Jarod Alper: Colloquium: Evolution of Moduli
In the rich landscape of algebraic varieties, moduli spaces stand out as some of the most enchanting varieties, capturing the imagination of algebraic geometers with their profound elegance and deep connections to other branches of mathematics. Moduli, the plural of modulus, is a term coined by Riemann to describe a space whose points afford an alternative description as certain classes of geometric objects. We will trace the origins of moduli spaces through the discoveries of Riemann, Hilbert, Grothendieck, Mumford, and Deligne, as a means to explain many of the fundamental concepts of moduli spaces. We will then survey how the foundations of moduli theory have further evolved over the last 50 years.
Five-minute talks from junior participants
Shikha Bhutani (Michigan State):
On Kawamata-Viehweg vanishing for surfaces of del Pezzo type over imperfect fields
Daigo Ito (Berkeley)
A derived category analogue of the Nakai-Moishezon criterion
Ronnie Cheng (Stanford)
On the tangent bundle and the divisor theory of a general matroid
Spencer Dembner (Stanford)
Sheaf cohomology in o-minimal geometry
Ruoxi Li (Berkeley)
Twisted Fourier-Mukai partners of abelian varieties and point objects in twisted derived categories
Haoming Ning (Washington)
Higher Du Bois and higher rational pairs
Michael R. Zeng (Washington)
The Grothendieck group of the variety of spanning line configurations and Fubini word combinatorics
Lily McBeath (Dartmouth)
On cubic fourfolds and elliptic K3 surfaces
Rachel Pries: Four perspectives on rational points on one Shimura curve
In this talk, I describe the points on a special Shimura curve from four perspectives: a family of cyclic covers of the projective line, a hyperbolic triangle, quadratic forms, and a unitary similitude group. By leveraging these four perspectives, we generalize a result of Elkies about supersingular reduction of elliptic curves to the case of genus four curves having an automorphism of order five. This is joint work with Wanlin Li, Elena Mantovan, and Yunqing Tang.
Dan Kaplan: Quiver varieties and symplectic resolutions
Symplectic resolutions arise in representation theory (the Springer resolution), geometry (the Hilbert-Chow morphism), and mathematical physics (3D mirror symmetry). Given a singularity, a main goal is to classify its symplectic resolutions. For instance, all such symplectic resolutions of certain Nakajima quiver varieties are given by variation of stability parameter [Bellamy-Craw-Schedler].
In joint work with Travis Schedler, we extend the classification beyond quiver varieties. Surprisingly, local resolutions of conical neighborhoods extend and glue uniquely to a global resolution, provided they are monodromy-free and chosen compatibly. Using this local-to-global approach, we classify symplectic resolutions in a wide range of examples, including symplectic torus quotients, multiplicative quiver varieties, and symmetric powers of singular surfaces.
Phil Engel: Matroids and the integral Hodge conjecture for abelian varieties
We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new combinatorial invariant of regular matroids, which obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. In concert with a result of Voisin, one deduces (via the intermediate Jacobian) the stable irrationality of a very general cubic threefold. This is joint work with Olivier de Gaay Fortman and Stefan Schreieder.
Sheel Ganatra: Homological and enumerative mirror symmetry for Batyrev mirror pairs
I will give an overview of recent joint work with Hanlon-Hicks-Pomerleano-Sheridan proving Kontsevich’s homological mirror symmetry conjecture for a large class of mirror pairs of (compact) Calabi–Yau hypersurfaces in toric varieties. These mirror pairs were first constructed by Batyrev from dual reflexive polytopes, and our result holds in characteristic zero and in all but finitely many positive characteristics. I will also say a few words about how our result implies a new proof of genus-0 enumerative mirror symmetry (following joint work with Perutz-Sheridan and Sheridan), and how it might conjecturally imply all-genus mirror symmetry, for such pairs
Kristin DeVleming: Weighted projective degenerations of projective space
A weighted projective space, P(a0, a1, …, an), is an analogue of projective space where the coordinates are allowed to have degrees larger than 1. These spaces arise naturally in many aspects of algebraic geometry, but notably appear in the study of moduli of del Pezzo surfaces. It is known by work of Manetti, Hacking, and Prokhorov that any mildly singular degeneration of P2 is a weighted projective surface P(a0, a1, a2) or a partial smoothing. In this talk, we will review this result for P2 and extend the study to higher dimensional projective space. We will find new examples of weighted projective degenerations of Pn and use them to address several questions on moduli of hypersurfaces and complete intersections. This is joint work with Jennifer Li and Sebastian Torres.
Joshua Mundinger: Hochschild homology of algebraic varieties in characteristic p
Hochschild homology is an invariant of an associative algebra. When the associative ring is commutative, the Hochschild-Kostant-Rosenberg theorem gives a formula for Hochschild homology in terms of differential forms. This formula extends to the Hochschild-Kostant-Rosenberg decomposition for complex algebraic varieties. In this talk, we quantitatively explain the failure of this decomposition in positive characteristic.
Xinwen Zhu: Moduli space of Langlands parameters
It is well-known that d-dimensional representations of a finitely generated group, such as a surface group, form an algebraic moduli space known as the character/representation stack (or its coarse moduli space, called the character/representation variety). However, there is generally no algebraic moduli space that parameterizes continuous representations of pro-finite groups. As a result, traditional studies focus on the deformation spaces of a given continuous representation. I will explain that, when the pro-finite groups in question are the Galois groups of local and global fields, there indeed exist well-behaved algebraic moduli spaces that parameterize their continuous representations. Specifically, I will discuss, in the context of global function fields, how de Jong’s conjecture on the etale fundamental groups of algebraic curves (as proved by Gaitsgory) exactly verifies one important part of Artin’s axioms for representability of functors.