Noncommutative deformations and moduli
spacesMonday 19 November 2018 - Friday 23 November 2018
Lecture Hall(1F), Kavli IPMU

ProgramNoncommutative deformations and moduli spaces / Program

Saturday 21 September 2024

Noncommutative deformations and moduli spaces / Program

Noncommutative deformations and moduli spaces(Kavli IPMU, Nov. 19-23, 2018)
Schedule

Monday 19 November

9:30-9:50 Refreshment

9:50-10:00 Opening of the conference

10:00-11:00 Chris Brav - Nick Rozenblyum course 1

11:00-11:20 Coffee break

11:20-12:20 Yukinobu Toda

12:20-14:00 Lunch time

14:00-15:00 Mauro Porta

15:00-15:30 Photo session and tea break

15:30-16:30 Valerio Melani - Pavel Safronov course 1

16:30-16:40 Break

16:40-17:40 Yanki Lekili

Tuesday 20 November

9:30-10:00 Refreshment

10:00-11:00 Chris Brav - Nick Rozenblyum course 2

11:00-11:20 Coffee break

11:20-12:20 Atsushi Takahashi

12:20-14:00 Lunch time
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Saturday 21 September 2024

Noncommutative deformations and moduli spaces / Program

14:00-15:00 Hao Wen

15:00-15:30 Tea break

15:30-16:30 Valerio Melani - Pavel Safronov course 2

16:30-16:40 break

16:40-17:40 Isamu Iwanari

Wednesday 21 November

9:30-10:00 Refreshment

10:00-10:30 Kirill Salmagambetov

10:30-11:00 Vladimir Gavran

11:00-11:15 Coffee break

11:15-11:45 Svetlana Makarova

11:45-13:15 Lunch time

13:15-14:15 Agnieszka Bodzenta (joint with MS seminar)

15:30- MS seminar: Alastair Craw

Thursday 22 November

9:30-10:00 Refreshment

10:00-11:00 Chris Brav - Nick Rozenblyum course 3

11:00-11:20 Coffee break

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Saturday 21 September 2024

Noncommutative deformations and moduli spaces / Program

11:20-12:20 Sheel Ganatra

12:20-14:00 Lunch time

14:00-15:00 Andrew Macpherson

15:00-15:30 Tea break

15:30-16:30 Valerio Melani - Pavel Safronov course 3

16:30-16:40 Break

16:40-17:40 Shinnosuke Okawa

Friday 23 November

9:30-10:00 Refreshment

10:00-11:00 Chris Brav - Nick Rozenblyum course 4

11:00-11:20 Coffee break

11:20-12:20 Ludmil Katzarkov

12:20-14:00 Lunch time

14:00-15:00 Marco Roballo

15:00-15:30 Tea break

15:30-16:30 Valerio Melani - Pavel Safronov course 4

16:30-16:40 Break

16:40-17:40 Michel Van den BerghPage 4

Saturday 21 September 2024

Noncommutative deformations and moduli spaces / Program

Saturday 21 September 2024

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 theequivalence of categories between deformation functors in thiscontext and differential graded Lie algebras. Examples include deformations of schemes, of objects
in differential graded categories, and of differential graded categories themselves, with the latterbeing described by the Gerstenhaber Lie bracket on shifted Hochschild chains.

Yukinobu Toda: Birational geometry for d-criticalloci and wall-crossing in Calabi-Yau 3-folds
In this talk, I will discuss birational geometry for Joyce’s d-critical loci, by introducing notions suchas ‘d-critical flips’, ‘d-critical flops’, etc.
I will show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau
3-folds are described in terms of d-critical birational geometry, e.g. certain wall-crossing diagramsof Pandharipande-Thomas stable pair moduli spaces form a d-critical minimal model program. I will
also show the existence of semi-orthogonal decompositions of the derived categories under simple
d-critical flips satisfying some conditions. This is motivated by a d-critical analogue of Bondal-Orlov,
Kawamata’s D/K equivalence conjecture, and also gives a categorification of wall-crossing formulaof Donaldson-Thomas 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 Riemann-Hilbertcorrespondence, by generalizing the result of my paper arXiv 1703.03907 to more generalcoefficients. 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 non-archimedean 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 onderived algebraic stacks, with particular emphasis on examples coming from the moduli of bundles
and Poisson-Lie groups. We will start by introducing derived symplectic geometry, giving the main
definitions of symplectic and Lagrangian structures, together with important existence results. Evenif 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 thederived setting. The solution is then given by formal localization, a highly non-trivial 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 thetheory leading to deformation quantization of derived moduli spaces.
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Saturday 21 September 2024

Yanki Lekili: Homological mirror symmetry forhigher dimensional pants
We prove that the partially wrapped Fukaya category of the complement of(n+2)-generic hyperplanes in CP^n (n-dimensional 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 n-dimensional 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 andnoncommutative 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 Calabi-Yautype. If there is time, we will discuss future issues on “noncommutative universal unfoldings”,
noncommutative deformations associated to Landau-Ginzburg orbifolds.

Hao Wen (YMSC, Tsinghua University):
Landau-Ginzburg model via $L^2$ Hodge theory
Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X with compact
critical locus, satisfying a general asymptotic condition. We construct a suitable subspace ofsmooth polyvector fields on X which carries a dGBV structure with a trace map and satisfies theHodge-to-de Rham degeneration property. This construction is based on the establishment of aversion of L^2 Hodge theory on X, which puts Landau-Ginzburg B-model of the pair (X,f) into thesame setting as compact Calabi-yau manifolds. It leads to a Frobenius manifold by theBarannikov-Kontsevich construction and can be viewed as a generalization of Kyoji Saito's higherresidue theory and primitive forms for isolated singularities. This is a joint work with Si Li.

Isamu Iwanari (Tohoku university): Calculus of
infinity-categories
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
Hochschildchain.
Thisalgebraic
structureisencodedbytheso-called
calculus/Kontsevich-Soibelman operad. I would like to describe a construction of an algebraic
structure on the pair of Hochschild invariants of a stable infinity-category.

Kirill Salmagambetov: Traces, Hochshild homology
and derived free loop space
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Saturday 21 September 2024

Traces in symmetric monoidal categories is a categorical formalism generalizing the familiar notionof trace from linear algebra. Many important constructions in derived and noncommutative algebraic
geometry involve taking traces in a suitable symmetric monoidal category. Examples include freeloop 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 ofautomorphisms 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 non-commutative
Calabi-Yau geometry
In my talk I will give the definition of Calabi-Yau structure on a smooth DG-category and describe
how this notion provides a noncommutative analog of Calabi-Yau varieties.

Svetlana Makarova: Combinatorial constructions ofderived 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 ongrassmannians will follow.

Agnieszka Bodzenta: Categorified
non-commutative deformations and abelianenvelopes
I will discuss categorified non-commutative deformations of a finite collection of objects in anabelian category. The deformation functor is a non-commutative Deligne-Mumford stack. To such astack we assign a monad and a comonad. I will prove that the Eilenberg-Moore categories of theseprovide 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 ofcorrespondences
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Saturday 21 September 2024

Cohomology is bivariant, which means that to a morphism f it associates not only a pullback mapf^*, but also (under certain conditions) an Umkehr map in the opposite direction. These maps
satisfy a "push-pull" 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", whosemorphisms 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 todefine 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 AS-regular I-algebras
Abstract: Noncommutative projective planes and noncommutative quadrics are defined as abeliancategories associated to the so-called 3-dimensional AS-regular quadratic (resp. cubic) Z-algebras.
Moreover there is a bijective correspondence between such algebras and certain geometric dataconsisting of a genus one curve and a collection of line bundles on it. I will talk on a work inprogress 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 CyclesIn 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 dg-category of matrix factorisations of an LG-pair, and theconstruction of motivic vanishing cycles of the pair.

Michel Van den Bergh : A k-linear 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 non-trivial example of such a category by gluing two pre-triangulated 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 andStrickland. These examples are however not linear over a field. This is joint work with Alice
Rizzardo.

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