More specifically: Holomorphic and meromorphic functions of one variable including Cauchy's integral formula, theory of residues, the argument principle, and Schwarz reflection principle. Multivalued functions and applications to integration. Meromorphic functions. The Fourier transform in the complex plane. Paley-Wiener theorem. Wiener-Hopf method and the Riemann-Hilbert problem. Entire functions including Jensen's formula and infinite products. Conformal mapping including the Riemann mapping theorem and the Schwarz-Christoffel integral. In the case of sufficient time, further topics include the Gamma and Zeta functions and an introduction to the theory of elliptic functions including the Weierstrass function.

We begin by considering PDEs posed on fixed domains. Topics include:

1. A brief review of basic FEM for elliptic and parabolic PDEs.
2. Development of modern unfitted FEM (CutFEM) theory for PDEs on embedded domains.
3. Introduction to elliptic and parabolic surface PDEs and surface (parametric) FEM.
4. Development of trace FEM (unfitted/CutFEM) for surface PDEs.

Throughout this part of the course, the emphasis will be on analysis, including well-posedness, stability, a priori error estimates, and the role of geometry approximation. Some software aspects will be discussed to illustrate the theory but will not be a central focus of the course. If time permits, we will also consider PDEs posed on moving (time-dependent) domains.

The second part of the course introduces shape derivatives as a tool to quantify how solutions of PDEs change with respect to variations in geometry. Topics include:

1. Shape derivatives of shape functionals and PDE solutions (Gâteaux and Fréchet derivatives).
2. An introduction to optimal control of PDEs, with applications to PDE-constrained shape optimization.
3. Geometry-dependent model reduction, including classical projection-based methods such as proper orthogonal decomposition (POD), and an introductory discussion of neural-network-based reduced-order models.

The emphasis in this part will be on the overall analytical framework and modeling concepts, with fewer complete proofs.

Homework will be assigned approximately every two weeks to reinforce the theoretical material; these may include running software simulations with code provided by the instructor. Each student will complete one small project that explores one topic from the course in greater analytical depth.

Topics include the linear algebra method in extremal combinatorics, set systems and intersection theorems, polynomial techniques, configurations in general position, and applications to graphs and discrete geometry. Additional topics may include tensor methods, inclusion matrices, and selected applications to theoretical computer science. The course prioritizes core techniques and their versatility over exhaustive coverage.

This course will provide a comprehensive introduction to the basic theory of matroids. This will include discussions of the basic definitions and examples, duality theory, and certain fundamental substructures of matroids. Particular emphasis will be given to matroids that arise from graphs and from matrices.