Central limit theorem for homology of simple closed curves
Vortrag von Dr. Pouya Honaryar
Datum: 09.10.24 Zeit: 13.30 - 14.30 Raum: Y27H28
Fix a hyperbolic surface $X$, and for $L > 0$, let $\mathcal{S}_L(X)$ denote the set of simple closed geodesics of length at most $L$ on $X$. Fixing a norm on $H_1(X, \mathbb{R})$, we may ask what is the statistics of the norm of homology class of $\alpha$, denoted by $[\alpha]$, when $\alpha$ is chosen randomly uniformly from $\mathcal{S}_L(X)$, as $L \rightarrow \infty$? For example, does one expect the norm of $[\alpha]$ to be of order $L$ or smaller? We answer this question by proving a CLT-type result for the norm of homology of a randomly chosen curve in $\mathcal{S}_L(X)$. We discuss the main steps to reduce the desired CLT to a CLT for the Kontsevich-Zorich cocycle obtained by Forni-Saqban. (Joint work with F. Arana-Herrera)