Graduate

Basic teaching techniques for teaching assistants, including responsibilities and rights; resource materials; computer skills; leading discussions or lab sessions; presentation techniques; maintaining class records; and grading. Examines research and professional training, including use of library; technical writing; giving talks in seminars and conferences; and ethical issues in science and engineering. (Formerly AMS 200.)

Accelerated class reviewing fundamental applied mathematical methods for all sciences. Topics include: multivariate calculus, linear algebra, Fourier series and integral transform methods, complex analysis, and ordinary differential equations. (Formerly AMS 211.)

Overview of advanced methods for the analytical solution of ODEs and PDEs, including asymptotic analysis (method of strained coordinates; method of multiple scales; matched asymptotics; WKB theory), dimensional analysis, self-similar solutions and Green’s function solutions of PDEs. Introduces students to variational calculus for the optimization of functionals. Students cannot receive credit for this course and AM 212B.

Focuses on analytical methods for partial differential equations (PDEs), including: the method of characteristics for first-order PDEs; canonical forms of linear second-order PDEs; separation of variables; Sturm-Liouville theory; Green's functions. Illustrates each method using applications taken from examples in physics. AM 211 or equivalent is strongly recommended as preparation. Students cannot receive credit for this course and AM 112. (Formerly AMS 211, Applied Mathematical Methods I.)

Covers perturbation methods: asymptotic series, stationary phase and expansion of integrals, matched asymptotic expansions, multiple scales and the WKB method, Pad approximants and improvements of series. (Formerly AMS 212B.)

Focuses on numerical solutions to classic problems of linear algebra. Topics include: LU, Cholesky, and QR factorizations; iterative methods for linear equations; least square, power methods, and QR algorithms for eigenvalue problems; and conditioning and stability of numerical algorithms. Provides hands-on experience in implementing numerical algorithms for solving engineering and scientific problems. Basic knowledge of mathematical linear algebra is assumed. (Formerly AMS 213A.)

Introduces the numerical solutions of ordinary and partial differential equations (ODEs and PDEs). Focuses on the derivation of discrete solution methods for a variety of differential equations, and their stability and convergence. Also provides hands-on experience in implementing such numerical algorithms for the solution of engineering and scientific problems using MATLAB software. The class consists of lectures and hands-on programming sections. Basic mathematical knowledge of ODEs and PDEs is assumed, and a basic working knowledge of programming in MATLAB is expected. (Formerly AMS 213B.)

Introduces continuous and discrete dynamical systems. Topics include: fixed points; stability; limit cycles; bifurcations; transition to and characterization of chaos; and fractals. Examples drawn from sciences and engineering; founding papers of the subject are studied. Students cannot receive credit for this course and AM 114 or MATH 145. (Formerly AMS 214.)

Application of differential equations, probability, and stochastic processes to problems in cell, organismal, and population biology. Topics include systems biology, cellular processes, gene-regulation, and population biology. Students may not receive credit for this course and AM 115. (Formerly AMS 215.)

Introduction to stochastic differential equations and diffusion processes with applications to biology, biomolecular engineering, and chemical kinetics. Topics include Brownian motion and white noise, gambler's ruin, backward and forward equations, and the theory of boundary conditions. (Formerly AMS 216.)

Covers fundamental topics in fluid dynamics at the graduate level: Euler and Lagrange descriptions of continuum dynamics; conservation laws for inviscid and viscous flows; potential flows; exact solutions of the Navier-Stokes equation; boundary layer theory; gravity waves. Students cannot receive credit for this course and AM 107. (Formerly AMS 217.)

Graduate-level course in dynamical systems theory, focusing on n-dimensional continuous dynamical systems. Covers fixed points, stability (linear and Lyapunov), normal forms, center manifold theorem, Liouville theorem, reversible and gradient flows, conservative systems, and quantitative characterization of chaos.

Advanced fluid dynamics course introducing various types waves and instabilities that commonly arise in geophysical and astrophysical systems, as well as turbulence. Topics covered include, but are not limited to: pressure waves, gravity waves, convection, shear instabilities, and turbulence. Advanced mathematical methods are used to study each topic. Undergraduates are encouraged to take this course with permission of the instructor. (Formerly Waves and Instabilities in Fluids.)

Focuses on recognizing, formulating, analyzing, and solving convex optimization problems encountered across science and engineering. Topics include: convex sets; convex functions; convex optimization problems; duality; subgradient calculus; algorithms for smooth and non-smooth convex optimization; applications to signal and image processing, machine learning, statistics, control, robotics and economics. Students are required to have knowledge of calculus and linear algebra, and exposure to probability. (Formerly AMS 229.)

Introduces numerical optimization tools widely used in engineering, science, and economics. Topics include: line-search and trust-region methods for unconstrained optimization, fundamental theory of constrained optimization, simplex and interior-point methods for linear programming, and computational algorithms for nonlinear programming. (Formerly AMS 230.)

Covers analysis and design of nonlinear control systems using Lyapunov theory and geometric methods. Includes properties of solutions of nonlinear systems, Lyapunov stability analysis, effects of perturbations, controllability, observability, feedback linearization, and nonlinear control design tools for stabilization. (Formerly AMS 231.)

Introduces optimal control theory and computational optimal control algorithms. Topics include: calculus of variations, minimum principle, dynamic programming, HJB equation, linear-quadratic regulator, direct and indirect computational methods, and engineering application of optimal control. (Formerly AMS 232.)

Computing the statistical properties of nonlinear random system is of fundamental importance in many areas of science and engineering. Introduces students to state-of-the-art methods for uncertainty propagation and quantification in model-based computations, focusing on the computational and algorithmic features of these methods most useful in dealing with systems specified in terms of stochastic ordinary and partial differential equations. Topics include: polynomial chaos methods (gPC and ME-gPC), probabilistic collocation methods (PCM and ME-PCM), Monte-Carlo methods (MC, quasi-MC, multi-level MC), sparse grids (SG), probability density function methods, and techniques for dimensional reduction. Basic knowledge of probability theory and elementary numerical methods for ODEs and PDEs is recommended. (Formerly AMS 238.)

Designed for STEM students and others. Through hands-on practice, this course introduces high-performance parallel computing, including the concepts of multiprocessor machines and parallel computation, and the hardware and software tools associated with them. Students become familiar with parallel concepts and the use of MPI and OpenMP together with some insight into the use of heterogeneous architectures (CPU, CUDA, OpenCL), and some case-study problems. (Formerly AMS 250.)

Introduces modern computational approaches to solving the differential equations that arise in fluid dynamics, particularly for problems involving discontinuities and shock waves. Examines the fundamentals of the mathematical foundations and computation methods to obtain solutions. Focuses on writing practical numerical codes and analyzing their results for a full understanding of fluid phenomena. (Formerly AMS 260.)

Studies the interaction of fluid motion and magnetic fields in electrically conducting fluids, with applications in many natural and man-made flows ranging from, for example, planetary physics and astrophysics to industrial metallurgic engineering. (Formerly AMS 275.)

Weekly seminar on mathematical and computational biology. Participants present research findings in organized and critical fashion, framed in context of current literature. Students present own research on a regular basis. (Formerly AMS 280A.)

Weekly seminar series covering topics of current research in applied mathematics and statistics. Permission of instructor required. Enrollment is restricted to graduate students. (Formerly AMS 280B.)

Weekly seminar/discussion group on geophysical and astrophysical fluid dynamics covering both analytical and computational approaches. Participants present research progress and findings in semiformal discussions. Students must present their own research on a regular basis. (Formerly AMS 280C.)

Independent completion of a masters project under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment is restricted to graduate students.

Thesis research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students.

Thesis research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students.

Thesis research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students.