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Tuesday, July 14, 2020

**Abstract:** Let $H$ be a graph with vertices $1,2,\ldots,n$ and edge-set $E$. We associate with it a functional that acts on bounded measurable (symmetric) functions $F: \: [0,1]^2 \to \mathbb{R}$, namely $$ t_H(F) \; = \; \int_{[0,1]^n} \prod_{\{i,j\} \in E} F(x_i,x_j) \: dx_1 dx_2 \cdots dx_n \; . $$ This notion arises from counting copies of $H$ in a large graph $F$.

We will review results and open problems in such areas as

Majorization ($H$ majorizes $G$ when $t_H(F) \geq t_G(F)$ for all $F$),

Positivity of $t_H$.

Convexity of $t_H$.

Please contact Sean English at SEnglish (at) illinois (dot) edu for the Zoom ID.