Not available at this time

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 introduces mathematical foundations and applications of game theory, with a focus on strategic decision-making in economics, finance, gambling, and international relations. Topics include non-cooperative and cooperative games, Nash equilibrium, repeated games, Bayesian games, evolutionary game theory, and mechanism design. Time permitting, we will discuss recent work in agentic AI and multi-agent game theory with AI agents.

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.

Classical theory of ordinary differential equations and some of its modern developments in dynamical systems theory. Linear systems and exponential matrix solutions. Well-posedness for nonlinear systems. Qualitative theory: limit sets, invariant set and manifolds. Stability theory: linearization about an equilibrium, Lyapunov functions. Autonomous two-dimensional systems and other special systems. Prerequisites: advanced calculus, linear algebra and basic ODE.

General theory of measure and integration and its specialization to Euclidean spaces and Lebesgue measure; modes of convergence, Lp spaces, product spaces, differentiation of measures and functions, signed measures, Radon-Nikodym theorem.

Introduction to groups, rings, and fields. Direct sums and products of groups, cosets, Lagrange's theorem, normal subgroups, quotient groups. Polynomial rings, UFDs and PIDs, division rings. Fields of fractions, GCD and LCM, irreducibility criteria for polynomials. Prime field, characteristic, field extension, finite fields. Some topics from Math 612 included.