LFU:online

Students who have completed that module have acquired particular knowledge of one or more branches of higher mathematics. They are able to develop innovative solutions for current problems of those branches of mathematics as well as to judge different approaches critically.
As a result they have developed learning strategies that enable them to acquire further mathematical matters autonomously.

We start with discussing a random walk on Z. It can be related to a Gaussian random walk via the Lindeberg’s swapping argument. Using the Fourier series, we will show that the latter converges to the Brownian motion. Altogether, this results in a simple non-computational proof of the seminal Donsker’s theorem.

We then continue with the Brownian motion in 2D. We will revisit some fundamental results from analysis: the Dirichlet problem and conformal invariance of the Brownian motion.

Next, we consider random height functions: distributions on maps real-valued functions on Z^2. As for the random walks, we can use the Fourier series to show convergence of Gaussian random height function. The limit is called the Gaussian free field and constitutes a fundamental object in mathematics and physics.

One way to view the Gaussian free field is as the standard one-dimensional Brownian motion, but with two-dimensional time. We will discuss its Markov property.

The course will finish by describing some of integer-valued height functions (analogues of a simple random walk) and a recent progress in their study. Remarkably, convergence to the Gaussian free field remains a conjecture.

This course gives a beautiful way to apply the material learnt in the Probability course. Quite a few ideas are coming from other branches of mathematics: Mathematical Physics, Functional Analysis, Complex Analysis, Conformal Geometry.

No previous knowledge of Physics/Mathematical Physics is needed. A course on Higher stochastics is recommended (but not mandatory).

Continuous assessment (based on regular written and/or oral contribution by participants).

Oral examinations are examinations that require responding to questions verbally; according to § 6, statute section on "study-law regulations"

Mörters, Peres "Brownian motion" (Sec 1.4, Chapter 3, Sec 5.3)
https://people.bath.ac.uk/maspm/book.pdf

Berestycki, Powell "Gaussian free field and Liouville quantum gravity" (Chapter 1)
https://homepage.univie.ac.at/nathanael.berestycki/wp-content/uploads/2023/09/master.pdf

Stochastik I and II (bachelor course on Probability Theory)

The course will be more or less evenly divided into lectures and exercise sessions. The subject is perfectly suited for learning by solving problems. This will also allow to have more interaction and to learn from other students.