Course: MCS 591, Advanced Topics in Combinatorics: The Probabilistic Method Call no: 37067 Time: MWF 11:00-11:50pm Place: 316 2TH
Professor: Dhruv Mubayi Office: 620 SEO Tel: TBA E-mail: mubayi@uic.edu Course Web Page: http://www.math.uic.edu/~mubayi/591/Fall2015/ProbMethodFall15info.html Office Hours: TBA
Grading Policies: You grade will be based on occasional homework assignments, class presentations, and discussions.
Disability Policy: Students with disabilities who require accommodations for access and participation in this course must be registered with the Office of Disability Services (ODS). Please contact ODS a 312/413/-2183 (voice) or 312/413-0123 (TTY).
Prerequisite: An undergraduate course in combinatorics or probability and the mathematical maturity of a graduate student.
Course Description: The probabilistic method is a powerful tool for tackling many problems in mathematics, statistics and computer science. The basic idea of the method is as follows: in order to prove that a structure with certain desired properties exists, once defines an appropriate probability space of structure
s and then shows that the desired properties hold in this space with positive probability. Introduced by Erdos in the 1940s to solve a basic problem in Ramsey theory, it has since proven useful in many
different areas of mathematics (e.g. geometry, number theory, combinatorics) and more recently in computer science
and statistics as well. The course should therefore be interesting and useful for students in a
variety of fields.
This course will be a thorough introduction to the probabilistic method. While we will focus on applicati
ons in combinatorics, problems in different areas will often be explored.
A sampling of topics: Ramsey numbers, random graphs, Local Lemma, second moment method, correlation inequa
lities (the four function theorem and the FKG inequality), martingales, circuit complexity, discrepancy, geometric problems, derandomization.
Text: The Probabilistic Method by Alon and Spencer (second edition)
Optional Texts: Random Graphs by Bollobas (second edition), Random Graphs by Janson, Luczak, Rucinski
Homework 1, Due Friday September 11
Homework 2, Due Friday September 25
Homework 3, Due Friday October 9
Homework 4, Due Friday October 23
Homework 5, Due Wednesday November 4 (no class on Friday Nov 6)
Homework 6, Due Friday November 20
Homework 7, Due Friday December 4