Valuations, rings of integral elements, ideal theory in algebraic number fields of algebraic functions of one variable, Dirichlet unit theorem and Riemann-Roch theorem for curves. Prerequisites: Math 611, 612 or equivalent.

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.

This course explores the interdisciplinary field of computational and mathematical biology, focusing on the development and application of computational models to solve complex biological problems. It blends mathematical modeling, algorithm development, and data analysis within a biological framework.

This course covers various topics Scientific Computing including: basic numerical techniques of linear algebra and their applications, data formats and practices, matrix computations with an emphasis on solving sparse linear systems of equations and eigenvalue problems. Students will learn about the state-of-the-art programming practices in numerical linear algebra, and will be introduced to numerical parallel algorithms and parallel programming with OpenMP, MPI and hybrid.

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.