Thursday, October 3, 2019
Time  Items 

All day 

4:00pm 
10/03/2019  4:00pm For a graph $H$, its homomorphism density in graphs naturally extends to the space of twovariable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by~$t_H(W)$. One may then define corresponding functionals~$\W\_{H}:=t_H(W)^{1/e(H)}$ and $\W\_{r(H)}:=t_H(W)^{1/e(H)}$ and say that $H$ is (semi)norming if $\.\_{H}$ is a (semi)norm and that $H$ is weakly norming if $\.\_{r(H)}$ is a norm. We obtain some results that contribute to the theory of (weakly) norming graphs. Firstly, we show that `twisted' blowups of cycles, which include $K_{5,5}\setminus C_{10}$ and $C_6\square K_2$, are not weakly norming. This answers two questions of Hatami, who asked whether the two graphs are weakly norming. Secondly, we prove that $\.\_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. This answers another question of Hatami, who estimated the modulus of convexity and smoothness of $\.\_{H}$. We also prove that every graph $H$ without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of $H$ when studying graph norms. Based on joint work with Frederik Garbe, Jan Hladky, and Bjarne Schulke. Location:
DL 431
10/03/2019  4:15pm Quantitative arguments in dynamical systems are the subject of recent investigations in various contexts. We will discuss an effective eqdistribution theorem in homogeneous dynamics and how it relates to questions about positive definite quadratic forms. Location: 10/03/2019  4:15pm It is conjectured that in order to test Kpolystability of a Fano variety, it is enough to consider only equivariant test configurations with respect to a finite or connected reductive group action. This has been proved for Fano manifolds and in singular cases, only for torus actions. In this talk, I will talk about a valuative criterion of equivariant Kstability with respect to an arbitrary group action. This generalizes parallel results for usual Kstability. Location:
LOM 214
