Mathematics Colloquium, Spring 2008
SPECIAL LECTURE

Flag Algebras

A substantial part of extremal combinatorics studies relations existing between densities with which given (fixed size) combinatorial structures may appear in unknown (and presumably very large) structures of the same type. Using basic tools and concepts from algebra, analysis and measure theory, a general framework is developed that allows to treat all problems of this sort in an uniform way and reveal mathematical structure that is common for most known arguments in the area (the central objects in this theory are called Flag Algebras whence comes the title.) In this talk I hope to give a general impression, illustrated by simple examples, of how things look like in this framework, and discuss its advantages and drawbacks compared to the "naive" approach. I will also mention some concrete results obtained with the help of these methods.