Pre-seminar 1:30

Title: Towards an Intersection Chow Cohomology Theory with a Primer on Toric Stacks

Abstract: We give a general introduction to stacks, focusing in particular on toric stacks. We discuss applications to the following two problems: (i) extending the Hirzebruch-Riemann-Roch theorem to stacks, and (ii) constructing an intersection Chow cohomology theory.

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Main seminar 2:30

Title: Interpolating Between the Batyrev-Manin and Malle Conjectures

Abstract: The Batyrev-Manin conjecture gives a prediction for the asymptotic growth rate of rational points on varieties over number fields when we order the points by height. The Malle conjecture predicts the asymptotic growth rate for number fields of degree d when they are ordered by discriminant. The two conjectures have the same form and it is natural to ask if they are, in fact, one and the same. We develop a theory of point counts on stacks and give a conjecture for their growth rate which specializes to the two aforementioned conjectures. This is joint work with Jordan Ellenberg and David Zureick-Brown.