Speaker: Mark Behrens (University of Chicago)
Title: Resolutions Arising from Thom Complexes
I'm going to summarize Mark Mahowald's paper "Ring spectra which are Thom complexes", Duke Math. J. (1979). In this paper he gives a geometric Thom isomorphism for a ring spectrum E which is also a Thom spectrum, and he uses this isomorphism to compute the d_1 in the E-Adams resolution. I'll then describe how Hopkins and Mahowald, through clever choice of the "right" E, are able to use the results of the paper to describe the E_2 term of the Adams-Novikov spectral sequence for computing the homotopy groups of the spectrum EO_2.
Speaker: Jack Calcut (University of Maryland)
Title: (Non)compact Exotica
The relationship between compact and noncompact exotica in 3 and 4 dimensions is not fully understood. Exotic R^4s are noncompact and it is unknown whether one can be used to obtain a fake 3 or 4sphere. Related is a problem of V.I.Arnol'd on the possible existence of an algebraic exotic R^4 in R^5. Topological and real algebraic aspects of these and related problems will be presented.
Speaker: Vlad Chernysh (University of Notre Dame)
Title: Homotopy Properties of the Space of Metrics of Positive Scalar Curvature.
We study deformations of metrics of positive scalar curvature. In particular, we show that, under certain conditions, the space of psc metrics on a given manifold is homotopy equivalent to a space of metrics that restrict to a fixed metric near a given submanifold of codimension greater or equal than 3.
Speaker: Marilyn Daily (North Carolina State University)
Title: L-Infinity Algebras
An L-infinity algebra is a differential graded vector space which has a Lie algebra structure up to (chain) homotopy. In particular, the Jacobi identity holds up to homotopy. In this talk, I will use some "small" finite-dimensional examples to illustrate this concept.
Speaker: Mike Dekker (University of Notre Dame)
Title: Positive Scalar Curvature and Index Theory
Positive scalar curvature makes an interesting connection between topological and geometric characteristics of manifolds, but the question "Which manifolds admit metrics of postive scalar curvature?" has not been fully answered. However, there exist bordism groups which contain an obstruction to the existence of a positive scalar curvature metric on a manifold, and we will use the "index map" out of these groups to examine them.
Speaker: Florin Dumitrescu (University of Notre Dame)
Title: Obstructions to Having a Positive Scalar Curvature Metric
We will address the question: "When does a manifold M admit a metric of positive scalar curvature?". There are certain obstructions (of topological nature), like, for example, when M is spin and of dimension divisible by four, the A-hat genus must be zero, and the answer might depend on subtle things like the differentiable structure. It was noticed however that the answer only depends on a bordism class of M, so the problem lends well to the field of stable homotopy. All the results presented are known.
Speaker: Benjamin Himpel (Indiana University)
Title: A Gauge Theoretic Discussion of Casson's Invariant for Homology 3-Spheres
In 1988 Taubes defined an invariant for homology 3-spheres by extending the Euler characteristic to certain infinite dimensional manifolds arising from gauge theory. Then he proved that his invariant is actually equal to the topologically defined Casson's invariant. I will explain how one can make a proof work, that uses an extension of Liviu Nicolaescu's splitting formula for the spectral flow of a family of Dirac operators to perturbed Dirac operators (joint work with Paul Kirk and Matthias Lesch).
Speaker: Elke Markert (University of Notre Dame)
Title: The Landweber Exact Functor Theorem
The theorem provides a functor which produces a homology theory out of an MU_*-module M (i.e. out of a coefficent ring and a formal group law), iff M satisfies certain conditions, the so-called "Landweber criterion." This can in particular be used to construct examples of "elliptic theories," using the formal group laws of elliptic curves.
Speaker: Chris Mitchell (Purdue University)
Title: The Action of an Operad on a Space and Some Results
The purpose of the talk will be to define the notion of an operad acting on a topological space and to provide some known results. Of particular interest are the results of Stasheff and May on the spaces which admit certain operad actions.
Speaker: Corbett Redden (University of Notre Dame)
Title: A Conjecture Concerning Ricci Curvature
There is a conjecture by Höhn and Stolz that if a string manifold M admits a metric of positive Ricci curvature then the Witten genus should vanish. We will look at evidence for this conjecture and an outline of a potential proof based on Witten's heuristic calculation of the Witten genus as the S1-equivariant index of the Dirac operator on LM.
Speaker: Jonathan Rogness
(University of Minnesota)
Title: Homotopy Theory of Exact Couples
Exact couples have been used for many years to construct spectral sequences, but they may be useful for other reasons. I will provide a short summary of an earlier construction (not my own) of a homotopy theory of exact couples which allows us to view them as a generalization of stable homotopy theory. Then I will describe how the category EC of exact couples satisfies most of the axioms of a closed model category, perhaps allowing us to use more modern homotopy theory techniques. Enlarging the category EC in a natural way results in an abelian category, so that we could also compute derived functors in a classical sense.
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http://www.nd.edu/~jbergner/Abstracts.html Last modified: April 8, 2003