# Valuative Tree Seminar

Monday, September 23, 2019 — 2:30 PM EDT

**Note time and room change**

Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo

"Introduction to the valuative tree"

If $X$ is a variety and $\varphi:X\rightarrow X$ is a morphism, we can ask some basic questions about the orbit $\{x,\varphi(x),\varphi^2(x),\ldots\}$. For example: when is it possible for an orbit to be Zariski-dense? Or how often does an orbit intersect a proper subvariety of $X$? These dynamical problems can be approached by analyzing the induced map $\varphi^*$ on the space of valuations on the coordinate ring $R=K[X]$ --- this space is called the Berkovich analytification $X^{\mathrm{an}}$. In the case $X=\mathbb{A}^2$ and $R=K[x,y]$, Xie proved the above two conjectures by studying certain $\varphi^*$-invariant subspaces of $X^{\mathrm{an}}$. This will be a recap of the three lectures we had in August.

MC 5413

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