This course is an introduction to the theory of convex polytopes and their applications to algebraic combinatorics. The course will cover basic facts and properties of polytopes including faces of polytopes, valuation theory, Ehrhart theory, and triangulations. We will also cover the following families of polytopes with interest in other fields: root polytopes, flow polytopes, polytopes from partially ordered sets, associahedra, and generalized permutahedra.

In this course, we will review, develop, and evaluate some computational biology methods. We will implement most of these methods in Python. Although programming skills, machine learning, or computational biology background are preferred, they are not required for this course. Importantly, this is a research-based course; it is an introduction on how to do research in computational biology. We all work as a team to learn novel methods in computational biology and hopefully find ways to improve them.

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.

A modern treatment of probability theory based on abstract measure and integration. Random variables, expectations, independence, laws of large numbers, central limit theorem, and general conditioning using the Radon-Nikodym theorem. Introduction to stochastic processes: martingales, Brownian motion.

Not available at this time

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis. Prerequisites: MATH 233 and 235, or consent of instructor, and knowledge of a scientific programming language.

Introduction to computational techniques used in science and industry. Topics selected from root-finding, interpolation, data fitting, linear systems, numerical integration, numerical solution of differential equations, and error analysis. Prerequisites: MATH 233 and 235, or consent of instructor, and knowledge of a scientific programming language.