Pieter Belmans: Deformations of Hilbert schemes of points through derived categories

Hilbert schemes of points on surfaces, and Hilbert squares of higher-dimensional varieties, are important and basic constructions of moduli spaces of sheaves. As such they provide a class of interesting yet tractable varieties. In a joint work with Lie Fu and Theo Raedschelders, we explain how one can (re)prove results about their deformation theory by studying their derived categories, via fully faithful functors and Hochschild cohomology, which describes both classical and noncommutative deformations.

Agnieszka Bodzenta: Categorifying noncommutative deformations

I will categorify non-commutative deformation theory by viewing underlying spaces of infinitesimal deformations on n objects as abelian categories with n simple object. When the deformed collection in simple, I will prove ind-representability of the deformation functor. For an arbitrary collection I will describe the ind-hull of the deformation functor and I will present it as a non-commutative Artin stack. This is joint work with A. Bondal.

Matt Booth: The derived contraction algebra

Given a simple threefold flopping contraction, one can associate to it a finite-dimensional noncommutative algebra, the contraction algebra, which controls the noncommutative deformation theory of the flopping curves. If the threefold was smooth, the contraction algebra is conjectured to determine the complete local geometry of the base. I'll talk about a new invariant, the derived contraction algebra (which has an interpretation in terms of derived deformation theory), and explain (via singularity categories) why a derived version of the above conjecture holds. Time permitting, I'll talk about the flop-flop autoequivalence and indicate some aspects of the theory for surfaces.

Tom Bridgeland: Geometry from Donaldson-Thomas invariants

It is well known that the genus 0 Gromov-Witten invariants of a smooth projective variety can be encoded in a Frobenius structure on its total cohomology. I will try to explain what should be the analogous story for Donaldson-Thomas invariants. From our point-of-view, the key property of these invariants is the Kontsevich-Soibelman wall-crossing formula, which describes how they vary under changes of stability parameters. The geometric structure we have in mind lives on the space of stability conditions, and involves a pencil of non-linear connections on the tangent bundle.

Francesca Carocci: Towards a modular desingularisation of the moduli space of genus 2 stable maps

In this brief talk I will try to explain the Ranganathan—Santos-Parker—Wise modular interpretation of the Vakil-Zinger desingularization of genus 1 maps and how, inspired from their approach, we started thinking about the genus 2 case. I will describe the components of the moduli space in the genus 2 case, discuss smoothability and speculate on how we expect the modular desingularization will look like.

Ana-Maria Castravet: Exceptional collections on moduli spaces of stable rational curves

A question of Orlov is whether the derived category of the Grothendieck-Knudsen moduli space of stable, rational curves with n markings admits a full, strong, exceptional collection that is invariant under the action of the symmetric group permuting the n markings. I will present an approach towards answering this question. This is joint work with Jenia Tevelev.

Daniele Faenzi: Coble cubics and moduli of Fano threefolds of genus 10

The period map sends the moduli space of Fano threefolds of genus 10 to the moduli space of curves of genus 2. I will show that the fibre of the period map at a curve Γ is the smooth locus of the Coble- Dolgachev sextic associated with Γ.

Daniel Huybrechts: Lagrangian fibrations of hyperkähler fourfolds

The base surface B of a Lagrangian fibration X→B of a projective, irreducible symplectic fourfold X is shown to be isomorphic to P^2. This is joint work with Chenyang Xu.

Dmitry Kaledin: Spectral algebras and Hodge-to-de Rham degeneration

Non-commutative, or categorical, algebraic geometry might be an emerging field, but so far, it has very few really general results. One of these is Hodge-to-de Rham Degeneration Theorem in characteristic 0. Several proofs are known (a couple by me, and a recent one by A. Mathew), they all work by reduction to positive characteristic, but more than that, they also use algebraic topology. I will present a slightly streamlined and simplified version of my proof, and try to explain why topology is indeed necessary and helpful. If time permits, I will also discuss possible generalizations (e.g. to Z/2-graded setting).

Manfred Lehn: Quintic rational curves on cubic fourfolds

TBA

Yucheng Liu: Stability conditions on products of projective varieties

We will review the Bridgeland's defnition of stability conditions and Abramovich and Polishchuk's construction of global heart. Then we will construct a natural polynomial stability function associated with the global heart constructed by Abramovich and Polishchuk. Then we use the HN structures of this polynomial to refine the positivity lemma established by Macrì and Bayer. These results will allow us to construct a rational stability condition on D^b(coh(X x C)) from a rational stability condition on D^b(cohX), where C is a smooth projective curve.

Valery Lunts: Triangulated subcategories of D^b(cohX) for an affine Noetherian scheme X

I will try to survey some known facts about subcategories of affine schemes and also discuss some recent new results obtained jointly with Alexey Elagin.

Emanuele Macrì: Bridgeland stability and the genus of space curves

I will give an introduction to various notions of stability in the bounded derived category of coherent sheaves on the three-dimensional projective space. As application I will show how to possibly use these techniques towards the study of space curves. This is joint work with Benjamin Schmidt.

Alex Perry: Stability conditions and cubic fourfolds

Kuznetsov showed the derived category of a cubic fourfold contains a special subcategory which can be thought of as a "noncommutative K3 surface". The goal of this talk is to describe the structure of moduli spaces of Bridgeland stable objects on this noncommutative K3 surface, together with several applications. This is joint work with Bayer, Lahoz, Macrì, Nuer, and Stellari.

Laura Pertusi: Elliptic quintics on cubic fourfolds and O’Grady spaces

The aim of this talk is to explain the modular interpretation of some hyperkähler manifolds, classically associated to a cubic fourfold. In particular, we show that elliptic quintic curves in a generic cubic fourfold give rise to a moduli space of semistable objects in the Kuznetsov component, whose resolution is a hyperkähler tenfold, equivalent by deformation to O’Grady’s example. This is a joint work in progress with Chunyi Li and Xiaolei Zhao.

Alexander Polishchuk: Pairs of 1-spherical objects and noncommutative orders over curves

The simplest example of a pair of 1-spherical objects in a triangulated category is a pair of line bundles on an elliptic curve. One would expect that all such pairs from degenerations of this example. However, it turns out that as soon as the difference of degrees of line bundles is bigger that 1, this is not the case: instead one has to consider noncommutative orders over singular curves. The goal of the talk is to explain this correspondence. If time permits I will discuss the connection with the Yang-Baxter equation and Fukaya categories of open surfaces.

Jørgen Vold Rennemo: Homological projective duality for symmetric rank loci

We consider the (projective) variety S_{k,n} of symmetric (n x n)-matrices of rank at most k. This is singular, but by work of Špenko-Van den Bergh it admits a (possibly Brauer twisted) non-commutative crepant resolution S'_{k,n}. Following a physical duality proposed by Hori and a conjecture by Kuznetsov, we work out the homological projective duals of these resolved spaces. Modulo taking double covers and Brauer twists according to a prescription based on the parities of k and n, we find that S'_{k,n} is HP dual to S'_{n-k+1,n}. This is joint work with Ed Segal.

Alice Rizzardo: Pre-triangulated A_n-categories give triangulated categories

I will show that the homotopy category of a pre-triangulated A_n-category is automatically triangulated. This is a much weaker notion than the data of a DG enhancement, and it is a crucial ingredient in constructing a triangulated category over a field that does not admit an enhancement. This is joint work with Michel Van den Bergh.

Pavel Safronov: Noncommutative Poisson geometry and the Kashiwara-Vergne problem

In this talk I will explain an interpretation of the Kashiwara-Vergne problem (a property of the Baker-Campbell-Hausdorff series) in terms of noncommutative geometry. Namely, one can reformulate it as a formality statement for the Calabi-Yau algebra of cochains on a Riemann surface equipped with a trivialization of the Euler class. In the talk I will also describe noncommutative versions of familiar concepts such as shifted Poisson structures and their unimodular versions. This is a report on work in progress with Florian Naef.

Špela Špenko: Comparing commutative and noncommutative resolutions of singularities

Quotient varieties for reductive groups admit the canonical Kirwan (partial) resolution of singularities, and quite often also a noncommutative resolution. The two can be compared via derived categories (in terms of the Bondal-Orlov conjecture). This is a joint work with Michel Van den Bergh.

Yukinobu Toda: On categorical Donaldson-Thomas theory for local surfaces

I will introduce the notion of categorical Donaldson-Thomas theories for moduli spaces of stable sheaves on the total space of a canonical line bundle on a smooth projective surface. They are defined to be the Verdier quotients of derived categories of coherent sheaves on derived moduli stacks of coherent sheaves on the surface, by the subcategory of objects whose singular supports are contained in the unstable locus. Our construction is regarded as a certain gluing of matrix factorizations via the linear Koszul duality. I will also prove that the moduli stack of D0-D2-D6 bound states on the local surface is isomorphic to the dual obstruction cone over the moduli stack of framed sheaves on the surface. This result is used to define the categorical Pandharipande-Thomas theory on the local surface. Several conjectural wall-crossing formulas of categorical PT theory are proposed, motivated by d-critical analogue of D/K conjecture in birational geometry. Among them, I prove the wall-crossing formula of categorical PT theory with irreducible curve classes.

Michel Van den Bergh: Hochschild cohomology of semi-co-categories

We introduce a new tool to establish and compare higher structure on Hochschild-type complexes, inspired by Keller’s arrow category which he famously used to obtain derived invariance for dg categories. As an application, we compare the Hochschild complex of an A algebra with the co–Hochschild complex of its Bar resolution considered as a coalgebra in the category of A bimodules. This is joint work with Wendy Lowen.

Michael Wemyss: Stability and Monodromy for 3-fold flops

I will first explain how to construct a hyperplane arrangement from the data of an E_6 Dynkin diagram and choice of central vertex. The construction is a bit strange (!), but folding it naturally constructs a sphere, minus six points. The point is that this turns out to be the combinatorics of flops: I will explain that the stability manifold is a regular covering of the hyperplane arrangement, and the Stringy Kahler Moduli Space (SKMS) is the punctured sphere obtained via the folding. I will then give a geometric description of monodromy on the SKMS, and in the process construct new autoequivalences. If there is time, I'll also describe some of the applications, mainly to deformation theory and curve-counting. The first part is joint with Yuki Hirano, the second part with Will Donovan.

Xiaolei Zhao: A refined Derived Torelli Theorem for Enriques surfaces

We prove a refined Derived Torelli Theorem for Enriques surfaces, stating that two Enriques surfaces are isomorphic if and only if their Kuznetsov components are Fourier-Mukai equivalent. As an application of our techniques, we study a conjecture by Ingalls and Kuznetsov about the derived categories of Artin-Mumford quartic double solids. This is a joint work in progress with Bernardara, Li, Nuer and Stellari.