Schedule
All talks will be held on Zoom. See the Registration tab for details. The schedule is in Central Daylight Time (US).
Abstracts
Plenary Talks (Click title for slides):
Benjamin Antieau (Northwestern University)
Title: On the Brauer group of the moduli stack of elliptic curves
Abstract: Mumford proved that the Picard group of the moduli stack of elliptic curves is a finite group of order 12, generated by the Hodge bundle of the universal family of elliptic curves. After giving background on Brauer groups and on the moduli of elliptic curves, I will talk about work with Lennart Meier, which computes the Brauer group of the moduli stack over various arithmetic base schemes and shows in particular that the Brauer group of the integral moduli stack vanishes. This talk will focus on the concrete computational and arithmetic aspects of the proof.
Laure Flapan (Michigan State University)
Title: Fano manifolds and hyperkahler manifolds
Abstract: For many Fano manifolds whose cohomology looks like that of a K3 surface, there are known geometric constructions of corresponding hyperkahler manifolds. In this talk, we discuss a reverse procedure by which, given a hyperkahler manifold, one can geometrically construct a corresponding Fano manifold. This is joint work with E. Macrì, K. O’Grady, and G. Saccà.
Angela Gibney (Rutgers University - New Brunswick)
Title: Vertex algebras of CohFT- type
Abstract: Representations of vertex algebras of CohFT-type can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable pointed curves. As I'll explain in the talk, the name comes from the fact that such bundles define semisimple cohomological field theories. I'll present motivation for why one may be interested in such bundles and their classes, as well as a few examples.
Max Lieblich (University of Washington)
Title: Moduli spaces in computer vision
Abstract: I will discuss some moduli spaces that naturally arise in computer vision. While these spaces were traditionally studied using classical projective geometry, a modern functorial approach yields stronger results. I’ll talk a bit about the rich history of interactions between photogrammetry and projective geometry and how modern algebraic geometry helps us understand these connections in new a fruitful ways. I’ll also discuss a key open problem on these moduli spaces that has potentially important practical implications.
David Zureick-Brown (Emory University)
Title: Canonical rings of stacky curves
Abstract: We give a generalization to stacks of the classical (1920's) theorem of Petri - we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. This is joint work with John Voight.
Research Talks:
Weiyan Chen (Tsinghua University)
Title: Choosing points on plane cubic curves: rigidity and flexibility
Abstract: Every smooth plane cubic curve has 9 inflection points, 27 sextactic points, and 72 points of type nine. Motivated by these classical algebro-geometric constructions, we study the following topological question: Is it possible to continuously choose n distinct unordered points on each smooth plane cubic curve for a given natural number n? We show that the answer is no unless n is a multiple of 9. We also show that such choices are rigid when n/9 is small, and flexible when n/9 is large. Part of the talk is on work joint with Ishan Banerjee.
Changho Han (University of Georgia)
Title: Compact moduli of lattice polarized K3 surfaces with Z/3Z group actions
Abstract: Observe that any construction of "meaningful" compactification of moduli spaces of objects involve enlarging the class of objects in consideration. For example, Deligne and Mumford introduced the notion of stable curves in order to compactify the moduli of smooth curves of genus g, and Satake used the periods from Hodge theory to compactify the same moduli space. After a brief review of the elliptic curve case (how those notions are the same), I will generalize into looking at various compactifications of Kondo's moduli space of lattice polarized K3 surfaces (which are of degree 6) with non symplectic Z/3Z group action, then describe the birational relations between those different compactifications. The main advantage of this approach is that we obtain an explicit classification of degenerate K3 surfaces, which is used to find geometric meaning of points parametrized by Hodge-theoretic compactifications. This comes from a joint works with Anand Deopurkar and a joint work in progress with Valery Alexeev, Anand Deopurkar, and Philip Engel.
Andrew Kobin (UC Santa Cruz)
Title: Stacky Curves in characteristic p
Abstract: As stacks continue to become an essential part of a modern algebraic geometer’s toolbox, researchers look to their local structure as a guide to their nature. Over the complex numbers, this local structure is called a complex orbifold, or ‘orbit space of a manifold’ under a cyclic group action. In this talk, I will survey the classification of stacky curves in characteristic 0 and introduce a new construction, called an Artin–Schreier root stack, which allows for similar classification results in characteristic p.
Alicia Lamarche (University of Utah)
Title: Derived Categories and Arithmetic
Abstract: In this talk, we will explore the extent to which the derived category of coherent sheaves can be used to answer rationality questions about a variety defined over an arbitrary field. In particular, we will give some examples of using the machinery of the derived category in an arithmetic setting. We will also discuss recent results of joint work with Matthew Ballard, Alex Duncan, and Patrick McFaddin concerning the behavior of the derived category under twisting by a torsor.
Wanlin Li (MIT)
Title: The Section Conjecture at the Boundary of M_g
Abstract: Grothendieck's section conjecture predicts that rational points on a smooth projective curve of genus >=2 correspond to splittings of the arithmetic fundamental group sequence. We construct infinitely many curves in each genus where this sequence does not split and so these curves satisfy the section conjecture. The main inputs to our result are the construction of two cohomology classes which provide obstruction to splitting and the computation of the order of their Gysin images on various boundary components of M_g. This is joint work in progress with Daniel Litt, Nick Salter, and Padmavathi Srinivasan.
Ananth Shankar (University of Wisconsin-Madison)
Title: Finiteness results for Hecke orbits over local fields
Abstract: I will talk about a finiteness result for reductions of Hecke orbits of abelian varieties defined over finite extensions of Qp. This is joint work with Mark Kisin, Joshua Lam and Padmavathi Srinivasan.
Libby Taylor (Stanford University)
Title: Derived equivalences for stacky genus 1 curves
Abstract: We will study the question of when two stacky genus 1 curves over a field have equivalent derived categories. Stacky genus 1 curves occur as connected components of the Picard stack of a genus 1 curve. Some questions we will answer: when are two genus 1 curves derived equivalent? Can we make derived equivalences explicit? And how do we connect derived equivalences to moduli problems?
Lightning Talks
Daniel Bragg (UC Berkeley): Derived equivalences and Hodge numbers in positive characteristic
Maryam Khaqan (Emory University): Elliptic curves and Moonshine
Soohyun Park (University of Chicago): Cut and paste relations and cubic hypersurfaces
Jack Petok (Dartmouth College): Moduli spaces of low degree fourfolds of K3^[2]-type
Aleksander Shmakov (University of Georgia): Cohomology of Local Systems on Moduli of Abelian Varieties
Sebastian Bozlee (Tufts University): Contractions of universal curves through log geometry
Manami Roy (Fordham University): On local data of rational elliptic curves with non-trivial torsion
Caroline Matson (University of Colorado at Boulder): The moduli stack of formal groups
Juliette Bruce (MSRI): The top weight cohomology of A_g.
Charlotte Ure (University of Virginia): Clifford algebras and the moduli stack of binary cubic forms
Huy Dang (Shing-Tung Yau Center of Southeast University): A Galois theory for deformations of covers