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.

Lecture 2. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 3. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 4. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 5. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 6. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 7. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 8. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 9. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 10. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 11. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 12. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 13. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 14. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 15. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 16. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 17. Not available.

Lecture 18. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 19. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 20. Not available.

Lecture 21. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 22. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 23. Not available.

Lecture 24. Not available.

Lecture 25. Not available.

Lecture 26. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 27. Not available.

Lecture 28. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 29. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 30. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 31. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 32. Download LaTeX file, .dvi file, .ps file, or .pdf file.

Lecture 33. Download LaTeX file, .dvi file, .ps file, or .pdf file.

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.

Click here to return to Igor Pak Home Page.

To e-mail me click here and delete .zzz
Put 18.317 in the Subject line.

Last updated 2/16/2002