PhD Coursework

The schedule for first-year PhD coursework in AY 2022-23 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.

Complex Variables

Analytic functions, harmonic functions, Schwarz lemma, contour integration, conformal mapping, Riemann mapping theorem, Mittag-Leffler theorem, analytic continuation, theory of series of Weierstrass.

Numerical Analysis

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

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.

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. Prerequisite: Real Analysis.