Petar Maymounkov's
18.317 Probability and Combinatorics on Groups
with Prof. Igor Pak, Spring 2006

Related links:

• Local copy of past (Fall 2001) course web site – 18.317: Probability and Combinatorics on Groups (Fall 2001)

Project:

• Generators of Wreath Products

Scripted Notes:

• Script (2/8): Overview
• Script (2/10): Basic Cycle Structure of S_n
• Script (2/15): Erdös-Turan Theorem (Original proof was wrong for many years. Pak fixed it!)
• Script (2/17): Dixon's Theorem (Part I)
• Script (2/21): Dixon's Theorem (Part II)
• Script (2/22): The Random Group Process
• Script (2/24): Lubotsky and Hall Theorems
• Script (2/27): Abelian and Solvability Tests (Part I)
• Script (3/1): Abelian and Solvability Tests (Part II), Test for G=S_n and Goldbach's Conjecture
• Script (3/3): Counting Groups
• Script (3/6): Erdös-Renyi Lemma for group isomorphisms, Group Membership Test
• Script (3/8): Product Replacement Graphs
• Script (3/10): 312-avoiding Permutations, Catalan Number on Graphs
• Script (3/13): Product Replacement Graphs
• Script (3/15): Dunwoody's Theorem, Generating Sets for Solvable Groups
• Script (3/17): Cayley Graph of A_n with 3 Generators, Product Replacement Algorithm
• Script (3/20): Lower-bounds on Generating Set Sizes
• Script (4/3): Random Walks (on Cayley Graphs) and Mixing Times
• Script (4/5): Mixing Time and Eigenvalues
• Script (4/7): Conductance, Strook-D.? Bound on Mixing Time
• Script (4/10): Dirichlet Forms (Part I)
• Script (4/12): Dirichlet Forms (Part II), Minimax Principle in Linear Algebra
• Script (4/19): Erdös-Renyi Machine, Self-avoiding Paths
• Script (4/21): Long Paths in Cayley Graphs
• Script (4/24): Cayley Graphs with Small Diameter over O(1) Generating Sets, Kassabov's Results, Conjecture
• Script (4/26): History of Expanders (Part I): Margulius, Kazhdan
• Script (4/28): History of Expanders (Part II): Alon, Bopana, Ramanujan Graphs
• Script (5/1): Bounded Generation, and S_¥
• Script (5/3): Growth of Groups
• Script (5/5): Definition of Grigorchuk's Group
• Script (5/8): Grigorchuk's Group has Super-polynomial Growth (The Lower Bound)
• Script (5/10): Grigorchuk's Group has Sub-exponential Growth (The Upper Bound)
• Script (5/12): Group of Upper-Trianglular Matrices and its Growth, Conjecture
• Script (5/15): Continued: Viennot Representation
• Script (5/17): Percolation of Finite and Infinite Cayley Graphs