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.
