# Current Activities

Tuesday activities take place between 13:00-14:45. When at KTH, then in room 3418, and when at SU, then in room 22, house 5. Divergence from this is marked with a "*" and completed by, if needed, neccessary information.

**Kristian Moi: On real THH**

**Time: **Tue 2017-10-03 13.00 - 14.00

**Location: **Room 3418, Lindstedtsvägen 25, Department of mathematics, KTH

**ABSTRACT**: Topological Hochschild homology (THH) was introduced by Bökstedt in order to make computations of algebraic K-theory of rings and ring spectra. For rings with anti-involution, Hesselholt and Madsen introduced real algebraic K-theory, which combines Hermitian and algebraic K-theory into a single genuine Z/2-equivariant spectrum. I will report on recent joint work with Dotto, Patchkoria and Reeh on real topological Hochschild homology, which is the corresponding equivariant refinement of THH. The talk will focus on basic ideas and computations.

**Fabian Hebestreit: Tautological classes of aspherical manifolds**

**Time: **Tue 2017-10-10 13.00 - 14.45

**Location: **Room 22, House 5, Kräftriket, Department of Mathematics, Stockholm University

**ABSTRACT**: Tautological classes, or generalised Morita-Miller-Mumford classes as they are often called, are characteristic classes of manifold bundles derived from tangential data. They have recently come to the foreground through the work of Madsen, Tillmann and Weiss on surface bundles, and then Galatius and Randal-Williams for higher dimensional fibres. Their results describe the cohomology of the diffeomorphism group in degrees linearly bounded by the genus of the manifold in terms of these classes and close relatives.

Aspherical manifolds are an interesting class of manifold that have vanishing genus and indeed, I will speak about a result of ours that shows the vanishing of tautological classes on them away from two obstructions: The consequences of the surface case and hyperbolisation. This is joint work with M.Land, W.Lück and O.Randal-Williams

**Hochschild reading seminar - Bashar Saleh: Hochschild homology and the free loop space, HH of group rings**

**Time: **Tue 2017-10-17 13.00 - 14.45

**Location: **Room 3418, Lindstedtsvägen 25, Department of mathematics, KTH

**ABSTRACT**: This is the first talk in the
reading seminar on Hochschild homology.

**Hochschild reading seminar - Jeroen Hekking: The B-operator and the Hochschild-Kostant-Rosenberg**

**Time: **Tue 2017-10-31 13.00 - 14.45

**Location: **Room 3418, Lindstedtsvägen 25, 4th floor, Department of mathematics, KTH

**ABSTRACT**: This is the second talk in the
reading seminar on Hochschild homology.

**Hochschild reading seminar - Clas Löfwall: Free models and cyclic homology of algebras of global dimension at most two**

**Time: **Tue 2017-11-07 13.00 - 14.45

**Location: **Room 22, House 5, Kräftriket, Department of Mathematics, Stockholm University

**ABSTRACT**: We give a formula for cyclic homology for free algebras which is used to compute cyclic homology of global dimension two algebras, in particular a free algebra modulo a symmetric quadratic form or not too many general symmetric quadratic forms which using Koszul duality gives a formula for the cyclic homology of a commutaive polynomial ring modulo sufficiently many quadratic relations. This talk is part of the
reading seminar on Hochschild homology.

**Lennart Meier: Unstable telescopic homotopy theory**

**Time: **Tue 2017-11-14 13.00 - 14.45

**Location: **Room 3418, Lindstedtsvägen 25, 4th floor, Department of mathematics, KTH

**ABSTRACT**: In the 1960s, Quillen found a Lie algebra model for rational homotopy theory. Together with later work by Sullivan, this led to a lot of computational work on rational homotopy theory. But rational homotopy theory is just the first of a whole ladder of approximations to the homotopy theory of spaces. For every prime p and every n, there is a telescopic homotopy theory of spaces (of which rational homotopy is the limit case n=0).

The Bousfield-Kuhn functor allows to compare the unstable telescopic homotopy theory with stable homotopy theory. I will describe how this allows to give a model of unstable telescopic homotopy as "Lie algebras in T(n)-local spectra". In the first 45 minutes, I will give an introduction to basic ideas of rational, chromatic and telescopic homotopy theory.

**Hochschild reading seminar - Greg Arone: Cyclic homology**

**Time: **Tue 2017-11-21 13.00 - 14.45

**Location: **Room 22, House 5, Kräftriket, Department of Mathematics, Stockholm University

**ABSTRACT**: This is the fourth talk in the
reading seminar on Hochschild homology.

**Lukas Brantner: Operations on Spectral Lie Algebras and a Conjecture of Ravenel**

**Time: **Tue 2017-11-28 13.00 - 14.00

**Location: **Room 3418, Lindstedtsvägen 25, 4th floor, Department of mathematics, KTH

**ABSTRACT**: We describe the operations acting on the E-theory of K(n)-local Lie algebras and discuss how they relate to a classical open problem in chromatic homotopy theory.

**Hochschild reading seminar - Ben Ward: Operations on Hochschild and cyclic cohomology**

**Time: **Tue 2017-12-05 13.00 - 14.45

**Location: **Room 22, House 5, Kräftriket, Department of Mathematics, Stockholm University

**ABSTRACT**: We will continue the Hochschild reading seminar with a recollection of the classical deformation theory of associative algebras and its relationship to Hochschild cohomology. There is a parallel story for cyclic cohomology which we will also discuss. This parallel story will motivate us to calculate the S^1 equivariant homology of the configuration spaces of points in the plane. This is the fifth talk in the
reading seminar on Hochschild homology.

**Dan Burghelea: The homotopy type of the classifying space of the group of automorphisms of compact manifolds away of prime 2 and in "stability range"**

**Time: **Tue 2017-12-12 13.00 - 14.45

**Location: **Room 3418, Lindstedtsvägen 25, 4th floor, Department of mathematics, KTH

**ABSTRACT**: I will review the description of the homotopy type of the classifying space of the group of automorphisms (diffeomorphisms / homeomorphisms) of a compact manifold M in "stability range" and "away of the prime 2" (known 40 years ago but apparently not well remembered today).

I will also review the relations with the "moduli space of Riemannian structures" on M and with the space of "non parameterized closed curves" on M.