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.
Grading:
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.
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Last updated 8/14/2008