Purdue University

Department of Mathematics
Topology Seminar


Fall 2012
Time and place: Thursdays 1:30-2:30 in BRNG 2291
Organizer: Ralph Kaufmann. Contact organizer.


08/16/12 John Harper (Purdue). Structured ring spectra and TQ-homology.
            Abstract: This is an introductory talk on what one might think of as homotopy theoretic commutative algebra. In more detail, we will discuss spectra, commutative ring spectra, and generalized algebra spectra, together with naturally occurring invariants of such algebra spectra called TQ-homology. We will finish with some recent results indicating that certain homotopical properties of ring spectra are detected by TQ-homology.

09/06/12 John Harper (Purdue). Localization and completion of nilpotent structured ring spectra.
            Abstract: Quillen’s derived functor notion of homology provides interesting and useful invariants in a wide variety of homotopical algebraic contexts. For instance, in Haynes Miller’s proof of the Sullivan conjecture on maps from classifying spaces, Quillen homology of commutative algebras (Andre-Quillen homology) is a critical ingredient. Working in the topological context of structured ring spectra, this talk will introduce several recent results on localization and completion with respect to topological Quillen homology of commutative ring spectra (topological Andre-Quillen homology), E_n ring spectra, and operad algebras in spectra. This includes homotopical analysis of a completion construction and strong convergence of its associated homotopy spectral sequence. The localization and completion constructions for structured ring spectra are precisely analogous to Sullivan's localization and completion of spaces (for which he recently won the Wolf prize), and Bousfield-Kan's version of Sullivan's localization and completion called the R-completion of a space with respect to a ring R. This is joint work with Michael Ching.

10/11 Ben Ward (Purdue). Homotopy Theory of Generalized Operads.
        Abstract:  First I will recall how the homotopical algebra of operads can be used to study homotopy invariant algebraic structures.  Then I will turn our attention to various generalizations of operads and give conditions for the existence of a meaningful model structure on the category of such objects valued in a model category.  Finally I will discuss some results which buttress the use of the word meaningful in the preceding sentence.

10/18 David Shea Vela-Vick (LSU). The equivalence of transverse link invariants in knot Floer homology.

        Abstract:  The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces.  Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined.  The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams.  The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo.  We show that these two previously defined invariant agree.  Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.

10/25 Ben McReynolds (Purdue). Arithmetic Manifolds: An Introduction with Topological Problems.

Abstract: I will give an introduction to arithmetic manifolds; this is be more detailed than the talk I gave earlier in this term in the Bridge to Research seminar. I will also provide some interesting topological problems for arithmetic manifolds. The talk will be 2 hours with a break at the half way point for rest and/or polite escape.
We will change to MATH 215 for the second part.

11/01 Matthew Stover (Michigan). Totally Geosdesic Surfaces and Cohomology of Complex Hyperbolic Manifolds.

Abstract: Let M be a compact quotient of complex hyperbolic 2-space containing lots of holomorphically embedded totally geodesic Riemann surfaces. I will describe why all of H1(M) comes from these surfaces. In particular, this applies to certain arithmetic complex hyperbolic manifolds. The proof uses the relationship between topology and algebraic geometry to apply a so-called `weak Lefschetz theorem' to M and a line bundle determined by a nice collection of surfaces on M. This is joint work with Ted Chinburg.

11/08 Matthew Thibault (U of Chicago).  Finite simplicial complexes via pro-posets.

Abstract: One has a pair of functors between finite topological spaces and finite simplicial complexes. Via this correspondence, McCord proves that finite topological spaces up to weak homotopy equivalence coincides with finite simplicial complexes up to homotopy equivalence. Since finite topological spaces coincide with finite posets, this allows one to convert problems in algebraic topology into problems in combinatorics. However, due to a dearth of maps in the category of finite spaces, one must enlarge this category in order to describe all homotopy classes of maps between (finite) simplicial complexes. In this talk, I will describe the homotopy category of finite simplicial complexes in terms of the category of pro-posets.

11/09 Matthew Hedden (Michigan State). Some recent progress on topologically slice knots

    Abstract: Modulo a 4-dimensional equivalence relation, the set of knots in the 3-sphere can be endowed with a group structure. The resulting group is called the concordance group and, perhaps surprisingly, the group depends on whether we work in the homeomorphism or diffeomorphism category. Particularly important to understanding this distinction is the set of topologically slice knots: those knots which bound topologically flat embedded disks in the 4-ball. These knots generate a fundamental subgroup of the smooth concordance group of knots which, for instance, can be used to demonstrate the existence of a 4-manifold which is homeomorphic, but not diffeomorphic, to euclidean 4-space. I'll give an introduction and overview of the concordance groups, and discuss recent work which provides the first examples of (two-) torsion elements in this topologically slice subgroup. The talk should be accessible to those who have taken a graduate course in algebraic or differential topology. The new results which I'll mention are joint work with Se-Goo Kim and Charles Livingston.

11/29 Martin Frankland (UIUC). The Homotopy of p-complete K-algebras
     Abstract: Morava E-theory is an important cohomology theory in chromatic homotopy theory. Rezk described the algebraic structure found in the homotopy of K(n)-local commutative E-algebras, via amonad on E*-modules that encodes all power operations. However, theconstruction does not see that the homotopy of a K(n)-local spectrumis L-complete (in the sense of Greenlees-May and Hovey-Strickland). Weshow that at chromatic height 1, the construction can be improved to amonad on L-complete E*-modules, and sketch why this should hold at every height. Joint with Tobias Barthel.