Noncommutative deformations and moduli spaces
(Kavli IPMU, Nov. 1923, 2018)
Schedule
Monday 19 November
9:309:50 Refreshment
9:5010:00 Opening of the conference
10:0011:00 Chris Brav  Nick Rozenblyum course 1
11:0011:20 Coffee break
11:2012:20 Yukinobu Toda
12:2014:00 Lunch time
14:0015:00 Mauro Porta
15:0015:30 Photo session and tea break
15:3016:30 Valerio Melani  Pavel Safronov course 1
16:3016:40 Break
16:4017:40 Yanki Lekili
Tuesday 20 November
9:3010:00 Refreshment
10:0011:00 Chris Brav  Nick Rozenblyum course 2
11:0011:20 Coffee break
11:2012:20 Atsushi Takahashi
12:2014:00 Lunch time
14:0015:00 Hao Wen
15:0015:30 Tea break
15:3016:30 Valerio Melani  Pavel Safronov course 2
16:3016:40 break
16:4017:40 Isamu Iwanari
Wednesday 21 November
9:3010:00 Refreshment
10:0010:30 Kirill Salmagambetov
10:3011:00 Vladimir Gavran
11:0011:15 Coffee break
11:1511:45 Svetlana Makarova
11:4513:15 Lunch time
13:1514:15 Agnieszka Bodzenta (joint with MS seminar)
15:30 MS seminar: Alastair Craw
Thursday 22 November
9:3010:00 Refreshment
10:0011:00 Chris Brav  Nick Rozenblyum course 3
11:0011:20 Coffee break
11:2012:20 Sheel Ganatra
12:2014:00 Lunch time
14:0015:00 Andrew Macpherson
15:0015:30 Tea break
15:3016:30 Valerio Melani  Pavel Safronov course 3
16:3016:40 Break
16:4017:40 Shinnosuke Okawa
Friday 23 November
9:3010:00 Refreshment
10:0011:00 Chris Brav  Nick Rozenblyum course 4
11:0011:20 Coffee break
11:2012:20 Ludmil Katzarkov
12:2014:00 Lunch time
14:0015:00 Marco Roballo
15:0015:30 Tea break
15:3016:30 Valerio Melani  Pavel Safronov course 4
16:3016:40 Break
16:4017:40 Michel Van den Bergh

Joint course by Christopher Brav and Nick Rozenblyum: Derived deformation theory
We introduce the basics of derived algebraic geometry in characteristic zero and then formulate the equivalence of categories between deformation functors in this
context and differential graded Lie algebras. Examples include deformations of schemes, of objects in differential graded categories, and of differential graded categories themselves, with the latter being described by the Gerstenhaber Lie bracket on shifted Hochschild chains. 
Yukinobu Toda: Birational geometry for dcritical loci and wallcrossing in CalabiYau 3folds
In this talk, I will discuss birational geometry for Joyce’s dcritical loci, by introducing notions such as ‘dcritical flips’, ‘dcritical flops’, etc.
I will show that several wallcrossing phenomena of moduli spaces of stable objects on CalabiYau 3folds are described in terms of dcritical birational geometry, e.g. certain wallcrossing diagrams of PandharipandeThomas stable pair moduli spaces form a dcritical minimal model program. I will also show the existence of semiorthogonal decompositions of the derived categories under simple dcritical flips satisfying some conditions. This is motivated by a dcritical analogue of BondalOrlov, Kawamata’s D/K equivalence conjecture, and also gives a categorification of wallcrossing formula of DonaldsonThomas invariants.

Mauro Porta: Tannaka duality in analytic geometry
In this talk I will survey recent work on the Tannaka duality in the analytic setting. More precisely I will describe a Tannakian reconstruction criterion for maps $X \to Y^{an}$, where X is a (derived) analytic space. I will next describe an application of this result to the RiemannHilbert correspondence, by generalizing the result of my paper arXiv 1703.03907 to more general coefficients. Finally, if time permits, I will sketch how combining these results with the punctured formal neighborhood technique we might hope to get an unconditional Tannakian reconstruction theorem in nonarchimedean geometry. Part of this work is in collaboration with J. Holstein.

Joint course by Valerio Melani and Pavel Safronov : Introduction to shifted symplectic and Poisson geometry
The purpose of these lectures is to describe the theory of symplectic and Poisson structures on derived algebraic stacks, with particular emphasis on examples coming from the moduli of bundles and PoissonLie groups. We will start by introducing derived symplectic geometry, giving the main definitions of symplectic and Lagrangian structures, together with important existence results. Even if building a theory of Poisson structures is classically not harder than building a theory of symplectic structures, the situation in derived algebraic geometry is completely different. We will explain the intrinsic problems preventing the naive definition of Poisson structure to work in the derived setting. The solution is then given by formal localization, a highly nontrivial procedure which we will try to explain during the lectures. We will then mention comparison results between derived symplectic and Poisson geometry. If time permits, we will explain an application of the theory leading to deformation quantization of derived moduli spaces.

Yanki Lekili: Homological mirror symmetry for higher dimensional pants
We prove that the partially wrapped Fukaya category of the complement of
(n+2)generic hyperplanes in CP^n (ndimensional pants) with respect to certain stops
is equivalent to a certain categorical resolution of the derived category of
the singular affine variety x_1x_2..x_{n+1}=0. By localizing, we deduce that the (fully) wrapped Fukaya category of ndimensional pants is equivalent to the derived category of
x_1x_2...x_{n+1}=0. This is joint work with A. Polishchuk. 
Atsushi Takahashi : Primitive forms and noncommutative deformation
An overview of Kyoji Saito’s theory of primitive forms is given.
In particular, we will see there the prototype of noncommutative Hodge structure of CalabiYau type. If there is time, we will discuss future issues on “noncommutative universal unfoldings”, noncommutative deformations associated to LandauGinzburg orbifolds. 
Hao Wen (YMSC, Tsinghua University): LandauGinzburg model via $L^2$ Hodge theory
Let X be a noncompact CalabiYau manifold and f be a holomorphic function on X with compact critical locus, satisfying a general asymptotic condition. We construct a suitable subspace of smooth polyvector fields on X which carries a dGBV structure with a trace map and satisfies the Hodgetode Rham degeneration property. This construction is based on the establishment of a version of L^2 Hodge theory on X, which puts LandauGinzburg Bmodel of the pair (X,f) into the same setting as compact Calabiyau manifolds. It leads to a Frobenius manifold by the BarannikovKontsevich construction and can be viewed as a generalization of Kyoji Saito's higher residue theory and primitive forms for isolated singularities. This is a joint work with Si Li.

Isamu Iwanari (Tohoku university): Calculus of infinitycategories
For a smooth manifold or an algebraic variety, its multivector fields and differential forms admit several algebraic operations which have been fundamental in geometry. As for a noncommutative associative algebra, an analogous structure appears on the pair of its Hochschild cochain and Hochschild chain. This algebraic structure is encoded by the socalled calculus/KontsevichSoibelman operad. I would like to describe a construction of an algebraic structure on the pair of Hochschild invariants of a stable infinitycategory.

Kirill Salmagambetov: Traces, Hochshild homology and derived free loop space
Traces in symmetric monoidal categories is a categorical formalism generalizing the familiar notion of trace from linear algebra. Many important constructions in derived and noncommutative algebraic geometry involve taking traces in a suitable symmetric monoidal category. Examples include free loop space of stack and Hochshild homology of large dg categories. I will give a friendly introduction to traces in symmetric monoidal categories, their functoriality and explain why traces of automorphisms come equiped with a functorial action of the circle group. The resulting formalism turns out to be useful for different comparison theorems. As an example, we will consider Dg category of quasicoherent sheaves on a stack X and see how the canonical S^1  action on it’s Hochshild homology can be interpreted in terms of derived geometry of the free loop space of X.

Vladimir Gavran: Introduction to noncommutative CalabiYau geometry
In my talk I will give the definition of CalabiYau structure on a smooth DGcategory and describe how this notion provides a noncommutative analog of CalabiYau varieties.

Svetlana Makarova: Combinatorial constructions of derived autoequivalences of grassmannians
In this talk, I will review a recent work on “magic windows” by several groups of people and consider an explicit example. A new (but more complicated) proof of fullness of Kapranov’s collection on grassmannians will follow.

Agnieszka Bodzenta: Categorified noncommutative deformations and abelian envelopes
I will discuss categorified noncommutative deformations of a finite collection of objects in an abelian category. The deformation functor is a noncommutative DeligneMumford stack. To such a stack we assign a monad and a comonad. I will prove that the EilenbergMoore categories of these provide abelian envelopes for the exact subcategory of the original abelian category generated by the deformed objects.

Sheel Ganatra: Structural results for wrapped Fukaya categories
I will describe some new structural results for wrapped and partially wrapped Fukaya categories of Stein manifolds, emphasizing relationships to structures in noncommutative geometry, mirror symmetry, and sheaf theory. This is joint work with J. Pardon and V. Shende.

Andrew Macpherson: A Yoneda philosophy of correspondences
Cohomology is bivariant, which means that to a morphism f it associates not only a pullback map f^*, but also (under certain conditions) an Umkehr map in the opposite direction. These maps satisfy a "pushpull" or "base change" identity. Everyone knows that this implies that cohomology can be thought of as a functor out of a certain category CORR of "correspondences", whose morphisms are "rooves" and whose composition law is defined by taking a fibre product of kernels. This fact is a crucial ingredient in the construction of cohomological 2D field theories.
In higher category theory, specifying objects by describing the morphism spaces and composition law explicitly  as we just did with correspondences  is rather inconvenient. Rather, it is better to define things via their universal properties. In this talk, I will give a universal interpretation for CORR in terms of "bivariant functors" into an (∞,2)category, which takes out the pain from constructing functors out of CORR.

Shinnosuke Okawa : Noncommutative del Pezzo surfaces as ASregular Ialgebras
Abstract: Noncommutative projective planes and noncommutative quadrics are defined as abelian categories associated to the socalled 3dimensional ASregular quadratic (resp. cubic) Zalgebras. Moreover there is a bijective correspondence between such algebras and certain geometric data consisting of a genus one curve and a collection of line bundles on it. I will talk on a work in progress with Tarig Abdelgadir and Kazushi Ueda which aims to generalize this story to obtain solid definition and classification of noncommutative del Pezzo surfaces of all other types as well.

Ludmil Katzarkov: Categorical Curve Complexes
In this talk, we will introduce a new categorical construction.
Applications will be considered. 
Marco Robalo: Matrix Factorisations and Motivic Vanishing Cycles
In this talk I will explain a result obtained in joint work with A. Blanc, B. Toen and G. Vezzosi. comparing the motive associated to the dgcategory of matrix factorisations of an LGpair, and the construction of motivic vanishing cycles of the pair.

Michel Van den Bergh : A klinear triangulated category without a model.
We give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG enhancement. To this end we develop the theory of pretriangulated A_n categories and we construct a nontrivial example of such a category by gluing two pretriangulated A_infty categories over an A_n functor which cannot be extended to an A_infty functor. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field. This is joint work with Alice Rizzardo.