Ph.D. Coursework

The schedule for Ph.D. coursework in AY 2026-27 is as follows:

Course Descriptions

MATH 8100: Real Analysis
Prereq: None
Description: This is a PhD-level course in real analysis covering measure, integration, and applications. Topics include: measure theory, the Lebesgue integral, L^p spaces, duality, representation theorems, the Radon-Nikodym and Fubini theorems, and if time allows differentiation of integrals, and a few facts from harmonic analysis. This class provides preparation for the real analysis portion of the Comprehensive Exam.

MATH: 8110 Advanced Linear Algebra
Prereq: None
Description: This is a PhD-level course that covers the 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, and norms and numerical issues. Further topics may include the Cayley-Hamilton Theorem, the Cauchy-Binet Formula, the Dunford decomposition, and the Theorem by Toeplitz-Hausdorff. This class provides preparation for the linear algebra portion of the Comprehensive Exam.

MATH 8120: Numerical Analysis
Prereq: None
Description: This course is at the level of PhD students before the qualifying exams. The topics covered are solving equations via iterations, polynomial interpolation, splines interpolation, numerical differentiation and integration, solving initial and boundary value problems of ordinary differential equations, and solving linear systems of equations. The goal is to gain skills for constructing, analyzing, evaluating and improving numerical algorithms in the context of the topics listed above.

MATH 8130: Complex Analysis
Prereq: MATH 8100
Description: This is a PhD-level course that rigorously covers core topics in classical complex analysis. These topics include: analytic functions; harmonic functions; the Schwarz lemma; contour integration; conformal mapping; the Riemann mapping theorem; the Mittag-Leffler theorem; analytic continuation. This class provides preparation for the complex analysis portion of the Comprehensive Exam.

MATH 8150: Probability
Prereq: MATH 8100
Description: This is a PhD-level course that is a measure-theoretic introduction to probability. Topics will include definition of probability spaces and associated analytic tools, law of large numbers, central limit theorems, martingales, ergodic theorems, and diffusions. Time permitting, computational methods associated with probability will be introduced.

MATH 8180: Computational Mathematics
Prereq: MATH 8110; MATH 8120
Description: This is a PhD-level course. 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.

MATH 8200: Functional Analysis
Prereq: MATH 8100; MATH 8110
Description: This is a PhD-level course that covers the Hilbert spaces, Banach spaces, Banach’s Fixed point theorem, convergence in topological vector spaces, dual spaces, Riesz representation theorem, Theorem of Hahn-Banach, open mapping theorem, closed graph theorem, principle of uniform boundedness, Spectral theory of linear operators in normed spaces, Compact linear operators, spectral properties of bounded self-adjoint linear operators, spectral theorem for (un)bounded operators, semigroups of linear operators, Fredholm operators and Fredholm index.

MATH 8250: Partial Differential Equation
Prereq: MATH 8100; MATH 8200
Description: This is a PhD-level course that covers a selection of the theory of linear Partial Differential Equations (PDEs). Classical solutions of the transport equation, the Laplace and Poisson equation, the heat equation and the wave equation are discussed. Initial-value problems and boundary-value problems are presented as well as energy integrals and maximum principles. In preparation of the treatment of weak solutions to linear PDEs, Sobolev spaces are introduced. Finally, weak solutions of linear second-order PDEs are discussed.

MATH 8260: Harmonic Analysis
Prereq: MATH 8100
Description: This is a PhD-level course. 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

MATH 8270: Riemannian Geometry
Prereq: MATH 8110
Description: This is a PhD-level course. Riemannian Geometry is a second course in Differential Geometry. This course will cover differentiable manifolds, Riemannian metrics, affine and Riemannian connections, geodesics and their minimizing properties, sectional curvature, Ricci curvature, scalar curvature, Jacobi fields as well as isometric immersions.

MATH 8300: Selected Topics in PDE
Prereq: MATH 8250
Description: This is a PhD-level course. This course is a continuation of MATH 8250. Topics include Nonlinear first-order PDE, the method of characteristics, Hamilton-Jacobi equations, conservation laws, Fourier transform, Laplace transform, and the Radon transform. In the context of similarity solutions plane waves, traveling waves, solitons, group velocity, and phase velocity will be introduced. The method of stationary phase will be discussed in the context of asymptotic solutions. Solutions based on power series and the Cauchy-Kovalevskaya Theorem are studied. The course concludes with the study of semigroups of linear operators. The method of semigroup will be applied to various linear and some non-linear equations and produce strong existence, uniqueness, and stability results.

MATH 9990: APMA Doctoral Seminar
Prereq: None
Description: APMA Doctoral Seminar Graduate requirement for the first 2 years of the APMA Program. In this seminar students will be exposed to different mathematical research areas, will engage in discussions, and will present on topics in applied mathematics.