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loci, equidistribution of Hecke translates of a curve in the moduli space of abelian varieties and equidistribution of some families of CM points in Shimura varieties. The results of this talk are joint work with Nicolas Tholozan.

We will consider the following problem: if (C,x_1,…,x_n) is a fixed general pointed curve, and X is a fixed target variety with general points y_1,…,y_n, then how many maps f:C -> X in a given homology class are there, such that f(x_i)=y_i? When considered virtually in Gromov-Witten theory, the answer may be expressed in terms of the quantum cohomology of X, leading to explicit formulas in some cases (Buch-Pandharipande). The geometric question is more subtle, though in the presence of sufficient positivity, it is expected that the virtual answers are enumerative. I will give an overview of recent progress on various aspects of this problem, including joint work with Farkas, Pandharipande, and Cela, as well as work of other authors.

https://harvard.zoom.us/j/97335783449?pwd=S3U0eVdyODFEdzNaRXVEUTF3R3NwZz09

Consider the set of Galois extensions L of Q whose Galois group is the quaternion group. For large X, Klüners counted extensions with |disc(L)| <= X. We discuss asymptotics when bounding invariants other than the discriminant.

One way that mathematics grows is by finding new questions to study beyond the standard topics of serious mathematical research. We give three examples, ranging from the recreational (how many digits in 6561101970383!, and how did I find this curious factorial? What’s the unusual feature of the “elementary” identity 12 + 21 + 27 + 38 + 44 + 53 = 2 + 5 + 11 + 22 + 33 + 48 + 74?) to a collaboration with faculty in the School of Public Health.