Lecture series

Boris Tsygan (Northwestern University) - Noncommutative differential calculus

I will review the current state of noncommutative differential calculus. The term stands for the theory that generalizes classical algebraic structures arising in differential calculus on manifolds to make them valid for any associative algebra (or, more generally, any differential graded category) instead of the algebra of functions on a manifold. The role of differential forms and multi-vector fields in this new theory is played by the Hochschild complexes of our algebra. The generalized algebraic structures from classical calculus are provided by the action of various operads on these complexes. I will summarize the current state of the subject as developed in the works of Kontsevich and Soibelman, Tamarkin, Willwacher, and other authors, as well as my own works in collaboration with Dolgushev, Nest, and Tamarkin.

Thomas Willwacher (University of Zurich) - Graph complexes

Graph complexes are differential graded vector spaces whose elements are linear combinations of combinatorial graphs. The differential is the operation of contracting an edge. These graph complexes exist in a variety of flavors (ribbon graphs, directed/undirected graphs etc.), each of which plays a central role in otherwise quite disjoint areas of mathematics like knot theory, geometric group theory and moduli spaces of curves. Despite the very elementary definition and its fundamental importance we know surprisingly little about what the graph homology actually is. The purpose of the course is to give an introduction to the problem and its origins, along with an overview of recent results.

Other talks

Ryszard Nest (University of Copenhagen) - Some applications of operations on cyclic complexes

In this lecture we will give some applications of the non-commutative differential calculus to results in cyclic homology, deformation theory and index theorems.

Niels Kowalzig (Univ. Sapienza di Roma) - Higher Structures on Modules over Operads

We show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild theory of associative algebras, for Poisson structures, but above all the calculus for general Hopf algebroids with respect to general coefficients (in which the classical calculus of vector fields and differential forms is contained); that is, a calculus structure on the pair of Ext- and Tor-groups over generalised bialgebras over noncommutative rings. Time permitting, we will also discuss Batalin-Vilkovisky algebras and their relationship to Cotor-groups.

Martin Doubek (Charles University, Prague) - Quantum open-closed homotopy algebras

Open-closed homotopy algebras first appeared in the work by Zwiebach on string theory in 1990's. In the "classical" case, the homotopy nature of these algebras was noticed by Kajiura and Stasheff. In the "quantum" case, these algebras can be succintly described in terms of an easily understood modular operad of 2-dimensional surfaces with boundary. We will explain this and some connections with more familiar structures. Finally, we discuss related graph complex and why it might help us to understand Koszul duality for modular operads.

Nathalie Wahl (University of Copenhagen) - Universal operations on the Hochschild complex of algebras over props

I will construct the complex of all "formal operations" on the Hochschild complex of algebras over a prop, and explain what universal property it satisfies. I will then compute this complex in the case of Frobenius algebras.

Pieter Belmans (University of Antwerp) - Derived categories of noncommutative quadrics and Hilbert schemes of points

The infinitesimal deformation theory of abelian categories is governed by their Hochschild cohomology, and provides the general framework for the notion of noncommutative planes (resp. quadrics) as developed by Bondal--Polishchuk and Van den Bergh. On the level of derived categories the presence of a full and strong exceptional collection yields a finite-dimensional algebra describing the noncommutative plane (resp. quadric).

By a recent result of Orlov the derived category of any finite-dimensional algebra can be embedded in the derived category of a smooth projective variety. This construction is far from canonical, but in the case of the plane Orlov shows that one can choose a deformation of the Hilbert scheme of two points. We show how similar results can be obtained for the more complicated case of noncommutative quadrics, and how more generally ``limited functoriality'' for Hochschild cohomology should give a relationship between the noncommutative deformations of a surface and geometric deformations of the Hilbert scheme of points.

This is joint work in progress with Theo Raedschelders and Michel Van den Bergh.

Benjamin Ward (Simons center for geometry and physics) - Hyper-commutative and hyper-Lie structures on Hochschild complexes

Eric Dolores Cuenca (Northwestern University) - Kontsevich Swiss Cheese Conjecture

Given an associative algebra A, there is an universal extension to an e_2 algebra, namely the Hochschild cochain complex Hoch(A), and the singular chain version of the 2-Swiss Cheese operad acts on the pair (A,Hoch(A)) ( Dolgushev, Tamarkin, Tsygan ).  

In this talk I'll discuss work of Justin Thomas, and my own work on generalization of the conjecture for e_d, d>2.

Daniela Egas-Santander (FU Berlin) - On the homology of Sullivan diagrams

In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. Finding non-trivial homology classes of these compactifications is related to finding non-trivial string operations. However, the homology of these complexes is largely unknown. In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams and give some computational results and further work on this direction.

Reiner Hermann (NTNU Trondheim) - The Lie bracket in Hochschild cohomology via the homotopy category of projective bimodules

We give an interpretation of the Lie bracket and the divided squaring operation in Hochschild cohomology of an associative algebra in terms of the (bounded below) homotopy category of projective bimodules, thus, in terms of the derived category of bimodules. Our approach combines results by Buchweitz on the fundamental group of a morphism in a triangulated category and by Schwede on the homotopy theory of extension categories. This is joint work with Johan Steen (NTNU, Trondheim).

Jacek Brodzki (University of Southampton) - The periodic cyclic homology of crossed products of finite type algebras

In this talk we will discuss the periodic cyclic homology groups of the cross-product of a
finite type algebra A by a discrete group Γ. In the case when  A is commutative and Γ is finite, our results are complete and given in terms of the singular cohomology of the strata of fixed points. These groups identify our cyclic homology groups with the orbifold cohomology of the underlying (algebraic) orbifold. The proof is based on a careful study of localization at fixed points and of the resulting Koszul complexes. We provide examples of Azumaya algebras for which this identification is, however, no longer valid. As an example, we discuss some affine Weyl groups.
This talk is based on joined work with Nistor and Dave.