Upcoming talks
The seminar will meet at 4:30pm in 2-132 unless otherwise noted.
Past seminars
A sequence of groups Gn (think
symmetric groups or GLnZ) exhibits homology
stability if the i-th homology group is independent of n for
large n. The usual approach goes through a sequence of
complexes with actions of the groups. I present a version
that isolates the conditions on the groups from the condition
on the complexes. In particular, I explain why stability is
slower for the alternating groups than the symmetric groups,
even though they use the same complexes.
05.08.07:
Boris Chorny (
ANU).
Brown representability for space-valued functors.
The seminar will be held at 3:00 on Tuesday,
May 8 in 12-102.
In this talk we will discuss two theorems which resemble the
classical cohomological and homological Brown
representability theorems. The main difference is that our
results classify small contravariant functors from spaces to
spaces up to weak equivalence of functors.
In more detail, we will show that every small contravariant
functor from spaces to spaces converting coproducts to
products, up to homotopy, and taking homotopy pushouts to
homotopy pullbacks is naturally weekly equivalent to a
representable functor. This theorem may be considered as a
contravariant analog of Goodwillie's classification of linear
functors, see his [Calculus II, III] papers. The
interpretation of the current result in terms of Homotopy
Calculus is still a challenge.
Homological representability theorem states: every
contravariant continuous functor from the category of finite
simplicial sets to simplicial sets is equivalent to a
restriction of a representable functor. This theorem is
essentially equivalent to Goodwillie's classification of
linear functors.
05.07.07:
Tyler Lawson (
MIT).
Topological Hochschild homology of ku and ko.
I will discuss joint work with Vigleik
Angeltveit and Mike Hill
computing the topological Hochschild homology of the
connective real and complex K-theory spectra. I'll start
with the initial work by Bokstedt and McClure-Staffeldt and
show how some of these computations are simplified by modern
machinery. I will then indicate—without actually doing
any computations—how the computation of THH(ku)
(resp. THH(ko)) can proceed from this knowledge by obtaining
knowledge from four (resp. seven) spectral sequences, after
giving some large-scale indications of the results.
Quasi-categories were introduced by Boardman and Vogt, then
more recently named and developed by Joyal, Lurie, and
others. The model theory of quasi-categories can be thought
of as a model of the homotopy theory of homotopy theories,
alongside e.g. complete Segal spaces and simplicially
enriched categories. Quasi-categories have the advantage,
however, of functioning theoretically much like ordinary
categories, as well as being functionally extremely simple
(they are just simplicial sets satisfying the inner
horn-filling condition).
It is the goal of this talk to discuss one such similarity,
namely the notion of quasi-category fibred in Kan complexes
(what one might call quasi-category fibred in
quasi-groupoids, and what in practice is known as a right
fibration). Not only do these objects generalize
Grothendieck's categories fibred in groupoids, the
quasi-category of right fibrations over a given base
simplicial set S can be constructed as having n-simplices
being all right fibrations over S×Δn.
Thus, the quasi-cateogry of right fibrations can be thought
of as a quasi-cateogry which arises “naturally”
(as opposed to as the coherent nerve of a category enriched
in Kan complexes, for example). Constructing the
quasi-category this way also means this quasi-category
classifies families of right fibrations up to
isomorphism (not just homotopy equivalence).
If there is time, we will also discuss how (homotopy) limits
and the Yoneda functor can be constructed directly using this
formalism.
We will discuss the notion of a string structure
on a manifold, which induces a spin structure on the free
loop space. Using a Hodge theoretic version of the Serre
spectral sequence (developed by Mazzeo-Melrose), we see that
a string structure and metric together give a canonical
3-form which trivializes the p1/2 form.
While computing algebraic K-theory groups is
often very difficult, understanding fixed point spectra of
topological Hochschild homology can aid in such
computations.Taking homotopy groups of these spectra, we
arrive at TR-groups, an integer-graded theory with a rigid
algebraic structure. By understanding TR-groups, one can
compute topological cyclic homology, and consequently,
algebraic K-theory. This homotopy theoretic approach to
algebraic K-theory has been very fruitful.
For many algebraic K-theory computations, however, it is
beneficial to further exploit the S1-equivariant
structure of topological Hochschild homology. The topological
Hochschild S1-spectrum has naturally associated
equivariant homotopy groups graded by the real representation
ring of S1, which provide an
RO(S1)-graded TR-theory.
In this talk we will review classical TR-theory, and
introduce the RO(S1)-graded analog and its
applications to algebraic K-theory. The main result will be
the explicit computation of the RO(S1)-graded
equivariant homotopy of THH(Fp). This computation
of TRnα(Fp;p) extends
a result of Hesselholt and Madsen to provide the first
complete computation of RO(S1)-graded
TR-groups. In particular, we compute the groups
TRnq+α(Fp;p) for q
even, and the order of these groups for q odd.
We review the contruction of M. Hopkins and G. Laures of K(1)-local tmf
as a finite E∞-cell algebra at the primes 2
and 3. Then we discuss some problems of generalizing this
construction to arbitrary primes. The seminar will be held at 3:00 on Tuesday,
Mar. 6 in 2-132. Let M be an S2 bundle over
S2. I will explain why the space of complex
structures on M compatible with a symplectic form is
contractible. This implies that the classifying space of a
symplectomorphism group Diff(M,ω) is the homotopy type
associated to an open substack of the stack of complex
structures on M. The latter statement allows for a fairly
good understanding of the homotopy type of
BDiff(M,ω). This is joint work with Miguel Abreu
and Nitu
Kitchloo.
The seminar will be held at 4:00 on Tuesday,
Mar. 6 in 2-131.Let M be a compact, connected 3-manifold with a
fixed boundary sphere d0M. For each prime manifold
P, we consider the mapping class group of the manifold
MPn obtained from M by taking a
connected sum with n copies of P. We prove that the
ith homology of this mapping class group is
independent of n in the range n>2i+1. Our theorem moreover
applies to certain subgroups of the mapping class group and
include, as special cases, homological stability for the
automorphism groups of free groups and of other free
products, for the symmetric groups and for wreath products
with symmetric groups. This is joint work with Allen
Hatcher. The seminar will be held at 3:00 in 2-146.In this talk I will show how one can sometimes
“uncomplete” the p-completed classifying space of
a finite group, to obtain the original (non-completed)
classifying space, and hence the original finite group. This
“uncompletion” process is closely related to
well-known local-to-global questions in group theory, such as
the classification of finite simple groups. The approach goes
via the theory of p-local finite groups. This talk is a
report on joint work with Bob
Oliver. 02.26.07:
André Joyal (
UQÀM).
On direct Quillen equivalences between quasi-categories, simplicial categories, Segal categories and complete Segal spaces.
Julia Bergner has established a chain of Quillen
equivalences between simplicial categories, Segal categories
and complete Segal spaces. We have added quasi-categories at
the end of the chain. Some of these equivalences have
opposite directions. We shall describe a direct Quillen
equivalence for each of the six pairs of model categories.
02.12.07:
Sunil Chebolu (
UWO).
Freyd's Generating Hypothesis in modular representation theory.
The Generating Hypothesis (GH) of Peter Freyd is
the conjecture that there are no non-trivial maps between finite
spectra which are zero on all stable homotopy groups. Despite serious
efforts of homotopy theorists over the last 40 years the conjecture
remains open; only partial affirmative results are known due to
Devinatz. In order to gain some insight into this deep problem it is
natural to investigate its analogues in some algebraic
settings.
In this talk, I will formulate the analogue of the GH in the stable
module category of a finite p-group. I will show that the only
p-groups for which this version of the GH holds are the cyclic group
of order 2 and 3. For the other groups, I will introduce an integer
invariant which measures the degree of the failure of the GH, and will
show to compute this invariant in some examples.
This is joint work with Dave Benson, Dan Christensen and Ján
Minác. Working over an arbitrary field k, I will
construct a map from the algebraic K-theory of k into the motivic
stable homotopy groups over k. This map is analogous to the complex
J-homomorphism of classical homotopy theory. It allows us to
construct a non-zero element of the motivic stable homotopy group in
dimension (4k-1,2k) for every k>1. I will also describe explicit
geometric models for non-zero elements in dimensions (1,1), (3,2), and
(7,4); these are analogous to the classical Hopf maps.
Motivated by the program of Waldhausen and Rognes
for obtaining a conceptual understanding of the K-theory of the sphere
spectrum via a chromatic filtration, Ausoni and Rognes computed
THH(l) (following McClure and Staffeldt) and THH(ku).
Hesselholt observed that the answers they obtained are related, and
moreover that this relationship is consistent with a conjectural
interpretation of the map l → ku as a “tamely
ramified” extension of commutative ring spectra. In this talk,
I will discuss ongoing work with Michael Mandell to verify this
picture.
11.29.06:
Nora Ganter (
UIUC).
Stringy power operations in Tate K-theory and Hecke operators in generalized Moonshine.
The seminar will be held at 4:00 on Wednesday,
Nov. 29 in 4-257.I will start with a short introduction to
Norton's generalized Moonshine conjecture and lay out how the geometry
of equivariant elliptic cohomology naturally gives rise to
Moonshine-like objects and to the Moonshine notion of "twisted Hecke
operators". This discussion results in a formula describing the
effect of Hecke operators on Hopkins-Kuhn-Ravenel characters.
Here I switch subjects and outline how, in the light of my thesis, the
work of Dijkgraaf, Moore, Verlinde and Verlinde dictates a definition
of power operations in Devoto's equivariant Tate K-theory (making the
Witten genus an H∞ map) and how the work of Lupercio,
Uribe and Xicotencatl provides a motivation for these definitions in
terms of the theory of orbifold loop spaces.
The resulting Hecke operators are described by the same formula as the
twisted Hecke operators in generalized Moonshine. Further, they agree
with Matthew Ando's definition of Hecke operators on KTate,
making our power operations 'elliptic' in the sense of [Ando, Hopkins,
Strickland II] and putting the fact that the Witten genus is an
H∞ map into the same framework as the
Ando-Hopkins-Strickland result that the sigma-orientation of
E2 is an H∞-map.
I will explain how equivariant transfer maps can
be used to define field-theory type operations in several contexts in
string topology. Specific examples include the string topology of
classifying spaces, and the S1-equivariant operations
defined by Chas and Sullivan.
Many triangulated categories arise from chain
complexes in an abelian category by passing to chain homotopy classes
or inverting quasi-isomorphisms. Such examples are called
‘algebraic’ because they have underlying abelian (or at
least additive) categories. Stable homotopy theory produces examples
of triangulated categories by quite different means, and in this
context the underlying categories are usually very
‘non-additive’ before passing to homotopy classes of
morphisms. Because of their origin I refer to these examples as
‘topological triangulated categories’ (not to be confused
with topologically enriched categories).
In this talk I want to explain some systematic differences between
these two kinds of triangulated categories. There are certain
properties—defined entirely in terms of the triangulated
structure—which hold in all algebraic examples, but typically
fail in topological examples. These differences are all torsion
phenomena, and rationally there is no difference between algebraic and
topological triangulated categories.
11.06.06:
Pokman Cheung (
MIT).
Categories of field theories and generalized cohomology.
For each integer n, a small category
SEFTn is constructed. The objects of SEFTn are
certain 1-dimensional supersymmetric quantum field theories. The
classifying space of SEFTn has the homotopy type of the
n-th space in the periodic K- (or KO-)theory spectrum. The A-head
genus can be described as functors Cn →
SEFTn, where each Cn is a small category whose
classifying space is MSpinn. One can also define categories
AFTn whose objects are certain 2-dimensional field
theories, such that their classifying spaces form the elliptic
spectrum associated to the Tate curve. My goal is to define categories
whose objects are 2-dimensional conformal field theories and compare
their classifying spaces with tmf.
10.30.06:
Veronique Godin (
Harvard).
String operations parameterized by the homology of the mapping class groups.
Chas and Sullivan defined a product on the
homology of the loop space LM of an oriented manifold. I will discuss
how this product is part of a much bigger structure. More precisely,
the entire homology of the moduli space of bordered Riemann surfaces
parameterizes operations on the homology of LM.
I will describe how the rational homology of a
space closely related to the space of embeddings of a manifold in a
Euclidean space can be studied using orthogonal calculus of functors.
In particular, under appropriate dimensional assumptions, the
orthogonal calculus tower for this space splits into the product of
its layers. Equivalently, the rational homology spectral sequence
associated to this tower collapses at E1. One consquence
is that the rational homology groups of this space of embeddings are
determined by the rational homotopy type of the manifold. The main
tools in the proofs are embedding calculus of functors and
Kontsevich's formality of the little balls operad.
When G is a finite group and X is a G-spectrum,
the homotopy orbit spectrum and the associated homotopy orbit spectral
sequence are familiar constructions in homotopy theory. When G is
profinite and for certain types of G-spectra, we give a construction
that we call the homotopy orbit spectrum. Using this construction,
when G is countably based, for certain G-spectra, we are able to
construct a homotopy orbit spectral sequence whose E2-term
is continuous group homology. This yields a homotopy orbit spectral
sequence for the Gn-homotopy orbits of the K(n)-local
Spanier-Whitehead dual of the Lubin-Tate spectrum, where Gn
is the extended Morava stabilizer group.
Given any compact ANR fibration, Becker and
Shultz have given an axiomatic characterization of the Becker-Gottlieb
transfer associated to it. In joint work with Wojciech
Dorabiała, we have verified these axioms for the composite of
the assembly map followed by Waldhausen's A-theory transfer and then
the trace map. In particular, this is related to an elegant result of
Dwyer, Weiss, and Williams, who show that the assembly map forms a
natural transformation from the Becker-Gottlieb transfer to the
A-theory transfer for a smooth fiber bundle. We will start with an
introduction of the cast of characters, followed by an overview of our
proof. I will describe an attempt at understanding the
Deligne-Mumford compactification of the moduli space of genus g curves
from a homotopy theoretical point of view. This is joint work with
Ya. Eliashberg.
The seminar will be held at 4:30 on Thursday, Sept. 7 in
2-135.I will describe an extension of the category of simplicial
sets, to be called that of "dendroidal" sets. I will explain how some
notions for simplicial sets extend to the dendroidal context, and in
particular explain the comparison of ratios categories: simplicial
sets = operads : dendroidal sets.
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