# AG@PUI virtual seminar

This is a biweekly virtual seminar for algebraic geometers at primarily undergraduate institutions. In Fall 2024, it runs on Fridays at 3 - 4 pm (Eastern), 2 - 3 pm (Central), 1 - 2 pm (Mountain), and 12 - 1 pm (Pacific). The current organizers are Han-Bom Moon, Julie Rana, and David Swinarski.

This event is open to everyone, not only to the faculty at PUI. If you want to receive a biweekly announcement email, please contact Julie.

The seminar is broadcasted via zoom. Check the announcement emails for the link.

Bring a cup of coffee or tea!

## Fall 2024

Date: September 20.

Speaker: Adam Boocher (University of San Diego)

Title: From Classical Commutative Algebra to Some Special Diophantine Equations

Abstract: One of the first theorems one learns in a commutative algebra class is the "Principal Ideal Theorem" which essentially says that in a system of polynomial equations, the codimension of the corresponding variety is never more than the number of equations. In this talk, I'll share a bit of history and explain how this Theorem is actually the start of a rich story. The focus will be on two rather mysterious conjectures about betti numbers and I will explain an attempt to study them in a special case, using the relatively new techniques of Boij-Soederberg Theory. This approach leads to some interesting diophantine equations, which may shed light on the original conjectures. This is joint work with my two undergraduate students Noah Huang and Harrison Wolf.

Date: October 4.

Speaker: John Little (Holycross)

Title: Rational “multi-lump” solutions of the KP equation from cuspidal algebraic curves

Abstract: We discuss an application of algebraic geometry to the construction of certain solutions of a PDE modeling solitary wave (soliton) phenomena in two space dimensions and time (the Kadomtsev-Petviashvili, or KP equation). The connection between the theta functions associated with the Jacobians of (smooth) algebraic curves and solutions of this equation was a major development of the 1970's; in the 1980's, this provided significant progress toward a solution of the Schottky problem. In this talk we will discuss recent work on obtaining rational "multi-lump" KP solutions from cuspidal singular curves in a parallel way building on results obtained by the speaker together with Daniele Agostini, and Turku Ozlum Celik. Through discussions of the context and detailed explicit examples, we will strive to make this understandable for non-experts in this area.

Date: October 18.

Speaker: Ursula Whitcher (Mathematical Reviews)

Title: The divisors under the rainbow

Abstract: Adinkras are decorated graphs that encapsulate information about the physics of supersymmetry. If we color the edges of an Adinkra with a rainbow of shades in a specific order, we obtain an associated algebraic curve. We use this structure to characterize height functions on Adinkras, then show how to encapsulate the same information using data from our rainbow. We give a complete description of the map in the case of the 5-dimensional hypercube Adinkra. This talk describes joint work with Amanda Francis.

Date: November 1.

Speaker: Gabriel Dorfsman-Hopkins (St. Lawrence University)

Title: TBA