WAKE FOREST UNIVERSITY
DEPARTMENT OF MATHEMATICS & STATISTICS
University of Virginia
Thursday, March 28, 2019
Manchester Hall, Room 018
“Moduli problems and root constructions in algebraic geometry”
Many classification problems in math – usually stated as “describe all isomorphism classes of (blank)”, or “how many (blanks) are there?” – have a natural parametrization or geometric structure to them. A moduli problem is, loosely speaking, such a classification problem with natural geometric structure. In the first part of this talk, I will interpret several problems in geometry as moduli problems and show how algebraic geometry can be used to solve them. Among these are: the number of lines on a cubic surface (spoiler: there are exactly 27) and the theory of modular forms. In the second part, we will discuss how to take n-th roots — not just of numbers, but of more general objects in math, and in particular of line bundles. Then, I will explain how modular forms over the complex numbers relate to the problem of taking nth roots of certain line bundles over a curve. Time permitting, I will introduce a new construction that I am using to shed light on these ideas in characteristic p.