topics include the anatomy and function of the primary visual cortex (V1) and its connection to convolutional neural networks (CNNs), as well as machine learning with spiking neural networks (SNNs).

Given a collection of polynomial equations defined over Q, we can associate to it a family of generating functions, called zeta functions. They are built using topological construction based on local information (specifically solutions of these equations modulo prime powers). But there are deep and important conjectures predicting that these functions encode global information --- the Q-solutions of these equations.

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