# Courses

Today analysis is a central ingredient in most areas of mathematics and its applications, as well as a fascinating area of study in its own right. Whether you are planing to focus on probability, dynamical systems, geometry, partial differential equations or even number theory in your 3rd and 4th year, analysis will be most helpful if not essential to master these subjects.

While its roots go back to Newton, Leibniz and beyond, modern analysis plays a crucial role in much of modern mathematics. For example, the analysis of partial differential equations was central to Perelman's recent resolution of the Poincare conjecture, a conjecture about the topology of three dimensional manifolds. Although analysis has an infinitesimal nature, it also plays a key role in the study of discrete structures such as the distribution of primes among integers (via the Riemann zeta function). Finally, analysis has been successfully employed recently in the study of random processes (or use percolations) in statistical mechanics such as the Ising model.

To be a bit more concrete, in the first two years you have studied:

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

The goal of years 3 and 4 is not only to learn "new" analysis and look at the above topics in a systematic approach towards a more general theory from a more general point of view but also to unravel some of the deep interconnections between the various branches of analysis. You can learn about infinite dimensional normed vectorspaces whose elements are functions and maps between such spaces (

**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

In your analysis course you have met the Riemann integral. One of its main drawbacks is that it behaves poorly under limits: The limit of a sequence of Riemann integrable functions does not have to be Riemann integrable. This is bad news for doing analysis which at the end of the day is all about taking limits!

The main goal of this course is to define a new type of integral, the

*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*and then

*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.While the first three quarters of the course will be mainly concerned with measure and integration in the familiar Euclidean

*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

The central object of Functional Analysis are infinite dimensional vectorspaces which enjoy an additional analytical structure, e.g. a norm or an inner-product. A typical example is the space of continuous functions on the interval [0, 1] equipped with the supremum-norm. Key to the subject is the analysis of linear maps between such function spaces. For instance, in linear algebra you learned how to diagonalise symmetric matrices, i.e. linear operators from

**R**(finite dimensional) to itself. Can one do something analogous for linear operators between innite dimensional spaces? What would that mean?^{n}Historically, Functional Analysis evolved from the study of linear partial differential equations: A linear differential operator acts on the elements of a function space to produce elements of another function space. Functional Analysis therefore also connects intimately to Fourier Analysis and the Theory of Distributions and to Measure and Integration.

**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

Abstract concepts from the course on measure and integration find an application in probability theory, therefore this course can be viewed as a continuation of a measure theory course. Probability is defined as measure, and random variables as measurable functions. However, probability theory differs from measure theory by adding to the latter the concept of independence.

In this course on the foundations of probability we will discuss, in particular: probability measures, random variables, independence, sequences and sums of independent random variables (including various types of convergence, law of large numbers, central limit theorem), characteristic functions.

### Random matrices

The course is an introduction to the theory of random matrices. Generally, a random matrix is a finite dimensional matrix whose elements are random variables. The theory aims to describe properties of the spectrum of such matrices in the limit when their dimension is large. Foundations of the theory were laid in 1950-60's, but it still remains a very active research area. We will discuss the basics of the theory and some of the recent results. A good background in complex analysis and some knowledge of probability and functional analysis (which can be gained by attending the corresponding courses in parallel) is suffcient to take this course.

Random matrices were invented as a model to describe large systems of particles whose precise law of interaction is unknown and can be considered random. They have therefore applications in several branches of physics. Random matrices also have intriguing conjectural connections to the Riemann zeta-function.

**Syllabus:**

Unitary ensembles. Distribution of eigenvalues. Correlation functions. Orthogonal polynomials. Determinantal point processes. Gaussian Unitary Ensemble. Sine kernel and Airy kernel. Equilibrium measures. Zeros of orthogonal polynomials and eigenvalues of random matrices. Circular unitary ensembles. Beta-ensembles. Wigner ensembles; semicircle law.

### Fourier analysis and theory of distributions

Fourier analysis is a fundamental part of modern mathematics. Loosely speaking, Fourier analysis is the study of general functions as a superposition of harmonic "waves" of different frequencies. It has important applications across mathematics, science and engineering, including (but not limited to): Number theory; PDE theory; Probability; Numerical analysis; Quantum mechanics; Fluids; Radio transmission; Acoustics; Optics and Signal processing. Much modern technology would be impossible without our understanding of Fourier analysis.

In this course we shall study the mathematical underpinnings of Fourier analysis. We will give a rigorous foundation to concepts that you may have already seen, such as Fourier series, and establish important results, such as the Fourier Inversion Theorem and the Plancherel Theorem, that you will encounter in many places. Along the way, we shall introduce the theory of distributions, which generalises the idea of a function and allows us to give meaning to concepts such as the "Dirac delta function".

**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

Complex analysis is the study of the functions of complex numbers. It appears in a wide range of topics including dynamical systems, algebraic geometry, number theory, and quantum field theory to name a few. On the other hand, as the separate real and imaginary parts of any analytic function satisfy the Laplace equation, complex analysis is widely employed in the study of two-dimensional problems in physics such as hydrodynamics, thermodynamics, Ferromagnetism, and percolations.

While you have become familiar with basics of functions of a complex variable in the complex analysis course, here we look at the subject from a more geometric viewpoint. We shall look at geometric notions associated with domains in the plane and their boundaries, and how they are transformed under holomorphic mappings. In turn, the behavior of conformal maps is highly dependent on the shape of their domain of definition. The following is the syllabus for the course.

**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

This is the course with the introduction to the theory of pseudo-differential operators and its applications to partial differential equations. We will review basics of the Fourier analysis and then consistently develop elements of the theory that would allow us to solve partial differential equations with variable coefficients.

For further information on this topic we refer to Chapters 1 and 2 here:

### 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