Title: Flag Algebras
A substantial part of extremal combinatorics studies relations existing between densities with which given combinatorial structures (fixed size ``templates'') may appear in unknown (and presumably very large) structures of the same type.
Using basic tools and concepts from algebra, analysis and measure theory, we develop a general framework that allows to treat all problems of this sort in an uniform way and reveal mathematical structure that is common for many known arguments in the area. The backbone of this structure is made by certain commutative algebras depending on the problem in question.
Once understood, it gives rise to the possibility of computer-aided theorem proving in this area based upon semi-definite programming.
In this talk I will give a general impression of how things work in this framework, and we will pay a special attention to concrete applications of our methods.