The interplay of the geometric and the spectral properties of Riemannian manifolds are highly influential in essentially all areas of mathematics, but are far from being fully understood. Some of the recent advancements in understanding this relation could be achieved by means of transfer operators. I shall overview some recent developments in this area with a focus on hyperbolic surfaces, automorphic functions, resonances and the dynamics of the geodesic flow and with an emphasis on insights and heuristics. In particular, I will, at the example of Hecke triangle groups, discuss how an outer symmetry in the dynamics translates into a parity for automorphic functions.

Location: https://meet.google.com/you-qymk-ybu

A special case of Hironaka's QUESTION F, named F', asks about the strong factorization of birational maps between reduced nonsingular algebraic schemes, which is still open. Suppose that $\varphi : X\dashrightarrow Y$ is such a map, and let $U\subset X$ be the open subset where $\varphi$ is an isomorphism. This problem asks if there exists a diagram $$\xymatrix{ & Z \ar[dl]_{\varphi_{1}}\ar[dr]^{\varphi_{2}}\X \ar[rr]^{\varphi} & & Y}$$ where the morphisms $\varphi_{1}$ and $\varphi_{2}$ are sequences of blow-ups of non-singular centers disjoint from $U$. In this talk, we will discuss how strong factorization can be simplified by providing a complete answer to the problem of toroidalization of morphisms, while we introduce the strong Oda conjecture.

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Meeting ID: 9086116889
Passcode: 362880