# Abstracts

**Mini-courses**

#### Alexander Kuznetsov (Steklov): *Homological Projective Duality*

Homological Projective Duality is a relation between a pair of (noncommutative) algebraic varieties (with some additional data) which on one hand generalizes classical projective duality, and on the other hand, captures homological properties of linear sections of the dual varieties. I will explain the general statement of HPD, discuss the proof, and show as many examples as possible.

#### Dmitri Kaledin (Steklov): *Homological methods in Noncommutative geometry*

I am going to give a brief introduction to the homological theory of DG algebras and DG categories, the basis of modern algebraic approach to non-commutative geometry. Among specific topics I want to cover are: generalities on homologically smooth and homologically proper DG algebras, finiteness conditions (this material is mostly due to B. Toen), Hochschild and cyclic homology for DG categories, Hochschild cohomology and deformations.

#### Alexander Efimov (Steklov): *TBA*

## Talks

#### Dmitri Kaledin (Steklov): *Hochschild-Witt complex*

The "de Rham-Witt complex" of Deligne and Illusie is a functorial complex of sheaves WΩ^*(X) on a smooth algebraic variety X over a finite field, computing the cristalline cohomology of X. I am going to present a non-commutative generalization of this: even for a non-commutative ring A, one can define a functorial "Hochschild-Witt complex" with homology WHH^*(A); if A is commutative, then WHH^i(A)=WΩ^i(X), X = Spec A (this is analogous to the isomorphism HH^i(A)=Ω^i(X) discovered by Hochschild, Kostant and Rosenberg). Moreover, the construction of the Hochschild-Witt complex is actually simpler than the Deligne-Illusie construction, and it allows to clarify the structure of the de Rham-Witt complex.

#### Alexander Kuznetsov (Steklov): *Categorical resolutions of singularities*

A categorical resolution of singularities of an algebraic variety Y is a triangulated category T with an adjoint pair of functors between D(Y) (the derived category of quasicoherent sheaves on Y) and T, such that the composition is the identity endofunctor of D(Y). If X→Y is a usual resolution, the derived category D(X) with pullback and pushforward functors is a categorical resolution only if Y has rational singularities. However, I will explain that even if Y has nonrational singularities, still one can construct a categorical resolution of D(Y) by gluing derived categories of appropriate smooth algebraic varieties. This is a work in progress, joint with Valery Lunts.

#### Artan Sheshmani (Max Planck): *Donaldson-Thomas invariants of torsion 2 dimensional sheaves and modular forms*

We study the Donaldson-Thomas invariants of the 2-dimensional stable sheaves in a smooth projective threefold. The DT invariants are deﬁned via integrating over the virtual fundamental class when it exists. When the threefold is a K3 surface ﬁbration we express the DT invariants of sheaves supported on the ﬁbers in terms of the the Euler characteristics of the Hilbert scheme of points on the K3 surface and the Noether-Lefschetz numbers of the ﬁbration. Using this we prove the modularity of the DT invariants which was predicted in string theory. We develop a DT-theoretic conifold transition formula through which we compute the generating series for the invariants of Hilbert scheme of points for singular surfaces. We also use our geometric techniques to compute the generating series for DT invariants of threefolds given as complete intersections such as quintic threefold.

#### Timothy Logvinenko (Warwick): *Spherical DG-functors*

Seidel-Thomas twists are autoequivalences of the derived category D(X) of an algebraic variety X. They are the mirror symmetry analogues of Dehn twists along Lagrangian spheres on a symplectic manifold. Given an object E in D(X) with numerical properties of such a sphere, Seidel and Thomas defined the spherical twist of D(E) along E and proved it to be an autoequivalence.

It was long understood that this should generalise to the notion of the twist along a spherical functor into D(X). In full generality this was long obstructed by some well-known imperfections of working with triangulated categories. In this talk, I present joint work with Rina Anno, where we fix this by working with the standard DG-enhancement of D(X). We define the notion of a spherical DG-functor and give the braiding criteria for twists along such functors.

#### Barbara Fantechi (SISSA, Trieste): *Stacky viewpoint on Intersection theory*

We briefly review the basic features of Fulton-MacPherson Intersection Theory, focusing on the key definition of Gysin pullback via degeneration to the normal cone. We then show how introducing stacky language leads to a slight generalization of the morphisms for which Gysin pullback is defined and a natural introduction to virtual pullbacks.

#### Igor Netay (HSE, Moscow): *On A-infinity algebras of highest weight orbits*

I will present recent results on syzygy algebras. For any algebraic variety X --> P^n with an embedding to projective space the syzygy spaces have a natural structure of an A-infinity algebra. I will discuss the case of projectivization of highest weight orbits in irreducible representations of reductive groups.

#### Andrey Trepalin (HSE, Moscow): *Rationality of the quotient of P^2 by a finite group of automorphisms over an arbitrary field of characteristic zero*

It's well known that any quotient of P^2 by a finite group is rational over an algebraically closed field. We will prove that any quotient of P^2 is rational over an arbitrary field of characteristic 0.