Mathematical Conversations

A main theme in extremal combinatorics is about asking when the random construction is close to optimal. A famous conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if $H$ is a bipartite graph, then the random graph with edge density $p$ has in expectation asymptotically the minimum number of copies of $H$ over all graphs of the same number of vertices and edge density. It turns out it is quite difficult to prove / disprove this conjecture. I will talk about the analytic version of this inequality, whose variants turn out to appear in other fields such as random matrix theory. I will discuss some tools used to address this conjecture and some related results.