Home › Speakers and abstracts: Graduate homotopy conference

Speakers and abstracts: Graduate homotopy conference

(The program is available here)

Julia Goedecke: A Comparison Theorem for Simplicial Resolutions

(Joint work with Tim Van der Linden) In 1969, Barr and Beck introduced the notion of comonadic homology in abelian categories, which essentially uses a functorial way to form projective simplicial resolutions of an object in an arbitratry category, and transforms these simplicial resolutions into a chain complex in an abelian category, which can be used to form homology. This notion can be generalized to semi-abelian categories, and we look at the particular question of how this homology theory depends on the given comonad. This involves a comparison theorem for simplicial resolutions in an arbitrary category, but with conditions on the comonad, and also the fact that simplicially homotopic maps in semi-abelian categories give rise to the same map on homology.

Ramesh Satkurunath: An algorithmic approach to Dold-Puppe complexes

I am interested in the construction of Dold-Puppe complexes, which are
complexes used in the construction of the derived functors of a non-linear functor $ F $ (e.g. of the $ k^{th} $ symmetric power functor or the $ k^{th} $ exterior power functor). These derived functors play a central role in Köck's approach to Adam's operations in Algebraic K-Theory. For a chain complex $ C. $ a Dold-Puppe complex is a complex of the form $ NF\Gamma(C.) $, i.e. the image of $ C. $ under the composition of the functors $ \Gamma $, $ F $ and $ N $; where $ \Gamma $, and $ N $ are the functors given by the Dold-Kan correspondence. The Dold-Kan correspondence gives a pair of functors $ \Gamma $ and $ N $ that provide an equivalence between the category of of bounded chain complexes and the category of simplicial complexes. The definition of $ \Gamma $ is quite abstract and combinatorial. Half of my work has been in creating an algorithm which streamlines the calculation of $ \Gamma(C.) $. The other half has been in designing algorithms that allow the explicit calculation of the Dold-Puppe complex in terms of the cross-effect functors of $ F $.

Fredrik Nordström: Flatness of functors indexed by $ Sing_A(M) $

Abstract

Charles Vial: Operations in Milnor K-theory

I will give a brief account on Milnor K-theory of a field and describe the operations $ K^M_i/p \rightarrow K^M_*/p $ over any field $ k $.

Steve Bennoun: Tannaka Duality for Comodules over a Hopf Algebra

I shall define the category of comodules over a bialgebra
A and prove that it is autonomous when A is a Hopf algebra. Then,
starting with this category, I shall use Tannaka duality to reconstruct
the bialgebra A. I shall then be able to explain how we can freely
adjoin an invertible antipode to a bialgebra.

Dustin Mulcahey: Generalized Homotopy Theory

I will give an overview of the recent work by Dwyer, Kan, Hirschorn,
and Smith, who defined homotopical categories and used the new
formalism to prove a strong homotopical completeness result for model
categories.

Mark Powell: The Knot Concordance Group

I will try to give a survey of slice knots, the higher dimensional knot concordance group which was worked out, in the 60s, and the classical knot concordance group. Recent (2000), work by Cochran, Orr and Teichner generalises higher dimensional, techniques to the classical case, taking account of the fundamental, group and giving an infinite filtration of the slice condition using, derived subgroups and gropes (this is one of the ways anyway). I am trying to understand this filtration, which "contains" all previously known concordance invariants; hopefully I can explain something about how the geometry and algebra fit together so wonderfully!

Piotr Beben: Decomposition of Looped Stiefel Manifolds, and Application to James numbers and Homotopy Exponents

For p fixed at an odd prime and n<=(p-1)(p-2) I obtain a decomposition of the p-local looped Stiefel manifold W(n,m) as a product p-1 factors, each of which is the looping of a space in a certain spherical resolution. This allows me to calculate upper bounds on the James numbers and p-exponents of W(n,m).

Alexander Berglund: Minimal Koszul models of dg-coalgebras

For a Koszul coalgebra $ C $ with Koszul dual algebra $ C^! $, I will show how a coderivation differential $ d $ on $ C $ gives rise to an $ A_\infty $-algebra structure on $ C^! $ quasi-isomorphic to the cobar construction $ \Omega (C,d) $. This generalizes the classical correspondence between $ A_\infty $-algebra structures on a graded module $ V $ and coderivation differentials on the tensor coalgebra $ T(sV) $. As an application, if $ (C,d) $ is a minimal model for the dg-coalgebra $ C_*(X) $ of chains on a simply connected space $ X $, then the loop space homology algebra $ H_*(\Omega X) $ is isomorphic to the algebra $ C^! $ with multiplication perturbed by the quadratic part of the differential $ d $.

Jonas Kiessling: Localizing subcategories of the derived category of a ring

I will discuss the classification of all localizing subcategories of the derived category of a Noetherian ring obtained by Neeman. I will also mention various results obtained by myself and Stanley concerning nullity and cellular classes, unstable analogs of localization classes.

Théophile Naito: Alexander-Whitney coalgebras

Let C(K) be the normalized chains on a simplicial set K. C(K) can be equipped with a coassociative co-multiplication which is almost cocommutative: the co-multiplication is cocommutative up to a sequence of homotopies. Such a coalgebra is called an Alexander-Whitney coalgebra. Actually, Alexander-Whitney coalgebras can be seen as coalgebras over a certain operad AW. I use this fact to define a minimal model for C(K).

Alex Shannon: Weighted limits in DG-categories

Categories enriched over the category DG of chain complexes of abelian groups which vanish in negative degree are examples of higher-dimensional categories that we can handle very explicitly. In particular, many sorts of homotopy limits and colimits can be conveniently described using the language of weighted limits and colimits from enriched category theory. Certain such limits and colimits, for example cones and cylinders, play an important role in homological algebra, and in many cases, their constructions in categories of chain complexes can be obtained in an essentially combinatorial way from the universal property they satisfy. Moreover, DG-categories which have such limits and colimits, and which satisfy a stability property, have a natural triangulated structure on their homotopy category.

Carl McTague: Elliptic cohomology equals bordism modulo flops

Abstract tba

Tarje Bargheer: Geometric Operads and String Topology

Via examples, an attempt will be made to clarify the use of operads as a method for introducing additional structure upon topological spaces -- as well as their homology. In particular, we shall introduce the cacti operad as a method for giving the homology of a free loop space on a manifold -- $ H_*(LM) $ -- the structure of a BV-algebra; also known as the Chas-Sullivan loop product. Time permitting, we will discuss problems related to generalizations into higher dimensions.

Thomas Gregersen: Thoughts on equivariant motivic cohomology

We construct a possible candidate for equivariant motivic cohomology.

Markus Severitt: Motivic Homotopy Types of Projective Curves

Since at least Motivic Cohomology and Algebraic K-Theory of smooth varieties are representable over the (Un)Stable Motivic Homotopy Category, it is interesting to study (un)stable motivic homotopy types of such varieties. This is analogous to the situation in topology where the homotopy types of spaces give cohomological information and vice versa.The talk gives a full classification of smooth projective curves and abelian varieties up to isomorphism in the unstable motivic homotopy category H(k). Furthermore simple partial results of the stable case are given. Link

Andreas Holmstrom: Brown representability theorems in algebraic geometry

In topology, the Brown representability theorem says that every cohomology theory is representable by a spectrum. For cohomology theories in algebraic and arithmetic geometry, things are more complicated. I will say something about known representability results, and discuss some recent developments and ideas.

Dinesh Deshpande: Chow rings of classifying spaces

This talk is based on the paper by Burt Totaro by same title. I will define Chow rings of classifying spaces and talk about some examples where we can calculate it. Calculations show remarkable resemblance to complex cobordism. I will relate this to algebraic cobordism and point out that it is isomorphic to complex cobordism in these examples.

 

Participants not giving a talk:

  • Jonny Evans
  • Maurice Chiodo
  • Martin Blomgren
  • Philip Herrmann
  • Florian Strunk
  • Oscar Randal-Williams
  • Brandon Levin
  • Victoria Gregson
  • Alexander Palen Ellis
  • Jack Waldron