Conference: AGNES Summer School in Algebraic Geometry

Brown University, Providence RI

Investigators

Abstract

The AGNES Summer School on Intersection Theory on Moduli Spaces will be held at Brown University July 11-14, 2023. Algebraic curves are one-dimensional sets of points that can be described by polynomial equations. For example, the unit circle in the plane x^2 + y^2 = 1 defines an algebraic curve. A ubiquitous problem in the theory of algebraic curves is to understand how many algebraic curves there are satisfying a given list of conditions. This information corresponds to the so-called "intersection theory of their moduli space": a moduli space of curves is a space whose points correspond to algebraic curves, and its intersection theory describes how various loci in this space can meet (which corresponds to various conditions on the algebraic curve being satisfied simultaneously). The Chow ring of the moduli space packages this information of intersection theory into an algebraic structure. This summer school will introduce graduate students and postdocs to recent developments in computing Chow rings through four mini-courses, afternoon exercise sessions, and research groups. Students will participate in exercise sessions to reinforce the material from the mini-courses. They will also work in research groups to apply these techniques to compute new examples of Chow rings (both integral and rational) of moduli spaces of curves. More specifically, the four mini-courses will be on the following topics: (1) Equivariant intersection theory: this describes how intersection theory behaves in the presence of a group action; (2) Higher Chow groups: these invariants capture the failure of exactness of excision sequences of Chow rings; (3) The tautological ring of the moduli space of stable pointed curves: this describes a well-behaved subring of the Chow ring; and (4) Patching techniques and the integral Chow ring of the moduli of 2-pointed genus 1 curves: often enlarging moduli spaces to include curves with worse singularities can enable patching togetherinformation about Chow rings of strata to the Chow ring of the entire moduli space. https://sites.google.com/site/agneshomepage/brown-2023-agnes-summer-school This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.