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.

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

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.

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.

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 or Python.

The course covers the basic combinatorial techniques as well as introduction to more advanced ones. The topics discussed include elementary counting, the pigeonhole principle, counting spanning trees, Inclusion-Exclusion, generating functions, Ramsey Theory, Extremal Combinatorics, Linear Algebra in Combinatorics, introduction to the probabilistic method, spectral graph theory, topological methods in combinatorics.