# PhD Coursework

The schedule for first-year PhD coursework in AY 2022-23 is as follows:

Fall 2022 | Spring 2023 |
---|---|

Real Analysis | Numerical Analysis |

Advanced Linear Algebra | Functional Analysis |

Complex variables | Partial Differential Equations |

**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.