Monday, 16 April

Tuesday, 17 April

Wednesday, 18 April

Thursday, 19 April

Friday, 20 April

Jochen Heinloth:

Stability and existence of good moduli spaces for algebraic stacks

Recently Alper, Hall and Rydh gave general criteria when a moduli problem can locally be described as a quotient and thereby clarified the local structure of algebraic stacks.

We report on a joint project with Jarod Alper and Daniel Halpern-Leistner in which we use these results to show general existence results for good coarse moduli spaces. In the talk we will focus on two aspects that illustrate how the geometry of algebraic stacks gives a new point of view on classical methods. Namely we explain a version of Hilbert-Mumford stability and then show how Langton's proof of semistable reduction for coherent sheaves on projective varieties can be reformulated in terms of geometry. This allows to prove a semistable reduction theorem purely in terms of the geometry of the moduli problem.

On the cone of effective cycles on the symmetric products of curves

I will report on a joint work with F. Bastianelli, A. Kouvidakis and A. F. Lopez in which we study the cone of (pseudo-)effective cycles on symmetric products of a curve. We first prove that the diagonal cycles span a face of the pseudo-effective cone of cycles in any given dimension. Secondly, we look at the contractibility faces associated to the Abel-Jacobi morphism towards the Jacobian and in many cases we are able to compute their dimension.

The integral Hodge conjecture for threefolds

The Hodge conjecture predicts which rational cohomology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is known to be false in general (the first counterexamples were given in dimension 7 by Atiyah and Hirzebruch). I'll survey some known results on this conjecture, and present some new counterexamples in dimension three. This is joint work with Olivier Benoist.

Double ramification cycles from Jacobians

Let M_g,n be the moduli space of smooth n-pointed curves of genus g. For a given vector of integers (d1,...,dn) one can define a natural cycle of pointed curves (C, p_i) such that O_C(Σ d_i p_i) admits a nonzero global section. We discuss how this cycle can be extended to the moduli space of stable curves by interpreting it as the pullback of a cycle on (compactified) universal Jacobians. Because there are multiple ways to compactify the Jacobian, this leads to multiple cycles related by wall-crossing. We explain why this gives an effective approach for the computation of the cohomology class of these cycles (in terms of tautological classes). A joint work with Jesse Kass.

Eloise Hamilton:

Moduli spaces for Higgs bundles, stable and unstable

The aim of this talk is to illustrate using the example of Higgs bundles how a combination of classical and Non Reductive Geometric Invariant Theory (GIT) can be used to solve classification problems in algebraic geometry. This method can also be applied to the classification of sheaves and of curves.

I will start by defining Higgs bundles and explaining the classification problem for Higgs bundles. After introducing the stack of Higgs bundles, I will explain how it can be described geometrically. As we will see, the stack of Higgs bundles can be decomposed into disjoint strata, each consisting of Higgs bundles of a given 'instability type'. I will explain how classical GIT can be used to obtain a moduli space for the substack of semistable Higgs bundles, and how non-reductive GIT can be applied to obtain moduli spaces for the remaining unstable strata.

Homotopy invariants of singularity categories

I'll give a brief, but hopefully gentle, introduction to A1-homotopy invariants (e.g. homotopy K-theory) of DG-categories. I'll then discuss a strategy for computing these invariants for singularity categories of certain noetherian algebras. This strategy leads us to a description of the A1-homotopy invariants of the singularity categories of a class of 0-dimensional Gorenstein singularities. If time permits I'll speculate on how to use these ideas to produce new, and concrete, examples of phantoms. All of this is based on joint work with Sira Gratz.

The generalized Franchetta conjecture for hyperKaehler varieties

The original Franchetta conjecture, established by Harer, predicts that the restriction of a line-bundle on the universal family of smooth projective curves of given genus g>1 to a fiber is a multiple of the canonical line-bundle. Recently, O'Grady proposed an analogue of that conjecture for codimension-2 cycles on the universal family of polarized K3 surfaces of given degree. In this talk I will propose a version of the Franchetta conjecture for hyperKaehler varieties (and their powers) and provide some evidence, most notably by focusing on the Fano variety of lines on a smooth cubic fourfold. This is joint work with Lie Fu, Robert Laterveer and Mingmin Shen.

Hilbert schemes of points and various equivalences

In this talk, we analyze what happens when we take two k3 surfaces with the same derived category and their Hilbert schemes of points. We will show how the birational geometry of IHS manifolds forces these schemes to be non birational in many cases. This is part of a joint work with C. Meachan and K. Yoshioka.

Berkovich approach to degenerations of hyper-Kähler varieties

To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. In this talk I will show how the techniques of Berkovich geometry give an insight in the study of the dual complexes. In this way, we are able to determine the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-Kähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.

Derived categories of singular toric surfaces

I will talk about semiorthogonal decompositions of singular toric surfaces. If the class group of the surface is torsion free, I will construct a semiortohogonal decomposition with components being derived categories of explicit finite dimensional algebras. Otherwise, I will construct a similar semiorthogonal decomposition for the twisted derived category of the surface. This is a joint work with Joseph Karmazyn and Evgeny Shinder.

A logarithmic version of the derived McKay correspondence

I will talk about a "limit" version of the derived McKay correspondence of Bridgeland, King and Reid in the context of logarithmic geometry. The two sides of the derived equivalence in this case are given by objects called the "infinite root stack" and the "valuativization”, that correspond to different ways of enhancing the central fiber of some simple kind of degenerations. Our result also implies the rather surprising fact that the derived category of parabolic sheaves on a variety with a normal crossings divisor is invariant under certain blowups. This is joint work with Sarah Scherotzke and Nicolò Sibilla.

Moduli of sheaves on (noncommutative) surfaces

The universal object associated to a fine moduli space of sheaves on a surface gives a Fourier-Mukai functor from the derived category of the surface to the derived category of the moduli space. On the other hand we can consider the Hochschild cohomology of both varieties, and their associated Hochschild-Kostant-Rosenberg decomposition. If the functor is fully faithful we can relate the Hochschild cohomologies. There is an interesting interplay between the (commutative) deformations of the moduli space and deformations of the category of coherent sheaves on the surface. I will explain this interplay, and how studying noncommutative surfaces can lead to a better understanding of deformations of moduli spaces of sheaves.

The crepant resolution conjecture for Donaldson-Thomas invariants

The crepant resolution conjecture for Donaldson-Thomas (DT) invariants is a conjecture in enumerative geometry originating from string theory. It relates the DT generating series of a certain type of three-dimensional Calabi-Yau orbifold to that of a crepant resolution of its coarse moduli space; this setting is a global version of the McKay correspondence.

We discuss joint work with John Calabrese and Jørgen Rennemo in which we interpret the conjecture as an equality of rational functions, and prove it using wall-crossing methods.

Categorifying non-commutative deformation theory

I will categorify non-commutative deformation theory by viewing underlying spaces of infinitesimal deformations of n objects as abelian categories with n simple objects. If the deformed collection is simple, I will prove the ind-representability of the deformation functor. For an arbitrary collection I will construct an ind-hull for the deformation functor and use it to present the deformation functor as a non-commutative Artin stack.

Stability data, irregular connections and tropical curves

I will outline the construction of isomonodromic families of irregular meromorphic connections on P^1 with values in the derivations of a class of infinite-dimensional Poisson algebras, and describe two of their scaling limits. In the "conformal limit" we recover a version of the connections introduced by Bridgeland and Toledano-Laredo, while in the "large complex structure limit" the connections relate to tropical curves in the plane and, through work of Gross, Pandharipande and Siebert, to tropical/GW invariants. This is joint work with M. Garcia-Fernandez and J. Stoppa.

Noncommutative moduli

We discuss noncommutative thickenings of families of sheaves.