Valid from: Spring 2025
Decided by: Maria Sandsten
Date of establishment: 2024-11-26
Division: Mathematics
Course type: Third-cycle course
Teaching language: English
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.
Knowledge and Understanding
For a passing grade the doctoral student must Understand how combined methods from different fields leads to profound results on random walks.
Competences and Skills
For a passing grade the doctoral student must Be familiar with applying methods from group theory, geometry and ergodic theory to analyse random walks.
Judgement and Approach
For a passing grade the doctoral student must choose appropriate methods for analysing problems on random walks.
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.
Lalley, Steven P.: Random walks on infinite groups.. Springer Verlag, 2023.
Types of instruction: Seminars, self-study literature review
Examination formats: Oral exam, seminars given by participants
Grading scale: Failed, pass
Examiner:
Admission requirements: Master studies in mathematics at LTH or NF, or equivalent.
Assumed prior knowledge: Basic knowledge in algebra, probability and geometry.
Start date: 2025-01-07.
Start date is approximate.
End date: 2025-06-08
Course pace: Not specified
Course coordinator: Tomas Persson <tomas.persson@math.lth.se>