# Combinatorics, Probability and Computations on groups

Instructor: Igor Pak, MS 6240, pak@math.

Class Schedule: MWF 1:00-1:50, MS 6201.

## Brief outline

We will give an introduction to the subject, covering a large number of classical and a few recent results. The emphasis will be on the main ideas and techniques rather than proving the most recent results in the field. The idea is to to present a variety of results the field and their applications.

The prerequisites for the course are somewhat diverse, but not terribly difficult. I will use a number of standard results from Group Theory, and many of them will be stated and used without reference to their proof. I will need a number of basic results from Probability, mostly on Markov chains, but will re-prove some of these. I will need almost nothing from CS. In each case I will fully explain what is needed so the course will be largely self-contained.

For more info on this course, see lecture notes from an earlier version of the course.

## Content:

• Probability on groups:
• Symmetric group, distribution of orders, Erdos-Turan Theorem
• Probability of generation, Dixon Theorem, generalizations
• Enumeration of finite groups of given order (Higman-Sims)
• Random subproducts, Erdos-Renyi Theorems
• Diameters and expansion of random Cayley graphs
• Hamiltonicity of Cayley graph

• Random walks:
• Examples
• Various definitions of mixing time, relations between them
• Strong uniform time approach, examples, coupling arguments
• Conductance, multicommodity flows
• Comparison technique, rate of escape lemma
• Kazhdan's property (T), expanders (explicit constructions)

• Algorithms:
• Permutation groups management (Sims)
• Testing properties (abelian, nilpotent, solvable)
• Recognition of symmetric and linear groups
• Jerrum's Markov chain

• Generating random elements in finite groups:
• Babai Algorithm, improvements
• Product replacement algorithm:
• Connectivity issues
• Bias (following Babai-IP)
• Mixing time results
• Cooperman's algorithm
• Applications

Note: this is a bit too ambitious for a quarter course. I will prune the material as we go.

There are no homeworks. If you are taking this course for credit, there will be a research project at the end.

### Course Textbooks:

I will not follow any particular textbook, but instead try to make the lecture as self-contained as possible. I ordered one textbook:

David A. Levin, Yuval Peres and Elizabeth L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, 2008.

Various results from this book will be used, while others will be proved. In general, this book is a good supplementary reading and I will refer to it on several occasions. Additional reading sources will include:

Akos Seress, Permutation Group Algorithms, Graduate Texts in Mathematics 202, Cambridge.
Persi Diaconis, Group Representations in Probability and Statistics, Institute of Mathematical Statistics.
Simon R. Blackburn, Peter M. Neumann, Geetha Venkataraman, Enumeration of Finite Groups, Cambridge.

I believe all these books are available in the math library, from Amazon.com and other retailers.