Gäller från och med: Spring 2025
Beslutad av: Maria Sandsten
Datum för fastställande: 2024-11-26
Avdelning: Mathematics
Kurstyp: Ren forskarutbildningskurs
Undervisningsspråk: 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.
Kunskap och förståelse
För godkänd kurs skall doktoranden Understand how combined methods from different fields leads to profound results on random walks.
Färdighet och förmåga
För godkänd kurs skall doktoranden Be familiar with applying methods from group theory, geometry and ergodic theory to analyse random walks.
Värderingsförmåga och förhållningssätt
För godkänd kurs skall doktoranden 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.
Undervisningsformer: Seminarier, litteraturkurs som självstudier
Examinationsformer: Muntlig tentamen, seminarieföredrag av deltagarna
Betygsskala: Underkänd, godkänd
Examinator:
Förkunskapskrav: Master studies in mathematics at LTH or NF, or equivalent.
Förutsatta förkunskaper: Basic knowledge in algebra, probability and geometry.
Startdatum: 2025-01-07.
Startdatumet är ungefärligt.
Slutdatum: 2025-06-08
Kursfart: Not specified
Kursansvarig: Tomas Persson <tomas.persson@math.lth.se>