First Copenhagen Topology Conference

The University of Copenhagen

September 1-3 2006

Titles and Abstracts

Kasper Andersen

The solution of the Steenrod problem

Abstract: Around 1960 Steenrod began asking the following basic question: Which graded polynomial algebras can be realized as the cohomology of a space? I will report on joint work with Jesper Grodal, in which we give a solution to this problem for an arbitrary ground ring.

Dave Benson

The homotopy category of complexes of injective modules

Abstract: This talk describes joint work with Henning Krause. Let $k$ be a field of characteristic $p$ and let $G$ be a finite group. I shall talk about the homotopy category of complexes of injective $kG$-modules, and its relationship with the derived category of cochains on the classifying space $BG$. This is related to work of Dwyer, Greenlees and Iyengar on duality in topology and representation theory.

Marcel Bökstedt

The homology of the free loop space on a projective space

Tobias Colding

Extinction of Ricci flow

Søren Galatius

Homotopy theory of Deligne-Mumford space

Abstract: I will report on work in progress towards understanding the Deligne-Mumford compactification of the moduli space of genus g Riemann surfaces from a homotopy theoretical point of view. This is joint work with Ya. Eliashberg.

John Greenlees

Rational torus-equivariant cohomology theories

Abstract: The talk will discuss an complete algebraic model for rational torus-equivariant spectra. Using this model one may construct cohomology theories based on complex curves. In the case of genus 1 this gives circle-equivariant elliptic cohomology, and one may use the Weierstrass sigma function to construct an equivariant lift of the Ando-Hopkins-Strickland genus. This is joint work with Brooke Shipley (UIC) and Matthew Ando (UIUC). The talk will describe parts of this picture.

Hans-Werner Henn

K(2)-local homotopy theory at the prime 3

Abstract: In joint work with Goerss, Mahowald and Rezk we constructed a resolution of the K(2)-local sphere at the prime 3 which gives new insight into the K(2)-local homotopy category. In this talk we will survey more recent work of Karamanov as well as joint work with Karamanov and Mahowald. In this work the resolution is being used for effective calculations of central objects in the K(2)-local category like its Picard group or the homotopy groups of the K(2)-local Moore space (which had been computed before by Shimomura).

Peter Jørgensen

A ring theoretical application of Bousfield Localization

Henning Krause

Support varieties for triangulated categories

Abstract: The notion of support is a fundamental concept, first introduced in algebraic geometry for modules, sheaves, and complexes, but now widely used in various areas of modern mathematics. For instance, Hopkins, Neeman, and others used it to classify thick subcategories. To define the support of an object, one usually requires an abelian or triangulated categoy with a commutative tensor product. In my talk, I present an approach to define the support for objects in any triangulated category, which has small coproducts and is compactly generated. This approach covers the usual examples but has the potential to provide new insight, for instance in non-commuative situations. It is somewhat surprising, how little is needed to develop a satisfactory theory of support. The talk presents joint work with Dave Benson and Srikanth Iyengar.

Ib Madsen

Moduli spaces from a topological viewpoint

Abstract:The talk is a repeat of my ICM address in Madrid.It explains what topology at present has to say about a variety of moduli spaces currently under study in mathematics.This includes the classical moduli space of Riemann surfaces,the Gromov-Witten moduli space of pseudo-holomorphic curves in a background and the moduli space of graphs.The talk will sketch out the proof of Mumford's standard conjecture about the moduli space of Riemann surfaces and the corresponding standard conjecture for graphs.

Birgit Richter

Quasisymmetric functions from a topological point of view

Abstract: This talk is on joint work with Andy Baker.

It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as a topological model for the ring of quasisymmetric functions. I'll explain how to exploit topology to reprove the Ditters conjecture which says that the ring of quasisymmetric functions is a polynomial ring.

The loop map to BU gives a canonical Thom spectrum over $\Omega\Sigma\mathbb{C}P^\infty$. This spectrum is highly non-commutative. I'll talk about the homology of its topological Hochschild homology spectrum.

Ulrike Tillmann

Mapping braid to mapping class groups, and a conjecture by J. Harer

Abstract: (joint work with Yongjin Song) Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map $\phi: \beta_{2g} \to \Gamma _{g,1}$ from the braid group to the mapping class group. We prove that this map is trivial in homology with any trivial coefficients in degrees less than $g/2$. In particular this proves an old conjecture of J. Harer.