Abstract:

Moduli spaces of degree d dynamical systems on projective space are fundamental in algebraic dynamics. When the degree d is at least 2, these moduli spaces can be defined via geometric invariant theory (GIT), but when d = 1, there are no GIT stable linear maps. Inspired by the case of genus 0 curves, we show how to recover a nice moduli space by including marked points. Linear maps are the simplest dynamical systems, but with marked points, the moduli space becomes quite subtle. We construct the moduli space of linear maps with marked points, prove its rationality, and show that GIT stability is characterized by subtle dynamical conditions on the marked map related to Hessenberg varieties. The proof is a combinatorial analysis of polytopes generated by root vectors of the A_N lattice from Lie theory.

Abstract:

We prove that the categories of coherent (or, equivalently, of quasi-coherent) sheaves over two noetherian non-commutative schemes X and Y are equivalent if and only if there centers C(X) and C(Y) are isomorphic and there is a local progenerator in the category of coherent sheaves over X whose sheaf of endomorphisms is anti-isomorphic to the inverse image of the structure sheaf of Y under the isomorphism X->Y. To prove it, we combine the classical Morita theorem with the Gabriel’s theory of locally noetherian categories.

Abstract:

Arithmetic quotients of Type IV Hermitian symmetric domains have cohomology-valued modular forms whose coefficients are special cycles, by work of Borcherds. These can be interpreted as non-compact period spaces for K3-type Hodge structures. I will describe recent results (joint with P. Engel and S. Tayou) that give mock modular forms whose coefficients are compactified special cycles in a simplicial toroidal compactification. Next, I will discuss an application to the geometry of Severi curves associated to a rational elliptic surface.

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:

The minimal model program is an ambitious program that aims to understand the geometry of complex projective varieties (eg. manifolds defined by polynomial equations). In this talk we will discuss some recent results and challenges encountered trying to extend the minimal model program to the context of Kähler varieties.

We introduce two cohomological incarnations of the Non-Abelian Hodge Theorem for curves in characteristic p. Using the first one, we show that, in arbitrary characteristics, the cohomology of the moduli space of Higgs bundles is independent of the degree, as long as it is coprime to the rank. A closer analysis of the results above shows that, over the complex numbers, the Galois conjugation action on the character variety preserves the perverse Leray filtration on the cohomology of the moduli space of Higgs bundles. Everything above is based on joint works with Mark de Cataldo, Davesh Maulik, and Junliang Shen.

Abstract:

I will explain how to reinterpret invariant distributions on p-adic GL_n and its Lie algebra using symmetric functions. This reinterpretation, combined with some old results of Waldspurger, allows us to compute the Shalika germs of tamely ramified regular semisimple elements in GL_n and for example point-counts of compactified Jacobians on plane curve singularities. I will exposit the method and some interesting consequences such as how to t-deform many of the results using K-theory of Hilbert schemes of points on C^2. Time permitting, I will discuss the relation to triply graded knot homology. Based on joint work with Cheng-Chiang Tsai.

Abstract:

Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra. Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear. In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions. These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.

Abstract:

For any smooth affine log Calabi-Yau variety U, we construct the structure constants of the mirror algebra to U via counts of non-archimedean analytic disks in the skeleton of the Berkovich analytification of U. This generalizes our previous construction with extra toric assumptions. The technique is based on an analytic modification of the target space as well as the theory of skeletal curves. Consequently, we deduce the positivity and integrality of the mirror structure constants. If time permits, I will discuss further generalizations and virtual fundamental classes. Joint work with S. Keel.

Abstract:

I’ll discuss a number of interactions between algebraic geometry, low-dimensional topology, and arithmetic, arising from the study of local systems on moduli spaces of curves. I’ll explain how, in joint work with Aaron Landesman, we exploit these connections to resolve open questions of Prill, Esnault-Kerz, and others.

Abstract:

Given integers g, m, and n, the heavy/light moduli space Mbar_{g, m|n} is a compactification of the moduli space of smooth (m+n)-marked curves of genus g. These spaces are particular examples of Hassett’s moduli spaces of weighted stable curves. Their rational cohomology gives a rich family of representations of products of symmetric groups. I’ll discuss recent work on the structure of this family of representations, and how they relate to the S_n-representations determined by the cohomology of Deligne-Mumford compactifications. This talk is based on joint work with Stefano Serpente and Claudia Yun.

Abstract:

Abstract:

I will discuss recent work calculating the top weight cohomology of the moduli space A_g of principally polarized abelian varieties of dimension g for small values of g. The key idea is that this piece of cohomology is encoded combinatorially via the relationship between the boundary complex of a compactification of A_g and the moduli space of tropical abelian varieties. This is joint work with Madeline Brandt, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

Abstract:

A. Marian: I will discuss various aspects of the geometry of Quot schemes of torsion quotients on a smooth projective curve. I will describe in particular results on the cohomology of tautological vector bundles over the Quot scheme, which mirror a parallel picture in the geometry of the Hilbert scheme of points over a surface. The subject has an interesting combinatorial and computational flavor. The talk is partly based on joint work with D. Oprea and S. Sam.

T. Kinjo: Cohomological Donaldson-Thomas (CoDT) invariants were introduced by Kontsevich-Soibelman and Brav-Bussi-Dupont-Joyce-Szendroi as categorifications of the Donaldson-Thomas invariants counting objects in 3-Calabi-Yau categories. In this talk, I will explain applications of the CoDT theory to the cohomological study of the moduli of objects in 2-Calabi-Yau categories. Among other things, I will construct a coproduct on the Borel-Moore homology of the moduli stack of objects in these categories and establish a PBW-type statement for the Kapranov-Vasserot cohomological Hall algebras. This talk is based on a joint work in progress with Ben Davison.

Abstract:

The intersection theory of the moduli space of K3 surfaces polarized by a lattice is a subject of recent interest because of its deep connections with a wide variety of mathematics, including the intersection theory of moduli spaces of curves and the study of modular forms. Oprea and Pandharipande conjectured that the tautological rings of these moduli spaces of K3 surfaces are highly structured in a way that mirrors the picture for the moduli space of curves. I will discuss the proof of this conjecture in the case of K3 surfaces polarized by a degree 2 line bundle. This is joint work with Dragos Oprea and Rahul Pandharipande.