## Joonkyung Lee, Some Advances in Sidorenko’s Conjecture

Joonkyung Lee
University of Oxford, UK
2014/09/04 *Thursday* 4PM-5PM
Room 1409
Sidorenko’s conjecture states that for every bipartite graph $$H$$ on $$\{1,\cdots,k\}$$
\begin{eqnarray*}
\int \prod_{(i,j)\in E(H)} h(x_i, y_j) d\mu^{|V(H)|}
\ge \left( \int h(x,y) \,d\mu^2 \right)^{|E(H)|}
\end{eqnarray*}
holds, where $$\mu$$ is the Lebesgue measure on $$[0,1]$$ and $$h$$ is a bounded, non-negative, symmetric, measurable function on $$[0,1]^2$$. An equivalent discrete form of the conjecture is that the number of homomorphisms from a bipartite graph $$H$$ to a graph $$G$$ is asymptotically at least the expected number of homomorphisms from $$H$$ to the Erdos-Renyi random graph with the same expected edge density as $$G$$. In this talk, we will give an overview on known results and new approaches to attack Sidorenko’s conjecture.
This is a joint work with Jeong Han Kim and Choongbum Lee.

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