PhD Coursework

The schedule for PhD coursework in AY 2024-25 is as follows:

Real Analysis

Measure theory, Lebesgue integral, L^p spaces, duality, representation theorems, Radon-Nikodym and Fubini theorem, differentiation of integrals, a few facts from harmonic analysis.

Advanced Linear Algebra

Rigorous treatment of vector and inner product spaces, LU factorization, QR factorization, spectral theorem and singular value decomposition, Jordan form, positive definite matrices, quadratic forms, partitioned matrices, norms and numerical issues, Hilbert spaces, compact operators, diagonalization of self-adjoint compact operators, Fredholm alternative.

Numerical Analysis

Machine arithmetic, linear systems, root finding, interpolation and quadrature, eigenvalue problems, ordinary differential equations.

Partial Differential Equations

Linear and nonlinear first-order PDE, Sobolev spaces, distributions, Fourier transform, linear elliptic, parabolic and hyperbolic equations, initial value problems and boundary value problems.

Harmonic Analysis

The first part of the course concentrates on some fundamental results in Fourier analysis (Bochner’s theorem, Hardy-Littlewood maximal functions, Fefferman-Stein sharp functions, and the space of bounded mean oscillation) on Euclidean spaces. In the second part of this course, we are going to study singular integral operators through a few important example: Hilbert and Riesz transforms, the Szeg\{“}o projection operator on the Heisenberg group, and Cauchy integral on Lipschitz curves in the complex plane. Our goal is to explain some of the principal aspects of the great progress that has been made in the past thirty years or some toward understanding Calder\{‘}on-Zygmund operators.

Complex Analysis

Complex numbers. Analytic functions including exponential, logarithmic and trigonometric functions of a complex variable. Geometric and mapping properties of analytic functions. Contour integration, Cauchy’s theorem, the Cauchy integral formula. Power series representations. Residues and poles, with applications to the evaluation of integrals. Conformal mapping and applications as time permits.

Functional Analysis

Hilbert spaces, Banach spaces, convergence in topological vector spaces, dual spaces, Riesz representation theorem, Hahn-Banach theorem, open mapping theorem, closed graph theorem, principle of uniform boundedness, spectral theorem for (un)bounded operators, semigroups of linear operators, Fredholm operators and Fredholm index. Prerequisite: Real Analysis.

Computational Mathematics

This course will survey key elements of computational mathematics with a particular emphasis on methods applicable to data. Course topics will include optimization, matrix analysis and approximation, graphs and networks, and linear and non-linear methods in dimension reduction and function approximation. An emphasis will be placed on computation in high dimensions and associated theory. Students who lack a working knowledge of a scripting language such as R or Python are welcome, but should talk to the instructor prior to taking the course.

Probability

Topics include probability measures, independence and conditional probability, discrete and continuous random variables and their properties, joint distributions, moment generating functions, elements of Poisson processes, notions of convergence, Laws of Large Numbers, and the Central Limit Theorem. A working knowledge of multiple integrals and partial derivatives is essential for this course.