Abstract. Complexified complement of hyperplane arrangements are one of the most classical examples of study in singularity theory and algebraic geometry, while its symplectic properties are largely under exploration. In this talk, I’ll present our recent progress toward understanding symplectic geometry of hyperplane arrangements, which is connected to torus-equivariant topology of toric hyperKahler varieties. This is based on the recent joint work with Sukjoo Lee, Yin Li and Cheuk Yu Mak and work in preparation with Sheel Ganatra, Wenyuan Li and Peng Zhou.

Abstract. Lorentzian polynomials provide a natural generalization of log-concave sequences and have had striking applications to deep conjectures in combinatorics, in work of June Huh and others. We will define a class of closely related polynomials, `covolume polynomials’, and explore situations in intersection theory in which they occur naturally, specifically, their appearance in the context of Segre classes of subschemes of products of projective spaces. We will also describe an application of these considerations to the combinatorics of convex polyhedral cones.

Abstract. The tautological ring of the moduli space of abelian varieties was introduced and computed by van der Geer in the 1990s. I will explain recent progress regarding the existence of non tautological classes on the moduli space of abelian 6-folds. This is based on joint work with Samir Canning and Rahul Pandharipande. 

Abstract. Wall-crossing in enumerative geometry allows us to vary the counting problem and compare the resulting generating series with the original one. It has emerged as a powerful tool for computations and in the study of properties of generating series. In this talk, I will introduce wall-crossing based on some interesting applications and present some new developments, which allow wall-crossing in a more general context using localization of virtual classes. This is based on joint work with Nick Kuhn and Henry Liu.

Abstract. A complex manifold X is said to be Brody hyperbolic if it admits no entire curves, which are non-constant holomorphic maps from the complex numbers. When X is a smooth complex projective variety, Demailly introduced an algebraic analogue of this property known as algebraic hyperbolicity. We propose a conjecture on the algebraic hyperbolicity of generic sections of adjoint bundles on X, motivated by Fujita’s freeness conjecture and recent results by Bangere and Lacini on syzygies of adjoint bundles. We present some old and new evidence supporting this conjecture, including when X is any smooth projective toric variety or Gorenstein toric threefold. This is based on joint work with Joaquín Moraga.

Abstract. The Remodeling Conjecture of Bouchard-Klemm-Mariño-Pasquetti obtains all-genus open-closed Gromov-Witten invariants of a toric Calabi-Yau 3-fold from Chekhov-Eynard-Orantin topological recursion on the mirror curve. In this talk, I will discuss an extension of this framework to descendant mirror symmetry, that is, the all-genus descendant Gromov-Witten invariants can be obtained from the Laplace transform of topological recursion invariants. The result is based on a correspondence between equivariant line bundles supported on toric subvarieties and relative homology cycles on the mirror curve, which provides an identification of integral structures. I will also discuss an application to Hosono’s conjecture if time permits. This talk is based on joint work in progress with Bohan Fang, Chiu-Chu Melissa Liu, and Zhengyu Zong.

Abstract. Period integrals are a fundamental concept in algebraic geometry and number theory. In this talk, we will study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman.  We will give an overview of the non-archimedean SYZ program, which is a close analogue of the classical SYZ conjecture in mirror symmetry. Using the non-archimedean SYZ fibration, we will see how non-archimedean periods recover the complex analytic periods for log Calabi-Yau surfaces, verifying a conjecture of Kontsevich and Soibelman. This is joint work with Jonathan Lai.

Abstract. There are two major predictions for the quantum K-ring of a partial flag: One comes from quantum field theory, where the ring is the OPE ring associated to a 3D GLSM. The other comes from work identifying the quasimap version of the ring with the Bethe algebra associated to a quantum integrable system. These predictions have a mysterious set of equations in common, which can be regarded as a special case of Nekrasov-Shatashvili's gauge/Bethe correspondence. We give a purely geometric explanation for this coincidence in terms of the abelian/non-abelian correspondence and use it to prove both sets of predictions.

Abstract. In this talk, I will describe how to evaluate the virtual count of maps from a fixed-domain smooth curve to a hypersurface in a Grassmannian. We use the Quot scheme to compactify the space of maps and perform a virtual intersection-theoretic calculation. I will also discuss the conditions under which the virtual count is enumerative. This talk is based on joint work with Alina Marian.

Abstract. A theory of heights of rational points on stacks was recently introduced by Ellenberg, Satriano and Zureick-Brown as a tool to unify and generalize various results and conjectures about counting problems over global fields. In this talk I will present a moduli theoretic approach to heights on stacks over function fields inspired by twisted stable maps of Abramovich and Vistoli. For some well-behaved class of stacks, we obtain moduli spaces of points of fixed height whose geometry controls the number of rational points on the stack. I will outline an approach for more general stacks which is closely related to the geometry of the moduli space of vector bundles on a curve. This is based on joint work with Park and Satriano.

Abstract. How many isomorphism classes of genus g curves are there over a finite field F_q? In joint work with Samir Canning, Sam Payne, and Thomas Willwacher, we prove that the answer is a polynomial in q if and only if g is at most 8. One of the key ingredients is our recent progress on understanding low-degree odd cohomology of moduli spaces of stable curves with marked points.

Abstract. We discuss determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We also explore the ideals of relations among functions in the matrix entries that serve as building blocks for conformal correlators.

Abstract. An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. In this talk, we will restrict to the case of log Calabi-Yau pairs (X,B). We prove that if (X,B) admits a polarized endomorphism that preserves the boundary structure, then (X,B) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. This is joint work with Joaquin Moraga and Wern Yeong.

Abstract. The study of enumerative invariants dates back at least as far as Euclid’s work circa 300 BC, who observed that through two distinct points in the plane there is a unique line. In 1849, Cayley-Salmon found that there are 27 lines on a nonsingular cubic surface. In 1879, Schubert found that there are 2875 lines on a generic non-singular quintic threefold; Katz correctly counted 609250 conics in a generic nonsingular quintic threefold in 1986. In 1991, physicists Candelas-de la Ossa-Green-Parkes gave a generating function for genus 0 Gromov-Witten invariants of a generic non-singular quintic threefold by studying the mirror space. This observation represented a change in our approach to enumerative problems by counting rational degree d curves inside of the quintic threefold “all at once;” other landmark achievements in modern enumerative geometry include Kontsevich-Manin’s recursive formula for the number of rational plane curves. From the perspective of homological mirror symmetry, enumerative invariants come from the Hochschild cohomology of the Fukaya category. I’m interested in a different question, which asks what enumerative data can be gleaned from the bounded derived category of coherent sheaves. I’ll share results on giving presentations of derived categories, and if time allows, will describe Kalashnikov’s method to recover Givental’s small J-function and the genus 0 Gromov-Witten potential for CP^1 by viewing it as a toric quiver variety associated to the Kronecker quiver; i.e., from a presentation of the bounded derived category of coherent sheaves.