Abstract:

The classical question of determining which varieties are rational has led to a huge amount of interest and activity. On the other hand, one can take on a complementary perspective: Given a smooth projective variety whose nonrationality is known, how far is it from being rational? I will explain recent progress in this direction for complete intersections in projective space.

Abstract:

Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call "tautological bundles (classes)" of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

Abstract:

Let Y be a Calabi-Yau 3-fold containing a Hirzebruch surface E and let B be the homology class of the fibers of the Hirzebruch surface. Physics suggests that curve counting on Y should satisfy some symmetry relating curves in classes β and β’=β+(E.β)B. In this talk I’ll explain how to make such a symmetry precise with a new rationality result for the Pandharipande-Thomas invariants of Y. Mathematically, the symmetry is explained by a certain involution of the derived category of Y given as a composition of spherical twists; our proof is an instance of the general principle that automorphisms of the derived category should constrain enumerative invariants. This is joint work with Tim Buelles and it is highly inspired in the proof of rationality for the PT generating series of an orbifold by Beentjes-Calabrese-Rennemo.

Abstract:

I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.

Abstract:

The Chow ring of a variety packages information about its subvarieties and how they intersect (a sort of "algebraic version" of the cohomology ring). When studying Chow rings of moduli spaces, there are often certain "tautological" classes that arise naturally from the moduli problem. While determining the full Chow ring of a moduli space may be difficult, the subring generated by tautological classes may have a nicer structure. This motivates two questions: (1) What is the structure of the tautological ring? (2) To what extent are all classes tautological? I'll start by giving some background about these questions for the moduli space of curves M_g, and explain how they also motivated us to study Hurwitz spaces H_{k,g}, parametrizing degree k, genus g covers of P^1. Then, I'll give some results addressing (1) and (2) for the Hurwitz spaces H_{k,g} for k \leq 5. This is joint work with Samir Canning.

Abstract:

It is known that pushforwards of pluricanonical bundles under morphisms to abelian varieties have some positivity properties, which include global generation after pullback by an isogeny, the existence of the Chen—Jiang decompositions and so on. It is natural to ask what happens if we allow singularities. In this talk, I will discuss these positivity properties for pushforwards of klt pairs under morphisms to abelian varieties, which extend the previous results to the singular setting. I will also give some applications based on these properties.

Abstract:

The double ramification cycle -- roughly speaking, the cycle of curves admitting a rational function with prescribed ramification profile -- is an algebraic cycle in the moduli space of curves, intimately connected to Gromov-Witten theory and classical Abel-Jacobi theory. The DR cycle has been extensively studied in recent years; one of the outcomes of this study is a remarkable formula in terms of simple classes in the tautological ring of \bar{M}_{g,n}. However, for certain more delicate questions involving the DR, such as computing its higher dimensional analogues or its behavior under intersection, one must study certain refinements of the DR, for which the existing methods do not give analogous formulas. In this talk I will discuss joint work with Holmes, Pandharipande, Pixton and Schmitt on how one can obtain such formulas by studying the DR via compactified Jacobians.

Abstract:

Let $M(\beta,\chi)$ be the moduli space of one-dimensional semi-stable sheaves on a del Pezzo surface $S$, supported on an ample curve class $\beta$ and with Euler-characteristic $\chi$. Working over a non-archimedean local field $F$, we define a natural measure on the $F$-points of $M(\beta,\chi)$. We prove that the integral of a certain gerbe on $M(\beta,\chi)$ with respect to this measure is independent of $\chi$ if $S$ is toric. A recent result of Maulik-Shen then implies that these integrals compute the Donaldson-Thomas invariants of $M(\beta,\chi)$. A similar result holds for suitably twisted Higgs bundles. This is joint work with Francesca Carocci and Giulio Orecchia.

Abstract:

The classical Torelli theorem states that the Torelli map, sending a curve to
its Jacobian, is an injection on points. However, the Torelli map is not injective
on tangent spaces at points corresponding to hyperelliptic curves. This leads to
the natural question: If one restricts the Torelli map to the locus of
hyperelliptic curves, is it then an immersion?

We will give a complete answer to this question, starting out by describing the
classical history.

Abstract:

It is well-known that each cohomology group of a compact K\"ahler manifold carries a Hodge structure. If we consider a degeneration of compact K\”ahler manifolds over a disk then it is natural to ask how the Hodge structures of smooth fibers degenerate. When the degeneration only allows a reduced singular fiber with simple normal crossings (i.e. semistable), Steenbrink constructed the limit of Hodge structure algebraically. A consequence of the existence of the limit of Hodge structure is the local invariant cycle theorem: the cohomology classes invariant under monodromy action come from the cohomology classes of the total space. In this talk, I will try to explain a method using D-modules to construct the limit of Hodge structure even when the degeneration is not semistable.

Abstract:

Abstract:

Stable log varieties or stable pairs (X,D) are the higher dimensional generalization of pointed stable curves. They form proper moduli spaces which compactify the moduli space of normal crossings, or more generally klt, pairs. These stable pairs compactifications depend on a choice of parameters, namely the coefficients of the boundary divisor D. In this talk, after introducing the theory of stable log varieties, I will explain the wall-crossing behavior that governs how these compactifications change as one varies the coefficients. I will also discuss some examples and applications. This is joint work with Kenny Ascher, Giovanni Inchiostro, and Zsolt Patakfalvi.

Abstract:

In a letter to Weil, on November 9, 1959, Serre sketched an elegant proof of a Kähler analog of Weil’s Riemann hypothesis. The positive-characteristic analog of Serre’s result is thus called (by us) generalized Weil's Riemann hypothesis and semisimplicity. In this talk, I shall discuss how a dynamical approach can be applied towards these questions. In particular, I will describe the relationships between the above questions, the standard conjectures, and our dynamical conjectures. In the end, some applications to abelian varieties and Kummer surfaces are given.