This class covers topics of asymptotic methods and their applications. These include dominant balance, approximate solutions of differential equations, Frobenius expansions and divergent asymptotic expansions depending on type of singularities, complex integral representations & asymptotic integral evaluations through steepest descent path, saddle points, Stokes phenomena in analytic continuation, using the examples of special functions such as Airy, Gamma, Bessel, Parabolic Cylinder, Riemann-Zeta functions. This leads us to discuss WKB theory and applications to physics problems such as bound eigenstates and wave transmissions.

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.

A comprehensive introduction to the theory of nonlinear phenomena in fluids and plasmas, with emphasis on turbulence and transport. Experimental phenomenology; fundamental equations, including Navier-Stokes, Vlasov, and gyrokinetic; numerical simulation techniques, including pseudo-spectral and particle-in-cell methods; coherent structures; transition to turbulence; statistical closures, including the wave kinetic equation and direct-interaction approximation; PDF methods and intermittency; variational techniques. Applications from neutral fluids, fusion plasmas, and astrophysics.

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.

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.

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.

This course surveys the theory of combinatorial optimization. We cover the elementary min-max theorems of graph theory, such as Konig's theorems and Tutte's matching theorem, network flows, linear programming and polyhedral optimization, hypergraph packing and covering problems, perfect graphs, polyhedral methods to prove min-max theorems, packing directed cuts, the Lucchesi-Younger theorem, Packing T-cuts, T-joins and circuits, Edmonds' matching polytope theorem, relations with the four-color theorem, Lehman's results on ideal clutters, various further topics as time permits.