Eran Assaf

Dartmouth College

Computing Equations for Modular Curves

Modular curves are fundamental objects in number theory and lie at the core of several important open problems. We will discuss the state of the art methods for computing their equations, and my attempts to improve them.

Avi Kulkarni

Dartmouth College

Deep learning Gauss-Manin connections

In this talk, I will discuss how machine learning can aid in the computation of the periods of projective hypersurfaces. I will also report on the results of our large-scale computation to find the periods of all smooth quartics in P3 that are the sum of 5 monomial terms with unit coefficients. Joint work with Kathryn Heal and Emre Sertoz.

Carl Mautner

Dartmouth College

Category O for oriented matroids

The representation theory of Lie algebras has rich structure. Recently it was discovered that similar structure can be cooked up from the algebraic geometry of any of a whole zoo of varieties called symplectic resolutions. In joint work with Ethan Kowalenko we produce analogous structure coming from the combinatorics of (oriented) matroids.

Asher Auel

Dartmouth College

Sextic normal genus one curves

I’ll report on progress made with Marcello Bernardara on the problem of showing that every Brauer class is split by a genus one curve. After a brief overview of the problem, I’ll explain our new construction proving the index 6 case, which as a by-product gives a potentially useful presentation of the moduli space of sextic normal elliptic curves.

Jack Petok

Dartmouth College

Modular forms and moduli of hyperkählers

In my thesis and in ongoing work with Jen Berg, we use modular forms to study the birational geometry of some interesting moduli spaces. I'll review a bit of the algebraic geometry of these moduli spaces, and then I'll explain some methods, due to Gritsenko, Hulek, and Sankaran, for computing the Kodaira dimension of these moduli spaces using modular forms defined on high rank orthogonal groIn my thesis and in ongoing work with Jen Berg, we use modular forms to study the birational geometry of some interesting moduli spaces. I'll review a bit of the algebraic geometry of these moduli spaces, and then I'll explain some methods, due to Gritsenko, Hulek, and Sankaran, for computing the Kodaira dimension of these moduli spaces using modular forms defined on high rank orthogonal groups.

Ciaran Schembri

Dartmouth College

Torsion points on abelian surfaces with potential quaternionic multiplication

In a celebrated work Mazur classified which torsion subgroups can occur for elliptic curves defined over the rationals. In this talk I will give a brief introduction to the analogue question of which torsion subgroups can occur for abelian surfaces defined over the rationals which over some extension obtain endomorphisms by an order in a quaternion algebra.

Grant Molnar

Dartmouth College

The LCM Product and Grönwall's Theorem

In this talk, we define the LCM product, due originally to R. D. von Sterneck. We relate the LCM product to Dirichlet convolution, and then prove an analogue to Grönwall's Theorem for kth LCM powers of the identity function on the natural numbers.