Francesco Gallinaro (University of Leeds)

Algebraic flows on tori: an application of model theory

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A complex torus $\mathbb{T}$ is a Lie group which is obtained as a quotient of a finite dimensional complex vector space, $\mathbb{C}^g$, by a lattice; so there is a canonical projection map $\pi$ from $\mathbb{C}^g$ into $\mathbb{T}$. If we consider an algebraic subvariety $V$ of $\mathbb{C}^g$, then we can ask what the image of $V$ under $T$ looks like: Ullmo and Yafaev proved that if $V$ has dimension 1, then the closure of $\pi(V)$ in the Euclidean topology is given by a finite union of translates of closed subgroups of $\mathbb{T}$, and conjectured that this should hold in higher dimensions. Using model theoretic methods, Peterzil and Starchenko showed that this conjecture isn't quite true, but that a similar, slightly more complicated statement holds, describing the closure of $\pi(V)$ in terms of finitely many closed subgroups of $\mathbb{T}$. In this talk, I'll introduce the problem and describe the main ingredients of the Peterzil-Starchenko proof.