Upcoming talks
The seminar will meet at 4:30pm on Mondays in 2-131 unless otherwise noted.
This talk will take place at the normal time (4:30) in 4-270.
Past seminars
This will be a Tuesday talk, at 4:00 in 2-151.In this talk we will show that the space of almost commuting
elements in a compact Lie group splits after one suspension.
Given a smooth manifold M and two submanifolds A and B, their intersection need not be a smooth manifold. By Thom's transversality theorem, one can deform A to be transverse to B and take the intersection: the result, written A
B, will be a smooth manifold. Moreover, if A and B are compact, then there is a cup product formula in cobordism, integral cohomology, etc. of the form [A]∪[B]=[AB], where [-] denotes the cohomology fundamental class.
The problem is that AB is not unique, and there is no functorial way to choose transverse intersections for pairs of submanifolds. The goal of the theory of derived manifolds is to correct this defect.
The category of derived manifolds contains the category of manifolds as a
full subcategory, is closed under taking intersections of manifolds, and yet
has enough structure that every compact derived manifold has a fundamental
class. Even if the submanifolds A and B of M are not transverse (in which
case their intersection can be arbitrarily singular), their intersection
AxMB will be a derived manifold with [AxMB]=
[AB], and thus satisfy the above cup
product formula.
To construct the category of derived manifolds, one imitates the constructions
of schemes, but in a smooth and homotopical way. I will begin the talk by
explaining this construction. Then I will give some examples and discuss some
features of the category of derived manifolds. I will end by sketching the
Thom-Pontrjagin argument which implies that compact derived manifolds have
fundamental classes. 04.27.09:
Stefan Hornet (
Harvard University).
A generalization of a theorem of Ravenel and Wilson.
This will be a Wednesday talk, at 4:00 in 2-143.A second subspace of a product is the generalized moment-angle complex first defined in generality by Neil Strickland. Definitions, examples, as well as connections will be addressed.
One notable case is given by subspaces of products of infinite dimensional complex projective space 'indexed by a finite simplicial complex'. These spaces appearing in work of Goresky-MacPherson, Davis-Januskiewicz, Buchstaber-Panov-Ray, Denham-Suciu, Franz as well as many others encode information ranging from the structure of toric varieties in one guise, Stanley-Reisner rings, as well as 'motions of certain types of robotic legs' in another guise.
What do these spaces have to do with the motions of legs of a cockroach? This feature will be illustrated with slides.
Features of these spaces are developed within the context of classical homotopy theory based on joint work with A. Bahri, M. Bendersky, and S. Gitler.
This will be a Tuesday talk, at 4:00 in 2-151.The first basic example here is the configuration space of unordered k-tuples of distinct points in a space M. When specialized to the case of M given by the complex numbers, these spaces can be identified as the space of classical complex, monic polynomials of degree k which have exactly k distinct roots.
Elementary features of these spaces as well as their connections to spaces of knots, links, and homotopy groups of spheres will be addressed. These topics are based on joint work with R. Budney as well as J. Berrick, Y. Wong and J. Wu.
The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group G as well as quotients spaces given by the associated space of representations. These spaces as well as further spaces of representations admit the structure of a simplicial space at the heart of the work here. Features of geometric realizations will be developed.
What is the fundamental group or the first homology group of the associated space in case G is a finite, discrete group?
This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.
I will describe the stable (in genus) structure of the universal moduli space of flat connections on riemann surfaces. I will also introduce the category of 1-manifolds and 2-cobordisms endowed with flat connections. Using classical techniques of Atiyah-Bott, and more recent techniques introduced by Madsen-Weiss and coauthors, we will give a complete description of the classifying space of this category. This is joint work with R. Cohen and S. Galatius.
03.30.09:
Sam Isaacson (
Harvard Univeristy).
Cubical homotopy theory and monoidal model categories.
Let C be a model category. In a 2001 paper, Dan Dugger showed that if C is combinatorial, it can be realized as a left Bousfield
localization of simplicial presheaves on some small site. I'll
describe a variation of this theorem: by replacing simplicial sets
with a cubical model for the homotopy category, we can produce a
presentation for C when C is symmetric monoidal that retains the
monoidal structure of C as the Day convolution product.
This will be a Tuesday talk, at 4:00 in 2-151.For a given compact smooth manifold M we consider the space
Emb(M,Rk) of smooth embeddings of M into some large Euclidean space
Rk, or rather some geometric variant of it, which is a homotopy
invariant of M.
I will explain how Goodwillie's cutting method enables us
to understand the homotopy type of this space of emeddings.
I will then prove that the rational homology of that space is actually
an invariant of the rational homotopy type of M. The proof is based
on Kontsevich's theorem on the formality of the little cube operad
and Arone's description of the layers of Weiss' orthogonal tower for
the space of embeddings.
This is a joint work with Greg Arone and Ismar Volic.
In Haynes Miller's proof of the Sullivan conjecture on maps from classifying
spaces, Quillen's derived functor notion of homology (in the case of
commutative algebras) is a critical ingredient. This suggests that homology
for the larger class of algebraic structures parametrized by an operad will
also provide interesting and useful invariants. Working in the two contexts
of symmetric spectra and unbounded chain complexes, we establish a homotopy
theory for studying Quillen homology of modules and algebras over operads,
and we show that this homology can be calculated using simplicial bar
constructions. A key part of the argument is proving that the forgetful
functor commutes with certain homotopy colimits. A larger goal is to
determine the extra structure that appears on the derived homology and the
extent to which the original object can be recovered from its homology when
this extra structure is taken into account. This talk is an introduction to
these results with an emphasis on several of the motivating ideas.
Let R be an E∞ ring spectrum. Given a map f:X -> BGL1(R),
we can construct a Thom spectrum Xf. If f is a loop map, then there is an
A∞ R module structure on the Thom spectrum. I will consider various
examples of these Thom spectra and construct A∞ structures on them. I
will then use this identification to calculate Topological Hochschild
Homology.
For smooth manifolds P, Q, and N, let Link(P,Q;N) denote the space of smooth maps of P in N and Q in N such that their images are disjoint. I will discuss the connectivity of a "generalized linking number" from the homotopy fiber of the inclusion of Link(P,Q;N) into Map(P,N)xMap(Q,N) to a certain cobordism space of manifolds over a space which is a homotopy theoretic model for the intersections of P and Q. The proof of the connectivity uses some easy statements about connectivities in the world of smooth manifolds as a guide for obtaining similar estimates in a setting where the tools of differential topolgy do not apply. This is joint work with Tom Goodwillie.
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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