Midwest Topology Seminar
May 2 - 3, 2009
|Clark Barwick||Harvard University|
|William G. Dwyer||University of Notre Dame|
|Teena Gerhardt||Indiana University, Birmingham|
|John Klein||Wayne State University|
|Charles Rezk||University of Illinois at Urbana-Champaign|
|Victor Turchin||Kansas State University|
Operator Categories and Homotopy Coherent Algebra - Clark Barwick (Harvard University)
Abstract: I will introduce the new concept of an operator category, and I describe the theory of operads and algebras relative to a fixed operator category. Using this new notion, I give fully combinatorial homotopy-universal characterizations of various interesting operads. I then show how this notion can be combined with recent advances in the theory of infty-categories along with a powerful strictification theorem to prove some surprising results about the existence and unicity of En structures.
Power Operations in Morava E-Theory - Charles Rezk (University of Illinois at Urbana-Champaign)
Abstract: Structured ring spectra come with operations on their homotopy groups, called power operations. Classically, the Steenrod operations (on the mod p cohomology of spaces) and the Kudo-Araki-Dyer-Lashof operations (on the homology of infinite loop spaces) derive from power operations for the mod p Eilenberg Mac Lane spectrum. In this talk, I will survey what is known about power operations for Morava E-theory spectra (thanks to work of Ando, Strickland, and others), describe how they are to be understood in terms of isogenies of formal groups, and explain some recent results.
Long knots and operad maps - William Dwyer (University of Notre Dame)
Abstract: This is joint work with Kathryn Hess. We show that for m≥4 the space of framed long knots in Rm is weakly homotopy equivalent to the double loop space on the space X of (derived) operad maps from the associative operad to the m'th Kontsevich operad. (This Kontsevich operad is a version of the little m-disk operad.) The technique is to produce an economical cosimplicial formula for Ω2X and then observe that the formula matches Dev Sinha's cosimplicial formula for the long knot space.
Embedding spaces and the operad of little cubes - Victor Turchin (Kansas State University)
Abstract: The talk will cover three important connections between the operad of little cubes and embedding spaces. 1. Action of this operad on the spaces of knotted planes. 2. Relation between the rational homology of embedding spaces and the homology of this operad. 3. Delooping spaces of knotted planes.
Algebraic K-theory of the dual numbers - Teena Gerhardt (Indiana University, Bloomington)
Abstract: Nearly 30 years ago, Soul\'e showed that the abelian group Kn(Z[x]/x2, (x)) is finitely generated with rank 0 if n is even and 1 if n is odd. We compute this group for n odd and find its order for n even. Further, we generalize these results to the study of the algebraic K-theory of truncated polynomial algebras, Kn(Z[x]/xe, (x)). This is joint work with Vigleik Angeltveit and Lars Hesselholt.
Periodicity and Duality - John Klein (Wayne State University)
Abstract: Let K be a connected finite complex. I will consider the problem of whether one can attach a cell to some iterated suspension SjK so that the resulting space satisfies Poincare duality. When this happens, we say that SjK is a spine. I will give criteria for deciding when this is possible. I'll also explain how this leads to a new interpretation of the four-fold periodicity in the surgery obstruction groups.