Variations of canonical measures: Riemann surfaces, graphs and hybrid curves

Algebra Seminar
Wednesday, December 2, 2020 - 3:30pm for 1 hour (actually 50 minutes)
Noema Nicolussi
Justin Chen

In the last years, connections between graphs and Riemann surfaces have been
discovered on several different levels. In particular, graphs are closely related
to singular Riemann surfaces and the boundary in the Deligne–Mumford com-
pactification of moduli spaces. Moreover, in both settings there is a notion of a
canonical measure (the Arakelov–Bergman and Zhang measures) which reflects
crucial geometric information.
In this talk, we focus on the following question: what is the limit of the canon-
ical measures along a family of Riemann surfaces? Combining the canonical
measures on Riemann surfaces and metric graphs, we obtain a full description
and a new compactification of the moduli space of Riemann surfaces in terms
of hybrid curves.

Based on joint work with Omid Amini (École polytechnique).

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