Student Seminar
Randomness and geometry
of groups
Winter semester 2019/2020
In the first part, we study the geometry of a “random group”, in
particular we study Gromov’s density
model and aim to understand why a certain class of random groups
is typically hyperbolic.
In the second part, we investigate random processes on groups, in
particular the connection between behaviour of the process and the
geometry of the group.
The topic is at the intersection of group theory, geometry,
probability theory, combinatorics, and cryptography.
Target group: Master
students in Mathematics. Bachelor students with strong background in
geometric group theory or discrete probability theory and keen interest
in the topic may apply for admission.
Literature (preliminary):
- Yann Olliver: A January 2005 invitation to random groups
Ensaios Matemáticos [Mathematical Surveys], 10.
Sociedade Brasileira de Matemática, Rio de Janeiro, 2005.
http://www.yann-ollivier.org/rech/publs/randomgroups.pdf
- Diaconis, P. & Saloff-Coste, L.: Walks on generating sets of
Abelian groups.
Probab. Th. Rel. Fields (1996) 105: 393.
https://doi-org.emedien.ub.uni-muenchen.de/10.1007/BF01192214
- F. R. K. Chung, R. L. Graham: Stratified random walks on the
n‐cube.
Random Struct. Alg., 11: 199-222 (1997).
https://doi-org.emedien.ub.uni-muenchen.de/10.1002/(SICI)1098-2418(199710)11:3<199::AID-RSA1>3.0.CO;2-W
- Anna Ben-Hamou and Yuval Peres: Cutoff for a stratified random
walk on the hypercube
Electron. Commun. Probab., Volume 23, paper no. 32, 10 pp. (2018)
https://projecteuclid.org/euclid.ecp/1527300061 - S.P. Lalley: Percolation on Fuchsian groups
Annales de l’I. H. P., section B, tome 34, no 2 (1998), p. 151-177.
http://www.numdam.org/item?id=AIHPB_1998__34_2_151_0 - S. Martineau, F. Severo: Strict monotonicity of percolation thresholds under covering maps
Prepring arXiv:1803.09686 [math.PR] - Andrei Okounkov: Random matrices and random permutations
International Mathematics Research Notices, Volume 2000, Issue 20, 2000, Pages 1043–1095, https://doi.org/10.1155/S1073792800000532
Time: Fridays, 8:30 - 10:00 in room B251; dates as listed below.
Followed by breakfast in a nearby café (optional).
| Date
|
Topic | Topic |
| 15 Nov |
T.O. |
Random groups are hyperbolic I |
| 22 Nov |
L.Z. |
Random groups are hyperbolic II |
| 29 Nov |
A.R. |
Random groups are hyperbolic III |
| 06 Dec |
||
| 13 Dec |
D.K. |
Random walk on generating sets |
| 20 Dec |
||
| 10 Jan |
C.R. |
Stratified random walk on the n-cube |
| 17 Jan |
D.G. | Percolation on Fuchsian groups |
| 24 Jan |
||
| 31 Jan |
T.B. |
Percolation thresholds under covering maps |
| 07 Feb |
A.C. |
Random matrices and random permutations |