Calendar
Printer-friendly versionMonday, October 3, 2022
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| 4:00pm |
10/03/2022 - 4:00pm Given two hyperbolic structures m and m’ on a closed orientable surface, how many closed curves have m- and m’-length roughly equal to x, as x gets large? Schwartz and Sharp’s correlation theorem answers this question. Their explicit asymptotic formula involves a term exp(Mx) and 0<M<1 is the correlation number of the hyperbolic structures m and m’. In this talk, we will show that the correlation number can decay to zero as we vary m and m’, answering a question of Schwartz and Sharp. Then, we extend the correlation theorem to the context of higher Teichmuller theory. We find diverging sequences of SL(3,R)-Hitchin representations along which the correlation number stays uniformly bounded away from zero. This talk is based on joint work with Xian Dai. Location:
LOM 206
10/03/2022 - 4:30pm Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry, and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Hülya Argüz. Location:
LOM214
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