08/16/12 John Harper (Purdue). Structured
ring spectra and TQ-homology.
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
09/06/12 John Harper (Purdue). Localization
and completion of nilpotent structured ring spectra.
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
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
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
11/08 Matthew Thibault (U of Chicago). Finite simplicial complexes via
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
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
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.