# 18.317 Combinatorics, Probability and Computations on Groups (Fall 2001)

Instructor: Igor Pak

Class schedule: MWF 11-12, 2-255

Office Hours: 2-390, W 2:00-4:00

Grading: All listeners will have to take turns in writing up the lectures. Students wishing to receive a grade will have to do a research project.

Difficulty: This is a graduate level course, although it should be accessible to some advance undergraduate students. We assume familiarity with basic Group Theory, Discrete Probability and some Algebraic Combinatorics.

## Content (tentative)

• Probability on groups:
• Random subproducts, Erdos-Renyi Theorems
• Symmetric group, distribution of orders, Erdos-Turan Theorem
• Probability of generation, Dixon Theorem, generalizations
• Random groups? (enumeration, Higman-Sims)

• Random walks:
• 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
• Testing finiteness of linear groups (Babai-Beals-Rockmore)
• Jerrum's Markov chain

• Generating random elements in finite groups:
• Babai Algorithm, improvements
• Product replacement algorithm:
• Connectivity, T-systems
• Bias (following Babai-IP)
• Mixing time

## Lecture Notes:

Lecture 1. Download LaTeX file, .dvi file, .ps file, or .pdf file.

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Lectures 34-36. Survey on expanders. No scribe.

For those scribing lectures: get prepambule from Lecture 3. Check out this handout on how to use a similar preambule.