Abstract:

This talk is an invitation to derived methods in algebraic geometry. We will use derived algebraic geometry to give a solution to the problem of descending "bases" (semiorthogonal decompositions) on the category of perfect complexes on algebraic varieties. This generalizes, and gives a uniform treatment of the work of Elagin, Shinder, Bernardara, Bergh-Schürer, Auel-Bernardara and Ballard-Duncan-McFaddin.

Based on joint work with Ben Antieau in ArXiv:1912.08970

Abstract:

Abstract:

Recently there has been tremendous progress on constructing (projective) moduli spaces of Fano varieties using K-stability. In this talk, we will show that the K-moduli space of cubic fourfolds coincide with their GIT moduli space. In particular, all smooth cubic fourfolds are K-stable as well as those with simple singularities. The key ingredients are local volume estimates in dimension 3 due to Liu-Xu, Ambro-Kawamata non-vanishing theorem for Fano 4-folds, and degeneration of K3 surfaces.

Abstract:

If a complex, integral, projective curve C has only planar singularities, then its Jacobian admits a natural compactification with interesting topology. Work of Oblomkov, Shende, and others suggests the existence of a variety, stratified by algebraic tori and defined solely in terms of the topology of C, that retracts transcendentally onto this compactified Jacobian. We expect this retraction to be a new type of nonabelian Hodge correspondence: in particular, at the level of cohomology, it should map a (halved) weight filtration onto a filtration defined via perverse sheaves. I will construct a candidate for the larger variety using the combinatorics of braids and flag varieties, related to but ultimately different from a construction of Shende-Treumann-Zaslow. I will present evidence that the entire story is the SL_n case of a recipe that works for any semisimple group G, and that in a precise sense, these retractions should respect the isomorphism between the unipotent locus of G and the nilpotent locus of Lie(G). The key ingredient is a map from elements of the loop Lie algebra to conjugacy classes in a generalized braid group. The latter are the "algebraic braids" of the title.

Slides: https://math.mit.edu/~mqt/math/research/trinh_harvard-mit_20.pdf

Abstract:

Moduli spaces of stable sheaves on general type surfaces are typically singular. However, they carry a virtual class which can be used to define intersection numbers such as virtual Euler and Segre numbers. The former are part of SU(r) Vafa-Witten invariants. I give an overview of these invariants highlighting recent developments, such as a mathematical definition of SU(r)/Z_r Vafa-Witten invariants in terms of twisted sheaves, a new formula for SU(5) Vafa-Witten invariants in terms of the Rogers-Ramanujan continued fraction, and a virtual Segre-Verlinde correspondence. Joint works with Goettsche, Goettsche-Laarakker, and Jiang.

Abstract:

Using recent advances in the Minimal Model Program for moduli spaces of sheaves on the projective plane, we compute the cohomology of the tensor product of general semistable bundles on the projective plane. More precisely, let V and W be two general stable bundles, and suppose the numerical invariants of W are sufficiently divisible. We fully compute the cohomology of the tensor product of V and W. In particular, we show that if W is exceptional, then the tensor product of V and W has at most one nonzero cohomology group determined by the slope and the Euler characteristic, generalizing foundational results of Drézet, Göttsche and Hirschowitz. We also characterize when the tensor product of V and W is globally generated. Crucially, our computation is canonical given the birational geometry of the moduli space, providing a roadmap for tackling analogous problems on other surfaces. This is joint work with Izzet Coskun and John Kopper.

Abstract:

Stringy Hodge numbers are certain generalizations, to the singular setting, of Hodge numbers. Unlike usual Hodge numbers, stringy Hodge numbers are not defined as dimensions of cohomology groups. Nonetheless, an open conjecture of Batyrev's predicts that stringy Hodge numbers are nonnegative. In the special case of varieties with only quotient singularities, Yasuda proved Batyrev's conjecture by showing that the stringy Hodge numbers are given by orbifold cohomology. For more general singularities, a similar cohomological interpretation remains elusive. I will discuss a conjectural framework, proven in the toric case, that relates stringy Hodge numbers to motivic integration for Artin stacks, and I will explain how this framework applies to the search for a cohomological interpretation for stringy Hodge numbers. This talk is based on joint work with Matthew Satriano.

Abstract:

The generic vanishing theorem of Green-Lazarsfeld says that for general elements in the Picard variety of a projective manifold, their cohomology groups vanish in all degrees. Moreover, the cohomological jumping locus, that is, the locus where generic vanishing fails, is a union of torsion translated abelian subvarieties. If one replaces the Picard variety by the character variety of rank 1 local systems, then one can study a similar phenomenon, which are works by Simpson and Budur-Wang topologically and Esnault-Kerz arithmetically. In this talk, I will focus on the same phenomenon but from algebraic perspectives by using D-modules. More precisely, I will discuss zero loci of Bernstein-Sato ideals and explain why the zero loci can be treated as the algebraic analogue of topological jumping loci by using relative D-modules. Then I will prove a conjecture of Budur that zero loci of Bernstein-Sato ideals are related to the topological jumping loci in the sense of Riemann-Hilbert Correspondence. This is based on joint work with Nero Budur, Robin van der Veer and Peng Zhou.

Abstract:

Determining the computational complexity of matrix multiplication has been one of the central open problems in theoretical computer science ever since in 1969 Strassen presented an algorithm for multiplication of n by n matrices requiring only O(n^2.81) arithmetic operations. I will briefly discuss this problem and its reduction to deciding on which secant variety to the Segre embedding of a product of three projective spaces the matrix multiplication tensor lies. I will explain a recent technique to rule out membership of a fixed tensor in such secant varieties, border apolarity. Border apolarity establishes the existence of certain multigraded ideals implied by membership in a particular secant variety. These ideals may be assumed to be fixed under a Borel subgroup of the group of symmetries of the tensor, and in the simplest case, can consequently be tractably shown not to exist. When ideals exist satisfying the easily checkable properties, one must decide if they are limits of ideals of distinct points on the Segre. This talk discusses joint work with JM Landsberg, Alicia Harper, and Amy Huang.

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Given a variation of Hodge structures $V$ on a smooth complex quasi-projective variety $S$, its Hodge locus is the set of points $s$ in $S$ where the Hodge structure $V_s$ admits exceptional Hodge tensors. A famous result of Cattani, Deligne and Kaplan shows that this Hodge locus is a countable union of irreducible algebraic subvarieties of $S$, called the special subvarieties of $(S, V)$. In this talk I will discuss the geometry of the Zariski closure of the union of the positive dimensional special subvarieties. This is joint work with Ania Otwinowska.

Abstract:

I will discuss recent work on computing the top weight cohomology of A_g for g up to 7. We use combinatorial methods coming from the relationship between the top weight cohomology of A_g and the homology of the link of the moduli space of tropical abelian varieties to carry out the computation. This is joint work with Madeline Brandt, Juliette Bruce, Melody Chan, Margarida Melo, and Corey Wolfe.

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Abstract:

Positroid varieties are subvarieties of the Grassmannian obtained by intersecting cyclic rotations of Schubert varieties. We show that the "top open positroid variety" has mixed Hodge polynomial given by the q,t-rational Catalan numbers (up to a simple factor). Unlike the Grassmannian, the cohomology of open positroid varieties is not pure.

The q,t-rational Catalan numbers satisfy remarkable symmetry and unimodality properties, and these arise from the Koszul duality phenomenon in the derived category of the flag variety, and from the curious Lefschetz phenomenon for cluster varieties. Our work is also related to knot homology and to the cohomology of compactified Jacobians.

This talk is based on joint work with Pavel Galashin.

Abstract:

Moduli spaces of curves admit finite covers by moduli spaces which parametrize curves together with so-called level structures. In my talk, I will discuss how the cohomology of these spaces at infinite level is related to a profinite property of the mapping class group. I will then explain why tools from p-adic geometry yield vanishing statements for these cohomologies in high degree.