Upcoming talks
The seminar will meet at 4:30pm on Mondays in 2-131 unless otherwise noted.
Past seminars
This talk is to be held at 4pm in room 2-151.In this talk I will present recent work, joint with Radu Stancu, in which we obtain a bijection between saturated fusion systems on a finite p-group S and idempotents in the double Burnside ring of S satisfying a "Frobenius reciprocity relation". (These terms will all be defined in the talk.) The theorem and its proof are purely algebraic, so I will focus attention on implications in algebraic topology, answering long-standing questions on the stable splitting of classifying space and generalizing a variant of the Adams-Wilderson theorem, as well as the obvious implications for p-local finite groups.
This talk will begin at 5:00. Please note the time change.
I will discuss connections between the calculus of functors
and the Whitehead Conjecture, both for the classical theorem of Kuhn
and Priddy for symmetric powers of spheres and for the analogous
conjecture in topological K-theory. It turns out that key
constructions in Kuhn and Priddy's proof have bu-analogues, and there
is a surprising connection to the stable rank filtration of algebraic
K-theory.
11.17.08:
Bertrand Guillou (
UIUC).
Enriched and equivariant homotopy theory.
I will describe some joint work with J.P. May in which we investigate when enriched model categories can be modeled as enriched diagrams on a small (enriched) domain category. As an application, we are able to obtain a new model for the equivariant stable homotopy category of a compact Lie group.
This talk is to be held at 4pm in room 2-142.Carefully developing the homology and cohomology of ordered configuration spaces leads to a pretty model for the Lie cooperad. We use this model to unify the Quillen approach to rational homotopy theory with the theory of Hopf invariants. We will also share progress on a new approach to the cohomology of unordered configurations spaces (i.e. symmetric groups), which are of course relevant to homotopy theory at p.
The symmetric groups S_p are considered with the norm induced by the word
length (with respect to transpositions as generators). This gives a
filtration of their classifying spaces. Furthermore, using certain
deletion functions S_p ---> S_{p-1} the family of all symmetric groups can
be regarded as filtered simplicial object. we show: in its realization,
the stratum for norm equal to h has several components, each being
homoemorhic to a vector bundle over the moduli space M_g,_1^m
of genus g surfaces with one boundary curve and m punctures (for
h =3D 2g + m).
10.27.08:
Matthew Ando (
UIUC).
Parametrized spectra, Thom spectra, and twisted Umkehr maps.
Let R be an associative ring spectrum. I shall describe several new
constructions of the R-module Thom spectrum associated to a map f: X
-----> BGL_1 R. The space BGL_1 R classifies the twists of R-theory,
and to a fibration of manifolds g: Y -----> X I shall associated an
Umkehr map g_! from the fg-twisted R-theory of Y to the f-twisted R-
theory of X. In the case of K-theory, this twisted Umkehr map appears
in the study of D-brane charge. I shall review this story, and then
discuss the analogous construction for TMF.
In joint work with Keir Lockridge, we have been developing theories of
global and weak dimensions for ring spectra. We have good results for
ring spectra of dimension zero, and partial results but good
conjectures for the finite dimensional case.
10.15.08: Larry Smith (Georg-August-Universität Göttingen). Local cohomology, Poincare duality algebras, and Macaulay dual systems.
This is held in room 2-142!
The cohomology jumping loci of a space X come in two basic flavors: the characteristic varieties (the jump loci for cohomology with coefficients in rank 1 local systems), and the resonance varieties (the jump loci for the homology of the cochain complexes arising from multiplication by degree 1 classes in the cohomology ring of X). I will discuss various ways in which the geometry of these varieties is related to the formality, quasi-projectivity, and homological finiteness propoerties of the fundamental group of X.
We will describe how (multivariable) manifold calculus of
functors can be used for studying classical knots and links. In
particular, this theory yields a classification of finite type invariants
and Milnor invariants of knots, links, homotopy links, and braids.
Another novelty is that a certain cosimplicial variant of manifold
calculus provides a way for studying knots and links in a
homotopy-theoretic framework. Higher-dimensional analogs will also be
discussed. This is joint work with Brian Munson.
09.22.08: Student Holiday.
The moduli space of Riemann surfaces M is a classifying space for families of Riemann surfaces. It has a compactification Mbar, which is a classfying space for families of modal Riemann surfaces. A nodal Riemann surface is allowed to have singularities which look like the solutions to zw=0 in complex 2-space. I will describe how to decompose Mbar as a homotopy colimit of spaces which look more like M. Then I will use this to study part of the homology of Mbar, using what is known about the homology of M.
On every bimonoidal category with anti-involution, R, there is an
involution on the associated K-theory. This K-theory is the algebraic
K-theory of the spectrum associated to R. In the talk I will construct
this involution, discuss examples and indicate why the involution is
non-trivial in several examples.
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