# Courses

- the basic principles of multivariable calculus
- some elementary topology and the theory of metric spaces
- some introductory complex analysis

**Functional Analysis**) - this is at the heart of finding solutions to partial differential equations. The question "what is the size of a set?" will be revisited (

**Measure Theory**

**and Integration**) and eventually lead to the Lebesgue theory of integration, which replaces the Riemann integral. Both the most interesting function spaces mentioned above but even more so

**Probability Theory**require the language of measure theory. The course

**Geometric Complex**

**Analysis**looks at the calculus of functions of a complex variable from a geometric viewpoint, for instance, you will learn about how geometric quantities are transformed under holomorphic mappings. In

**Fourier Analysis**you will learn how any function can be studied by splitting it up into its component frequencies.

## More detailed course descriptions

### Measure and integration

*Lebesgue integral*, which remedies that deficiency. To prepare the stage we will begin by addressing the problem of measure, i.e. assigning a notion of volume to subsets of

*R*. This will turn out to be a non-trivial problem (as for instance the

^{n}*Banach-Tarski paradox*illustrates) and will lead us to the notion of

*measurable sets*

*measurable functions*. We will then construct the Lebesgue integral and we prove some key convergence theorems (\montone convergence theorem", "dominant convergence theorem"), from which the fact that the integral behaves well under limits becomes manifest.

*R*, we shall take a more abstract point of view in the last quarter. Here we will define general measure spaces with general measures (giving a notion of "size" to subsets of the general space). This opens the door to applications in Probability, Ergodic Theory and many more.

^{n}**Prerequisites: Second Year Analysis**

**Prepares for: Functional Analysis, Probability**

### Functional analysis

**R**(finite dimensional) to itself. Can one do something analogous for linear operators between innite dimensional spaces? What would that mean?

^{n}**Syllabus:**Brief review of metric spaces, completeness; Banach spaces, Hahn-Banach theorem and applications, Uniform Boundedness Theorem, Open Mapping Theorem, Weak Convergence, Hilbert Spaces, Spectral Theorem, Fredholm Alternative, PDE applications: The Dirichlet problem.

### Probability

### Random matrices

**Syllabus:**

### Fourier analysis and theory of distributions

**Syllabus:**Spaces of test functions and distributions; Fourier Transform (discrete and continuous); Bessel's, Parseval's Theorems; Laplace transform of a distribution; Solution of classical PDE's via Fourier transform; Basic Sobolev Inequalities, Sobolev spaces.

### Geometric complex analysis

**Part 1)**Elements of holomorphic mappings: Poincare metric, Schwarz-Pick lemma, Riemann mapping theorem, growth and distortion estimates, normal families, canonical mappings of multiply connected regions.

**Part 2)**Elements of Potential theory: Dirichlet problem, Green's function, logarithmic potential, Harnack inequality, harmonic measure.

**Part 3)**Elements of quasi-conformal mappings and elliptic PDEs, Beltrami equation, singular integral operators, measurable Riemann mapping theorem.

### Analytic methods in partial differential equations

### Riemann surfaces and conformal dynamics

This elementary course starts with introducing surfaces that come from special group actions (Fuchsian/Kleinian groups). It turns out that on such surfaces one can develop a beautiful and powerful theory of iterations of conformal maps, related to the famous Julia and Mandelbrot sets. In this theory many parts of modern mathematics come together: geometry, analysis and combinatorics.

**Syllabus:**

**Part 1:** Discrete groups, complex Mobius transformations, Riemann surfaces, hyperbolic metrics, fundamental domains

**Part 2:** Normal families of maps and equicontinuity, iterations of conformal mappings, periodic points and local normal forms, Fatou/Julia invariant sets, post-critical set