12.06.11:
Olga Stroilova (
MIT).
Level structures and automorphisms.
The generalized character map of HKR can be rephrased as saying
that the connected p-divisible group corresponding to the universal
formal group law over Morava E_n becomes constant over an appropriate
extension, L, of E_n. Here L is a colimit of inverted Drinfel'd rings
of level structures.
This ring L remembers E_n: it is faithfully flat over p^{-1}
E_n; furthermore, p^{-1} E_n can be recovered from L by takinginvariants with respect to a naturally occurring group action.
I will talk about this story and ask how it might be generalized to an
intermediate setting.
11.29.11:
Luis Pereira (
MIT).
Koszul duality of E_n operads.
One remarkable property of the $E_n$ operads is their apparent self-duality: it would seem that the Koszul dual of the $E_n$ operad is a $-n$ shifted version of itself. This is first suggested by examining the corresponding homology operad, which looks like an amalgamation of the commutative operad with the ($n-1$ shifted) Lie operad. A first proof at this (homology) level was first found by Getzler and Jones, with a much more recent proof at the chain complex level having been found by Fresse. A full proof at the most general level (Spaces/Spectra) seems however not to have yet been found.
In this talk we will, after defining the relevant concepts, discuss Getzler and Jones proof, which uses interesting compact models for the $E_n$ operads known as the Fulton McPherson operads, and, time permiting (and contigent on the speakers ability to both understand and be enlightning about them), the main ideas behind Fresse's more recent result.
11.15.11:
Dustin Clausen (
MIT).
The K(1)-local logarithm.
Sometimes a commutative ring R carries a ``logarithm'' map from the units
in R to the underlying additive group of R. In the case we'll be concerned
with, R is a K(1)-local commutative ring spectrum, and the existence of
such a logarithm follows abstractly from the Bousfield-Kuhn functor. A
priori this log is pretty opaque from a practical standpoint, but Rezk
managed to find a formula for it as an infinite series. This talk is about
Rezk's formula. Or I guess you could say it's about how Bott periodicity
manages to magically produce the p-adic logarithm and other interesting
series besides.
11.08.11:
Saul Glasman (
MIT).
Crystalline Cohomology.
Crystalline cohomology is the first robust headspace in which one can understand
the p-torsion in the cohomology of a variety over a field of characteristic p,
patching a notorious puncture in the great bicycle tyre that is Weil cohomology.
I'll begin by laying out a manifesto which states what we want to achieve, and
I'll go on to sketch an achievement of it, whose vivid chapters include the
functoriality of the crystalline topos and the isomorphism with de Rham
cohomology. If permitted by (i) time and (ii) the quantity of knowledge I can
guzzle in the next twenty-four hours, I'll serve a portion of the magnificent de
Rham-Witt complex, which explicitly computes crystalline cohomology.
11.01.11:
Rune
Haugseng (
MIT).
A Spectral Sequence for the Cohomology of an Infinite Loop
Space.
Taking the infinite loop spaces of the mod-2
Adams tower of a
spectrum gives a spectral sequence converging to the cohomology of the
infinite loop space of the spectrum, whose E_2-term can be identified with
certain algebraic derived functors. I'll explain how to set this up, then
say something about the computation of these derived functors.
10.25.11:
Tomer Schlank (
Hebrew University, Jerusalem).
A Projective Model Structure on Pro-categories , and the Relative
\'Etale Homotopy Type.
Isaksen showed that a proper model category
$C$, induces a model structure
on the pro-category $Pro(C)$.
In this talk I will present a new method for defining a model structure on
the pro-category $Pro(C)$. This method requires $C$ to satisfy a much
weaker condition then having a model structure. The main application will be
a novel model structure on pro-simplicial sheaves. We see that in this
model structure a "topological lift" of Artin and Mazur's \'Etale homotopy
type is naturally obtained as an application of some natural derived functor
to the terminal object of the \'etale topos. This definition can be
naturally generalized to a relative setting, namely- given a map of topoi T
\to S, we get a notion of a relative homtopy type of T over S which is a
Pro-simplicial object in S.
This definition turns out to be useful for the study of rational points on
algebraic varieties.
This is a joint work with Ilan Barnea
10.18.11:
Geoffroy
Horel (
MIT).
Topological Hochschild Homology.
In this talk we will give an overview
of THH (Topological Hochschild Homology) which is the analogue in the category
of spectra of good-old Hochschild homology for associative algebra over field.
We will give two different construction of THH. The first one through the cyclic
bar construction has the advantage of being a straightforward generalization of
the algebraic version. The second one through factorization homology is more
interesting as it describes THH as an example inside a large family of
constructions indexed by framed manifolds.
Finally if time permits we will introduce the Bockstedt spectral sequence and
make an explicit computation of THH(KU).
