In this talk I will introduce a generalization of Sullivan models that I propose to call `Koszul models'. These are commutative cochain algebras (A,d) whose underlying algebra A is a, not necessarily free, Koszul algebra. Koszul models share the computational benefits of Sullivan models, but are sometimes more tractable. Examples to illustrate their usefulness are provided by moment-angle complexes associated to simplicial complexes. These spaces have infinitely generated Sullivan models but admit finitely generated Koszul models. If a space X admits a finitely generated Koszul model then a consequence is that the homotopy Lie algebra π_{*}(ΩX) ⊗ Q is finitely generated and that the Poincaré series of the loop space homology H_{*}(ΩX;Q) is rational.
The strange fact is that this very naïve setup has homotopy theoretical content. The chromatic picture gives a hierarchy of cohomology theories according to how deep structure of stable homotopy theory the cohomology theory detects. Chromatic filtration zero, one and infinity have nice geometric interpretations through functions, vector bundles and bordisms, and the geometric origin is important for the analysis of key problems. Such geometric interpretations have been missing in higher finite filtrations.
Elliptic cohomology and topological modular are examples at chromatic filtration two, and Segal conjectured that there ought to be a geometric interpretation of elliptic cohomology through quantum field theories.
Recently it was shown that two-vector bundles give rise to a cohomology theory which is represented by the algebraic K-theory of topological K-theory ku. By results of Ausoni and Rognes K(ku) is of (a connected version of) chromatic filtration two. Hence we have a naturally defined geometric theory of the desired sort.
The connection to quantum field theories and the set-up of Stolz and Teichner is, however, still mysterious. There was a hope that an "integration of determinants through loops" construction would give a functor from two-vector bundles to quantum field theories, but this is unfortunately not the case.
Whereas commutative rings support determinants, this is not (in the most naïve sense) true for commutative ring spectra. In fact, neither the sphere spectrum nor topological K-theory support determinants. The latter is important for us since it rules out the conjectured integral functor to quantum field theories.
However the reason for its failure is very interesting: it stems from an observation that Rognes that Ausoni's calculations of K(ku) implies that the group of gerbes on the three dimensional sphere do not split off as a direct summand of the group of "virtual" two-vector bundles. This leaves one speculating about the geometry of two-vector bundles, even over very simple spaces.
Rationally, this problem vanishes, and Ausoni and Rognes have pushed through the program in this case: giving a virtual two-vector bundle on X is rationally the same as giving its virtual "dimesion bundle" and an "anomality bundle" on the free loop space of X.
Two-vector bundles is a theory in its infancy, and much is still left to explore. In particular the geometric and analytic aspects are so far largely terra incognita.
In my talk I will try to explain some of these ideas and results.
Recent results will be presented: Motivated by directed covering theory - important for applications - a new more convenient category for d-homotopy has been introduced and investigated (J.Roscicky and L.F.). The topology of the space of directed paths modulo reparametrizations, the space of traces of cubical complexes, has recently been described in terms closer to calculations (M.Raussen). A categorical framework for "components" of a (nice) d-space is about to be implemented (E.Goubault and E. Haucourt) for the purpose of static analysis of real life concurrent programs.
In the talk I will show how to approximate the C-Tate construction on THH(R) by a topological version of the algebraic Singer construction. As an application, I will report on work in progress that aims to calculate the (continuous) homology of the S^{1}-Tate construction on THH(MU).
This work is joint with J.Rognes.
However, the charts also make manifolds very fragile: it is easy to do something to a manifold that makes it no longer a manifold. Taking a quotient by a group action is one such, looking at mapping spaces is another. Often, specific operations can be fixed - orbifolds fix the quotienting, infinite dimensional manifolds fix the mapping spaces - but systematic case-by-case fixing is a little unsatisfying. Over the years there have been several attempts to build a suitable category of "smooth objects" generalising smooth manifolds. The general method is to take some property that all manifolds have, which can be defined in a more robust way than charts.
In this talk I shall review some of these attempts, focussing particularly on the similarities between them. I shall try to motivate my own favourite: Frölicher spaces. In addition, it is worth mentioning that the majority of these categories come under the heading of "sets with structure". There have also been attempts to do away with the "sets with" part of this and I shall talk about why one might wish to do this.
This is based partly on arxiv:0803.0611 [math.DG]
arXiv:0802.2225
[math.DG].
This is joint work with Robert Lipshitz and Peter Ozsváth.