English

If sufficient demand

The aim of the course is to enlighten and combine different areas, methods
and view points in mathematics. In particular, group theory, geometry, analysis,
probabilty theory, ergodic theory and dynamical system theory. The outcome
should be that students from different fields of mathematics will see that one
and the same area of research has different aspects and view points and that
general progress can only be made by combining many of the different methods and
ideas. On the other hand the students will learn that these combinations of
ideas will also contribute to the development of the single different areas. A
second point is that the topic of the course is a central field of research in
modern mathematics and some of the most prestigious mathematicians are working
on it. A welcome output of the course is to encourage the PhD students to look
out of their own special field of research and try to adopt methods and view
points from other even far away areas of mathematics. The students should see
that mathematics is not divided into different non-overlapping areas but rather
a unified combination of all areas.

Basic properties of random walks, the ergodic theorems of Birkhoff and Kingman, Markov operators, Dirichlet forms, isoperimetric inequalities, the Liouville property of a random walk, boundary of a random walk,
geometry of hyperbolic groups, and Gromov’s classification of groups of polynomial growth.

Understand how combined methods from different fields leads to profound results on random walks.

Be familiar with applying methods from group theory, geometry and ergodic theory to analyse random walks.

choose appropriate methods for analysing problems on random walks.

Seminars

Self-study literature review

Oral exam

Seminars given by participants

Failed, pass

Master studies in mathematics at LTH or NF, or equivalent.

Basic knowledge in algebra, probability and geometry.

Lalley, Steven P.: Random walks on infinite groups.. Springer Verlag, 2023.

FMA205F

2024-11-26

Maria Sandsten