## Introduction to Computer Science Research

## Linear Analysis with Applications

Part a: Covers the basic algebraic, geometric, and topological properties of normed linear spaces, inner-product spaces and linear maps. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations. Topics: Completeness, Banach spaces (l_p, L_p), Hilbert spaces (weighted l_2, L_2 spaces), introduction to Fourier transform, Fourier series and Sobolev spaces, Banach spaces of linear operators, duality and weak convergence, density, separability, completion, Schauder bases, continuous and compact embedding, compact operators, orthogonality, Lax-Milgram, Spectral Theorem and SVD for compact operators, integral operators, Jordan normal form. Part b: Continuation of ACM 107a, developing new material and providing further details on some topics already covered. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations.Topics: Review of Banach spaces, Hilbert spaces, Linear Operators, and Duality, Hahn-Banach Theorem, Open Mapping and Closed Graph Theorem, Uniform Boundedness Principle, The Fourier transform (L1, L2, Schwartz space theory), Sobolev spaces (W^s,p, H^s), Sobolev embedding theorem, Trace theorem Spectral Theorem, Compact operators, Ascoli Arzela theorem, Contraction Mapping Principle, with applications to the Implicit Function Theorem and ODEs, Calculus of Variations (differential calculus, existence of extrema, Gamma-convergence, gradient flows) Applications to Inverse Problems (Tikhonov regularization, imaging applications).

## Probability Theory and Computational Mathematics

## Mathematical Optimization

## Analysis and Design of Algorithms

This course develops core principles for the analysis and design of algorithms. Basic material includes mathematical techniques for analyzing performance in terms of resources, such as time, space, and randomness. The course introduces the major paradigms for algorithm design, including greedy methods, divide-and-conquer, dynamic programming, linear and semidefinite programming, randomized algorithms, and online learning. Not offered 2024-25

## Networks: Structure & Economics

## Machine Learning & Data Mining

## Topics in Learning and Games

This course is an advanced topics course intended for graduate students with a background in optimization, linear systems theory, probability and statistics, and an interest in learning, game theory, and decision making more broadly. We will cover the basics of game theory including equilibrium notions and efficiency, learning algorithms for equilibrium seeking, and discuss connections to optimization, machine learning, and decision theory. While there will be some initial overview of game theory, the focus of the course will be on modern topics in learning as applied to games in both cooperative and non-cooperative settings. We will also discuss games of partial information and stochastic games as well as hierarchical decision-making problems (e.g., incentive and information design). Not offered 2024-25.

## Advanced Topics in Computing and Mathematical Sciences

Advanced topics that will vary according to student and instructor interest. May be repeated for credit. Not offered 2024-25.

## Computing and Mathematical Sciences Colloquium

This course is a research seminar course covering topics at the intersection of mathematics, computation, and their applications. Students are asked to attend one seminar per week (from any seminar series on campus) on topics related to computing and mathematical sciences. This course is a requirement for first-year PhD students in the CMS department.

## Research in Computing and Mathematical Sciences

Research in the field of computing and mathematical science. By arrangement with members of the staff, properly qualified graduate students are directed in research.