Upcoming talks
The seminar will meet at 4:30pm on Mondays in 2-131 unless otherwise noted.
Past seminars
Thursday, 4:00pm. Location: 2-136.
08.20.08:
Mark Behrens (
MIT).
Homotopy fixed points of profinite Galois extensions.
I'll talk about some joint work with Daniel Davis, where we
investigate the homotopy theory of profinite Galois extensions of
E_infty ring spectra, in the sense of Rognes. I'll give some
K(n)-local applications.
08.13.08:
Ricardo Andrade (
MIT).
$E_n$-algebras from cosimplicial objects.
Following McClure and Smith, I will discuss a structure on a cosimplicial object which naturally gives rise to an action of a $E_n$-operad on its totalization.
08.06.08:
Inna Zakharevich (
MIT).
What is a homotopy limit, and why won't it leave me alone?.
I will be presenting several different definitions of a homotopy
limit, and (time permitting) explaining how they all fit together into
the definition in diagrams over a homotopical category.
I will talk about some known and unknown issues related to characteristic class theory for quadratic bundles, taking values in motivic cohomology.
07.23.08:
Martin Frankland (
MIT).
Classification of 2-types.
Here's a basic fact: The homotopy category of 1-types (i.e. connected spaces with only one non-trivial homotopy group in dimension 1) is equivalent to the category of groups. Namely, taking $\pi_1$, and the classifying space give inverse equivalences.
What about 2-types, i.e. if we allow $\pi_2$ to be non-trivial? Then the appropriate algebraic model is given by crossed modules. I will present this classic result that goes back to J.H.C. Whitehead, and explain why I care about it from a more modern point of view.
Time permitting, I will also say a few words about 3-types.
I will describe current work with Hopkins and Ravenel about the homotopy
groups of E_{f(p-1)}^{hG}, where G is a finite subgroup of the Morava
stabilizer group with order divisible by p. To ground the arguments, I will present a complete argument for f=2.
07.09.08:
Sam Isaacson (
Harvard).
Test categories and presheaves as models for homotopy types.
Tuesday, 4pm. Location: 2-146.I will describe the classification of 3-dimensional local topological field theories: corresponding to each dualizable object of a symmetric monoidal 3-category C, there is a 3-d local TFT with target C. The key technical ingredient is the classification of flag foliated singularities through dimension 4. As an application, I will introduce a local field theory corresponding to the conformal net of local fermions. This is joint work in progress with Arthur Bartels and Andre Henriques. (Note: A 3-dimensional local TFT has values for 0-, 1-, 2-, and 3-manifolds, and is sometimes called a 0+1+1+1-dimensional theory, or an extended theory.)
Friday, 4pm. Location: 2-131.We develop a theory of less commutative algebraic geometry where the role of commutative rings
is assumed by E_n-rings, that is, rings with multiplication parametrized by configuration spaces of
points in R^n. As n increases, these theories converge to the derived algebraic geometry of Toën-
Vezzosi and Lurie. The class of spaces obtained by gluing E_n-rings form a geometric counterpart to
E_n-categories, which are higher topological variants of braided monoidal categories. These spaces
further provide a geometric language for the deformation theory of general E_n structures. A version
of the cotangent complex governs such deformation theories, and we relate its values to E_n-Hochschild
cohomology. In the affine case, this establishes a claim made by Kontsevich. Other applications
include a geometric description of higher Drinfeld centers of E_n-categories, explored in work with
Ben-Zvi and Nadler.
Tuesday, 3pm. Location: 1-273.
04.28.08:
Peter Lee (
MIT).
Gröbner bases in rational homotopy theory.
The Mayer-Vietoris sequence in cohomology has an obvious Eckmann-Hilton dual which
characterizes the homotopy of a pullback, but there is no dual to the Eilenberg-Moore
spectral sequence characterizing the homotopy of a pushout. The main obstacle is the
lack of an Eckmann-Hilton dual to the Künneth theorem with which to understand the
homotopy of a coproduct.
This difficulty disappears when working rationally, and we dualize one particular
construction of the Eilenberg-Moore spectral sequence to produce a spectral sequence
converging to the homotopy of a pushout. We use Gröbner-Shirshov bases, an analogue
of Gröbner bases for free Lie algebras, to compute the E^2 term for pushouts of wedges
of spheres. We are able to mostly avoid
the use of minimal models, and instead approach simple but open problems in rational
homotopy theory using the combinatorial properties of Gröbner-Shirshov bases.
This talk will focus on the model categories of stable modules for
the rings Z/(p^2) and (Z/p)[\epsilon]/(\epsilon^2). Schlicting showed
these are not Quillen equivalent by using K-theory. We give another
proof using homotopy endomorphism ring spectra. Our considerations lead to
an example of two differential graded algebras which are derived equivalent but the associated K-theories and the associated model categories of modules are not Quillen equivalent. This is based on joint work with Dan Dugger.
I will discuss joint work in progress with David Nadler, in which we apply homotopical techniques
to study representations of real Lie groups. Our main result recovers the categories of Harish-Chandra modules for real Lie groups completely formally from their much simpler complex counterparts. In particular this allows
us to prove Langlands duality results for real groups by "base-change" from the known results for complex groups.
The Deligne conjecture states that the Hochschild complex of an associative algebra admits the action on the chain level of the little discs operad. McClure-Smith proved a topological version, with the Hochschild complex replaced by the totalization of an operad in based spaces. I will show that a cyclic operad in based spaces yields an action of the framed little discs.
I will present various examples and in particular an application to spaces of embeddings.
03.17.08:
Andrei Radulescu-Banu (
Brixnet).
Cofibration categories and homotopy colimits.
We will talk about homotopy colimits constructed from the axioms of an Anderson-Brown-Cisinski cofibration category, and some of the properties of these homotopy colimits leading up to the axioms of a Heller/Grothendieck derivator.
03.10.08:
Pokman Cheung (
MIT).
Vertex algebras and the Witten genus.
K3 spectra are much like elliptic spectra, just with elliptic curves replaced by K3 surfaces. In this talk, I will give an overview of K3 surfaces, focusing on similarities and differences between them and elliptic curves, and of the kind of spectra they are related to.
Derivators were introduced by Grothendieck in pursuing stacks and studied (sometimes indepently, and/or under various forms) later by A. Heller, J. Franke, B. Keller. Derivators describe the structure of homotopy categories in terms of homotopy limits and colimits. This talk will give an introduction to the subject. Illustrations will be given about the following fact: the derivator defined by classical homotopy theory of CW-complexes can be defined by an intrinsic universal property, giving rise to a nice theory of homotopy Kan extensions in derivators. This is the corner stone to understand the link between derivators and higher homotopical structures.
Algebraic K-theory of a ring encodes the "higher homotopy theory" of the
derived category. I will discuss recent work with Mandell which makes this
slogan precise by giving an explicit description of the K-theory space of a
Waldhausen category in terms of the Dwyer-Kan simplicial localization of the
category. This allows us to prove a very general criterion for functors to
induce an equivalence of K-theory, and also has applications in the study of
the K-theory of nonconnective ring spectra.
I'll explain work (joint with Greg Arone) to understand chain rules for Goodwillie's homotopy calculus. In order to express these chain rules, we produce nice models for the derivatives of various functors that have extra algebraic structure. In particular, we get a deeper understanding of where the operad structure on the derivatives of the identity comes from. These models should work in a much wider context than functors of topological spaces, and if there's time and interest, I'll mention some applications of these ideas to the algebraic K-theory of A∞ and E∞ ring spectra.
In this talk we will review a result of Bruno Vallette linking the notion
of Koszul duality of operads and Cohen-MacCauley posets. We'll present in
this context a joint work with F. Chapoton, where we compare two groups,
one built directly from operads, and another one associated to the
incidence Hopf algebra of a family of posets. This leads us to a new link
between the Hopf algebra of Connes and Kreimer in renormalisation theory
and operads built on rooted trees.
Babytop time and space (Tuesday, 4:00, room 2-135)Let G be a connected, semi-simple compact Lie group. In
this talk I will describe a stability theorem for the homology of the
moduli space, (Mg)G, of equivalence classes of triples (S, E,
ω), where S is a closed Riemann surface of genus g, E→S is
a principal G-bundle, and ω is a flat connection on E. The
theorem states that in a stable range (approximately g/2), the
homology of this moduli space is isomorphic to the homology of an
explicit infinite loop space. As a corollary, one has that in this
stable range, the rational cohomology of this moduli space is
generated by the Mumford-Morita kappa classes and H*(BG). When G
is the trivial group, this is a combination of the Mumford conjecture
as proved by Madsen and Weiss, and Harer's stability theorem. I
will also draw implications to moduli spaces of semi-stable
holomorphic bundles, and to representation varieties. Finally I
will describe a related result, identifying the homotopy type of the
cobordism category of surfaces with flat G-connections. This is
all joint work with S. Galatius and N. Kitchloo.
Classical obstruction theory seemingly produces uncountably many
A∞ structures on the Morava K-theory spectrum K(n). We show
that these A∞ structures are all equivalent, using a
Bousfield-Kan spectral sequence converging to the homotopy groups of
the moduli space of A∞ ring spectra equivalent to K(n). This
spectral sequence has infinitely many differentials, and to show that
all the relevant classes die we study the connective Morava K-theory
spectrum k(n) and use the theory of Postnikov towers and S-algebra
k-invariants developed by Dugger and Shipley.
The category of spectra can be thought of as the tangent
space of the category of spaces at the one-point space. I will extend
this point of view as far as I can, perhaps getting as far as jets,
connections and linear differential operators. It is no use forcing
the analogy, but it can be a useful organizing framework and also a
lot of fun.
Let G be a finite group and let k be a field whose characteristic
p divides the order of G. Freyd's generating hypothesis for the
stable module category of G is the statement that a map between
finite-dimensional kG-modules in the thick subcategory generated by
k factors through a projective if the induced map on Tate cohomology
is trivial. I will give an overview of joint work with Sunil Chebolu and
Ján Mináč in which we show that for groups with periodic
cohomology, the generating hypothesis holds if and only if the Sylow
p-subgroup of G is C2 or C3.
I will spend the last few minutes demonstrating GAP software written
by Peter Webb and myself which can be used to explore conjectures such
as the generating hypothesis.
This will be a leisurely exposition of different ways
of looking at at some E∞ spaces and E∞
ring spectra from a variety of perspectives, leading up
to a modern codification of classical structures in the
form of parametrized functors with smash product (PFSP's)
and their associated parametrized E∞ ring spectra.
10.22.07:
Grace Lyo (
MIT).
The algebraic K-theory of a characteristic p local field.
I will describe a conjecture of Carlsson's that provides an
explicit model of the homotopy type of the completed algebraic K-theory
spectrum (KF)p of an arbitrary field F. This is achieved using
exclusively the semilinear representation theory of the absolute Galois
group GF. Unlike other approaches which focus on the homotopy groups of
(KF)p, Carlsson's approach addresses the entire homotopy type. The
conjecture is known to hold in two special cases; in this talk I will
outline the proof of the most recently established case.
10.15.07:
Jacob Lurie (
MIT).
Moduli Problems for Ring Spectra.
A result of Hinich states that if x is a point of any "moduli space" X
defined over a field of characteristic zero, then a formal
neighborhood of x in X can be described by a differential graded Lie
algebra. In this talk I will explain what the above statement means,
and describe some extensions to other contexts (such as fields of
positive characteristic and moduli problems in noncommutative
geometry). At the end I will sketch a connection with some recent work
with Dennis Gaitsgory.
The notion of a Hall algebra has been a useful tool in
representation theory and has been shown to make important connections
between the category of representations and the quantum group associated
to certain Lie algebras. As a first step in strengthening this
relationship, Toen defines the derived Hall algebra associated to a
particular kind of stable model category. We'd like to extend his work to
more general stable homotopy theories, using the complete Segal space
model for homotopy theories. This talk will include background on Hall
algebras, a summary of Toen's work, and recent progress on extending to
this more general situation.
Seminar will occur in room 2-142 for this week only.This talk will describe algebraic K-theoretic obstructions to lifting fibrations to fiber bundles having compact smooth/topological manifold fibers. The surprise will be that a lift can often be found in the topological case. Examples will be given realizing the obstructions.
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