Algebraic Geometry Seminar

Abstract: To a symplectic algebraic variety Y, Rozansky-Witten theory associates a 2-category; if Y = T*X is a cotangent bundle, this is a 2-category of coherent sheaves of categories on X. The 3d mirror symmetry program aims to relate this 2-category to a dual 2-category defined from symplectic geometry. We will describe some structural features of this program, focusing on some basic examples of interest in algebraic geometry and representation theory.

Abstract: In this talk, we study arithmetic properties of the pentagram map, a dynamical system on convex polygons in the real projective plane. The map sends a polygon to the shape formed by intersecting certain diagonals. This simple operation turns out to define a discrete integrable system, meaning roughly that it can be viewed as a translation map on a family of real tori. We will explain how the real, complex, rational, and finite field dynamics of the pentagram map are all related by the following generalization: the pentagram map’s first or second iterate is birational to a translation on a family of Jacobian varieties of algebraic curves.

Abstract: In the 1980s, Donaldson and Uhlenbeck-Yau established the fundamental result that on a compact Kahler manifold, an irreducible holomorphic vector bundle admits a Hermitian metric solving the Hermitian-Yang-Mills equation if and only if the vector bundle is Mumford-Takemoto stable. Motivated by the characterization of supersymmetric B-branes in string theory and mirror symmetry, Collins-Yau asked if the following are equivalent: (i) a line bundle admits a solution of the deformed Hermitian-Yang-Mills (dHYM) equation, (ii) it is stable with respect to certain Bridgeland stability conditions. In this talk, we will discuss a partial answer to this question for a set of line bundles on a Weierstrass elliptic K3 surface. This is joint work with Tristan Collins, Jason Lo, and Shing-Tung Yau. 

Abstract: I will discuss my ongoing effort to comprehend, from a geometric viewpoint, the motivic monodromy conjecture for a "generic" complex multivariate polynomial f, namely any polynomial f that is non-degenerate with respect to its Newton polyhedron. This conjecture, due to Igusa and Denef--Loeser, states that for every pole s of the motivic zeta function associated to f, exp(2πis) is a "monodromy eigenvalue" associated to f. On the other hand, the non-degeneracy condition on f ensures that the singularity theory of f is governed, up to a certain extent, by faces of the Newton polyhedron of f. The extent to which the former is governed by the latter is one key aspect of the conjecture, and will be the main focus of my talk.

Abstract: Linear subvarieties are a special class of subvarieties of the moduli space of curves that arise in algebraic geometry and Teichmuller dynamics and are defined by linear relations among periods of differentials forms. One particular example of a linear sub variety are Hurwitz spaces of degree d covers of the projective line. Using recent advances on compactifcations of moduli spaces of differential forms by Bainbridge-Chen-Gendron-Grushevsky-Möller we explain how one can construct new birational models of Hurwitz spaces and time permitting we outline an approach to computing the cohomology class of Hurwitz spaces via intersection theory on the moduli space of multi-scale differentials. 

Abstract: The $\psi$-class intersection theory of the Deligne-Mumford-Knudsen compactification of the moduli space of curves is well known and most famously described via Witten's Conjecture, which gives the generating function of $\psi$-class intersection numbers as a $\tau$-function for the KdV hierarchy. In this talk, we relate the $\psi$-class intersection theory of a broad family of alternative compactifications of $\mathcal{M}_{g,n}$ (simplicially stable compactifications) to that of the usual compactification by stable curves via a (non-invertible) change of variables between the generating functions.  Our results also extend beyond intersection numbers to the level of cycles. This is joint work with Sebastian Bozlee.

Abstract: We construct a Legendrian link in R^3 from a “grid” plabic graph on R^2. We study a moduli space problem associated with the Legendrian link and construct a natural cluster structure on this moduli space using Legendrian weaves. The cluster seeds in the cluster structure can be interpreted as an algebraic invariant for exact Lagrangian fillings of the Legendrian link. We also construct Donaldson-Thomas transformations for these moduli spaces, and apply it to construct infinitely many fillings for some particular Legendrian links. In this talk, I will introduce the theoretical background and describe the basic combinatorics for constructing Legendrian weaves and the cluster structures from a grid plabic graph. This is based on a joint work with Roger Casals (arXiv:2204.13244).

Abstract: We study, using the extended isomonodromy deformation, the WKB approximation of Stokes matrices of a class of meromorphic linear ODE systems on the projective line of Poincare rank 1. This class of ODE systems appears in various contexts of geometry. We show that, via the degenerate Riemann-Hilbert map, the WKB approximation of Stokes matrices recovers the Gelfand-Tsetlin integrable systems whose action variables match with periods on spectral curves. If time permits, we will also discuss the ramifications to cluster theory and spectral networks. The talk is based on joint work with Anton Alekseev, Andrew Neitzke, and Xiaomeng Xu.

Abstract: What combinatorial properties are hidden in a configuration of vectors?  By building a vector bundle from such data, we will discuss how one can study the combinatorics via algebro-geometric methods.  In turn, by doing so we will meet various questions that probe the boundary between algebraic geometry and combinatorics.

Abstract: Albanese varieties provide a standard tool in algebraic geometry for converting questions about varieties in general, to questions about Abelian varieties.  A result of Serre provides the existence of an Albanese variety for any  geometrically connected and geometrically reduced scheme of finite type over a field, and a result of Grothendieck--Conrad establishes that Albanese varieties are stable under base change of field provided the scheme is, in addition, proper.   A result of Raynaud shows that base change can fail for Albanese varieties without this properness hypothesis.   In this talk I will discuss some recent results showing that Albanese varieties of  geometrically connected and geometrically reduced schemes of finite type over a field are stable under separable field extensions.  We also show that the failure of base change in general is explained by the L/K-image for purely inseparable extensions L/K.  This is joint work with Jeff Achter and Charles Vial.