The plan is to have some lectures on Geometric group theory and Probability on Graphs.
Tentative plan is to meet alternative wednesdays at 3.15 PM with one wednesday for "Geometric group theory" and the other for "Probability on Graphs".
LECTURES :
21/8 : Quasi-isometries - Basics and Milnor-Schwarz lemma. Section 7.1-7.4 of [1]. (by Akshay)
28/8 : Applications of Milnor-Schwarz lemma (by Akshay) ; Random walks and electrical networks - Intro. Section 2.1 of [2]. (by Rahul)
4/9 : Voltage as Green's function, Nash-Williams criterion for recurrence ; Sections 2.2-2.4 of [2]. (by Rahul).
23/9 : Finite energy criterion for transience, Rough isometries and random walks (Sections 2.5 and 2.6 of [2]). (by Yogesh.)
25/9 : Introduction to percolation on transitive graphs. Non-trivial phase transition and connections to growth and amenability of graphs (based on Part 2 of [4]) (by Yogesh).
1/10 : Amenability of groups. Quasi-isometry invariance of amenability, Amenability via invariant means. (parts of Ch. 9 of [5]) (by Akshay)
9/10 : Growth of groups. (parts of Ch. 6 of [5]) (by Akshay)
16/10 : Random walk on transitive graphs ; Spectral radius and return probabilities. (based on Ch 1 of Part 1 of [4]) (by Rahul).
23/10 : Gradient operator, isoperimetric constant and Kesten's criteria for non-amenability. (based on Ch 1 of Part 1 of [4]) (by Rahul).
30/10 : Carne-Varapoulos Bound. (based on Ch 2 of Part 1 of [4])(by Sarvesh)
REFERENCES :
[1] Office hours with a geometric group theorist : Clay and Margalit
[2] Probability on trees and networks : Lyons-Peres
[3] Probability and Geometry of Groups; G. Pete.
[4] Probability on transitive graphs. V. Tassion.
[5] Geometric Group Theory - Clara Leoh.
[6] Lectures on Ising and Potts Model on Hypercubic Lattice - Hugo Duminil-Copin