Saturday, May 3, at the University of Maryland

Gurbir Dhillon (UCLA)

Title: Ramification in geometric Langlands and representation theory.
Abstract:  Some celebrated recent work of Arinkin–Beraldo–Campbell–Chen–Faergeman–Gaitsgory–Lin–Raskin–Rozenblyum settles the unramified geometric Langlands conjecture, which is a certain equivalence of categories related to closed Riemann surfaces. After providing a gentle introduction to this circle of ideas, we will discuss a series of further conjectures, due to Beilinson–Drinfeld, Frenkel–Gaitsgory, and others, now involving punctured Riemann surfaces, and the highly non-trivial implications the latter circle of ideas has in representation theory. In the second half, we will discuss some results concerning the latter conjectures, their consequences, and some natural variants, based on joint works with Joakim Faergeman, Yau Wing Li, Ivan Losev, David Yang, Zhiwei Yun, and Xinwen Zhu.

Rachel Webb (Cornell)

Title: Finding the walls in GIT.
Abstract: Geometric Invariant Theory (GIT) can be used to construct familiar varieties, including many toric varieties, as well as flag varieties or more generally moduli of quiver representations. The version of GIT I will use in this talk depends on a complex reductive group G, a linear representation of G, and a homomorphism from G to the group of complex units. While complex reductive groups include general linear (type A) groups as well as orthogonal and symplectic (types B, C, D) groups, the examples of GIT mentioned in the first sentence all arise from representations type A groups. Even for representations of the general linear group, the GIT procedure sometimes “goes wrong” and produces an artin stack instead of a variety or orbifold.

Why don’t we have examples of GIT using symplectic or orthogonal groups? When the group is GLn (or more generally type A), can we tell when GIT will produce a variety or orbifold instead of an Artin stack? The answers to these questions both amount to “finding the walls” in variation of GIT. In these talks I will explain more about what it means to find the walls and why we care. I will also present some partial answers to the above questions. Some of these answers are joint with Hans Franzen, and others with Riku Kurama, Ruoxi Li, and Henry Talbott..

David Zureick-Brown (Amherst College)

Title: Elliptic curves, Galois theory, and Mazur’s “Program B”.
Abstract: The “N-torsion field Q(E[N])” of an elliptic curve E over the rationals is the field obtained by adjoining the coordinates of the N-torsion points of E. The Galois group of this field turns out to be an incredibly interesting and crucial object of study. In the first talk, I’ll explain several reasons why (for example, understanding this group is a generalization of Mazur’s theorem classifying possibilities for rational torsion). The problem of classifying all such Galois groups is called Mazur’s “Program B”.

In the second talk, I will discuss recent joint work with Jeremy Rouse and Drew Sutherland on “Program B”. This problem is equivalent to classifying rational points on certain modular curves XH. Our main result is a provisional classification of the possible images of l-adic Galois representations associated to elliptic curves over Q and is provably complete barring the existence of unexpected rational points on modular curves associated to the normalizers of non-split Cartan subgroups and two additional genus 9 modular curves of level 49.

I will also discuss various applications (for example: a very fast algorithm to rigorously compute the l-adic image of Galois of an elliptic curve over Q), and then highlight several new ideas from the joint work, including techniques for computing models of modular curves and novel arguments to determine their rational points, a computational approach that works directly with moduli and bypasses defining equations, and (with John Voight) a generalization of Kolyvagin’s theorem to the modular curves we study.