Fall 2021 Graduate Courses

MAT 321 / APC 321 (QCR): Numerical Methods

Nicholas Marshall

Introduction to numerical methods with emphasis on algorithms, applications and numerical analysis. Topics covered include solution of nonlinear equations; numerical differentiation, integration, and interpolation; direct and iterative methods for solving linear systems; computation of eigenvectors and eigenvalues; and approximation theory. Lectures include mathematical proofs where they provide insight and are supplemented with numerical demos using MATLAB.

MAT 377 / APC 377 (QCR): Combinatorial Mathematics

Staff

Introduction to combinatorics, a fundamental mathematical discipline as well as an essential component of many mathematical areas. While in the past many of the basic combinatorial results were at first obtained by ingenuity and detailed reasoning, modern theory has grown out of this early stage and often relies on deep, well-developed tools. Topics include Ramsey Theory, Turan Theorem and Extremal Graph Theory, Probabilistic Argument, Algebraic Methods and Spectral Techniques. Showcases the gems of modern combinatorics.

MAE 501 / APC 501 / CBE 509: Mathematical Methods of Engineering Analysis I

Luc Deike

Methods of mathematical analysis for the solution of problems in physics and engineering. Topics include an introduction to linear algebra, matrices and their application, eigenvalue problems, ordinary differential equations (initial and boundary value, eigenvalue problems), nonlinear ordinary differential equations, stability, bifurcations, Sturm-Liouville theory, Green's functions, elements of series solutions and special functions, Laplace and Fourier transform methods, and solutions via perturbation methods, partial differential equation including self-similar solution, separation of variables and method of characteristics.

APC 503 / AST 557: Analytical Techniques in Differential Equations           

Steven C. Cowley  Jong-Kyu Park

Asymptotic methods, Dominant balance, ODEs: initial and Boundary value problems, Wronskian, Green's functions, Complex Variables: Cauchy's theorem, Taylor and Laurent expansions, Approximate Solution of Differential Equations, singularity type, Series expansions. Asymptotic Expansions. Stationary Phase, Saddle Points, Stokes phenomena. WKB Theory: Stokes constants, Airy function, Derivation of Heading's rules, bound states, barrier transmission. Asymptotic evaluation of integrals, Laplace's method, Stirling approximation, Integral representations, Gamma function, Riemann zeta function. Boundary Layer problems, Multiple Scale Analysis.

MAT 522 / APC 522: Introduction to PDE

Sun-Yung Chang

The course is an introduction to partial differential equations, problems associated to them and methods of their analysis. Topics may include: basic properties of elliptic equations, wave equation, heat equation, Schr\"{o}dinger equation, hyperbolic conservation laws, Fokker-Planck equation, basic function spaces and inequalities, regularity theory for linear PDE, De Giorgi method, basic harmonic analysis methods, existence results and long time behavior for classes of nonlinear PDE including the Navier-Stokes equations.

APC 524 / MAE 506 / AST 506: Software Engineering for Scientific Computing

Gabe Perez-Giz

The goal of this course is to teach basic tools and principles of writing good code, in the context of scientific computing. Specific topics include an overview of relevant compiled and interpreted languages, build tools and source managers, design patterns, design of interfaces, debugging and testing, profiling and improving performance, portability, and an introduction to parallel computing in both shared memory and distributed memory environments. The focus is on writing code that is easy to maintain and share with others. Students will develop these skills through a series of programming assignments and a group project.

ORF 550 / APC 550: Topics in Probability: Probability in High Dimension

Ramon van Handel

An introduction to nonasymptotic methods for the study of random structures in high dimension that arise in probability, statistics, computer science, and mathematics. Emphasis is on developing a common set of tools that has proved to be useful in different areas. Topics may include: concentration of measure; functional, transportation cost, martingale inequalities; isoperimetry; Markov semigroups, mixing times, random fields; hypercontractivity; thresholds and influences; Stein's method; suprema of random processes; Gaussian and Rademacher inequalities; generic chaining; entropy and combinatorial dimensions; selected applications.

AOS 576 / APC 576: Current Topics in Dynamic Meteorology: Large-Scale Structure /Atmosphere

Stephen T. Garner

Dynamical concepts needed to develop a qualitative understanding of the large-scale structure of the atmospheric circulation. The control of the angular momentum budget by Rossby wave fluxes. Theories for the Hadley circulation in the tropics and the "macro-turbulence" of midlatitudes. Linear theories for deviations from zonal symmetry of the mean flow.