Category theory and Applications

Fridays, 15:30-17:00, room 215

The seminar is about categorical methods in algebraic and symplectic geometry, number theory, mathematical physics, mirror symmetry.

21 Sep 2018 at  15:30

Victor Przyjalkowski (HSE, MIRAN). Weighted complete intersections

We observe a classification and the main properties of one of the main class of examples of higher dimensional Fano varieties — smooth complete intersections in weighted projective spaces. We discuss their main properties and boundness results. We also discuss extremal examples from Hodge theory point of view and their relations with derived categories structures and their semiorthogonal decompositions. If time permits, we discuss mirror symmetry for the complete intersections and invariants of their Landau-Ginzburg models related to ones of the complete intersections.

14 Sep 2018 at  15:30

Ludmil Katzarkov (HSE). Categorical curve complexes

In this talk we introduce new categorical invariant. Classical and new examples will be considered.

30 Mar 2018 at 15:30 room 109
Okawa Shinnosuke (Osaka University, Japan).On the definition of noncommutative del Pezzo surfaces

Noncommutative projective planes and noncommutative  quadrics have clear definitions as the category qgr of 3-dimensional  Artin-Schelter regular quadratic (resp. cubic) Z-algebras. Other  noncommutative del Pezzo surfaces lack a definition of this sort, but  are constructed by blowing up noncommutative projective planes. In  this talk I will talk about an attempt toward a definition of noncommutative del Pezzo surfaces without using blowups. This is a  joint work in progress with Tarig Abdelgadir and Kazushi Ueda.

16 Mar 2018 at  15:30

L. Katzarkov (HSE). P=W conjecture and algebraic cycles

In this talk we will introduce a new point of view of P=W conjecture.
Connection with a classical questions in algebraic geometry will be discussed.

9 Feb 2018 at  15:30 , room 208

Darya Polyakova (NRU HSE). Formality of P-objects

19 January 2018 at  17:00, room 308

L. Katzarkov (NRU HSE). Some Brill Noether Categorical Invariants (continuation).

In this talk we will explain some A side categorical invariants. We will exploit the analogy with the Classical Brill Noether theory.

12 January 2018 at  15:30

L. Katzarkov (NRU HSE). Some Brill Noether Categorical Invariants.

In this talk we will explain some A side categorical invariants. We will exploit the analogy with the Classical Brill Noether theory.

26 December 2017 14:00 

Sergey Barannikov (Paris VIII). Batalin-Vilkovisky formalism and mirror symmetry for manifolds of general type.

The approach to studying functions with non-isolated singularities due to Batalin-Vilkovisky will be reviewed, with aim of applications to the mirror symmetry for manifolds of general type.

8 December 2017 at  14:45, room 110

Alexey Golota. Special Varieties and Kobayashi pseudometric

at 15:30, room 110

Chris Brav. Functions on moduli spaces from cyclic homology, part II

We discuss the 'moduli of objects' M_D in a dg category D and construct a map from cyclic homology of D to functions on the moduli space M_D. When D is a smooth, oriented dg category ('Calabi-Yau'), the cyclic homology HC(D) is endowed with a shifted Lie bracket ('algebraic string bracket') and the functions on M_D are endowed with a shifted Poisson bracket. We show that the map from cyclic homology to functions entwines the brackets. Examples include the Goldmann bracket of free loops on a surface, the string bracket of Chas-Sullivan, and the Hitchen system for Higgs bundles. This is joint work very much in progress with Nick Rozenblyum.

24 November 2017 at  15:30

A.Konovalov (NRU HSE). Topological K-theory of dg-categories

In the paper L. Katzarkov, M. Kontsevich, T. Pantev "Hodge theoretic aspects of mirror symmetry", the authors suggested existence of a functorial "noncommutative Hodge sctructure" on the periodic cyclic homology of a smooth proper dg-category. In particular, it was conjectured that periodic homology possess a functorial rational structure, analogous to the Betti one in the commutative case. Recently Anthony Blanc proposed a possible candidate, so-called topological K-theory of dg-categories. In the talk, I will remind the definition of this invariant, discuss some of its properties and tell about some cases when it can be shown that topological K-theory indeed provides us with a rational structure on HP.

17 November 2017 15:30 

Artan Sheshmani (Aarhus University, Harvard University). The theory of Nested Hilbert schemes on surfaces

We construct natural virtual fundamental classes for nested Hilbert schemes on a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as Poincare invariants of Duerr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In the case of the nested Hilbert scheme of points, we can express these invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. The virtual fundamental classes of the nested Hilbert schemes play a crucial role in the local Donaldson-Thomas theory of threefolds that I will talk about, in talk 2. This talk is based on arXiv:1701.08899.

10 November 2017 15:30 

Chris Brav (НИУ ВШЭ) Functions on moduli spaces from cyclic homology

We discuss the 'moduli of objects' MD in a dg category D and construct a map from cyclic homology of D to functions on the moduli space MD. When D is a smooth, oriented dg category ('Calabi-Yau'), the cyclic homology HC(D) is endowed with a shifted Lie bracket ('algebraic string bracket') and the functions on M_D are endowed with a shifted Poisson bracket. We show that the map from cyclic homology to functions entwines the brackets. Examples include the Goldmann bracket of free loops on a surface, the string bracket of Chas-Sullivan, and the Hitchen system for Higgs bundles. This is joint work very much in progress with Nick Rozenblyum.

Oct 27, 15:30 

Anton Fonarev (NRU HSE). Derived categories of curves as components of Fano varieties

We will show that the bounded derived category of a generic curve of genus g > 1 can be embedded as a semiorthogonal component into the bounded derived category of a smooth Fano variety. Namely, the moduli space of stable rank 2 vector bundles with fixed odd determinant. This is joint work with A. Kuznetsov.

Sep 29, 13:00 

Alessio Corti (Imperial College London). Computing ramification of Laurent polynomial

I will define Golyshev's ramification invariant of local systems.

I will then discuss methods of computation and, if time permits, some strategies to construct Laurent polynomials of small ramification

Sep 15, 15:30 

Alexander Efimov. Exotic t-structures on the product of two elliptic curves

In this talk I will recall the notion of a partially defined operation of a tensor product of t-structures on enhanced triangulated categories. In general, the existence of a product is very difficult to establish. Then I will sketch the proof of the following statement: such a tensor product exists in the case of two isogenous elliptic curves and t-structures associated with irrational slopes, which are related by a fractional-linear transform from PGL^-₂(Q).

Sep 8, 15:30 

D.Alekseeva. Presentations of symplectic mapping class group of rational 4-manifolds (blow-ups of CP²).

Известно, что в случае 4-мерного рационального многообразия X (l-кратного раздутия CP²) симплектическая группа классов отображений π₀(Symp(X, ω)) зависит только от класса когомологий [ω] симплектической формы. С другой стороны, для различных классов когомологий симплектическая группа классов отображений может существенно различаться.
Так же известно, что в случае lX, гладко изотопный тождественному, будет симплектически изотопен тождественному. Поэтому интересно найти случаи, когда симплектическая группа классов отображений π₀(Symp(X, ω)) "большая", и когда она допускает полное описание.
В своем докладе я опишу два специальных класса симплектических форм на 4-мерных рациональных многообразиях, которые называются D_l и E_l. Они характеризуются следующим свойством: существует такая конфигурация Лагранжевых сфер в (X, ω), для которых граф инцидентности есть граф Дынкина типа D_l или E_l соответственно. Случай E_l может быть охарактеризован как раздутие CP² в l<9 точках, а класс когомологий симплектической формы ω есть класс Черна c₁(X). Таким образом, случай E_l является симплектическим аналогом поверхностей дель Пеццо.
Для симплектических форм такого типа я опишу конструкцию, которая позволяет свести вычисление группы π₀(Symp(X, ω)) к вычислению фундаментальной группы дополнения определённого дивизора в пространстве модулей раздутий CP², таким образом получить естественное геометрическое копредставление симплектической группа классов отображений π₀(Symp(X, ω)). В наших случаях симплектическая группа классов отображений есть фактор-группа группы кос: Br(D_l) или Br(E_l) соответсвенно, а образующие суть симплектические скручивания Дена вдоль Лагранжевых сфер.
В случаях D_l и E₅ я дам описание полной системы соотношений, что и дает искомые копредставления.

June 30, 14:15 

L. Katzarkov. Central manifolds and filtrations

In this talk we will make a parallel between problems in classical geometry and some new categorical constructions

June 23, 14:15

A.Efimov, A.Fonarev, V.Golyshev. Wedge categories