Upcoming talks
The seminar will meet at 4:30pm on Mondays in 2-132 unless otherwise noted.
Past seminars
Ando constructed power operations for the Lubin-Tate cohomology
theories using the theory of finite subgroups of a formal group.
Moreover, he was able to produce a necessary and sufficient condition
for a complex orientation of these cohomology theories to be
compatible with the power operations. This result concerns the stable
homotopy category of spectra.
However, the Lubin-Tate spectra of Morava are very rigid objects.
Using ideas of Ando, Hopkins and Rezk, we can classify those
orientations of complex K-Theory that are compatible with Ando's power
operations, but on the point set level.
In this talk, we will show the equivalence of these two descriptions
for complex p-adic K-Theory. To achieve this goal, we use the
language of Bernoulli numbers attached to a formal group law and their
relationship with distributions on a p-adic Lie group. Many of these
tools were developed by N. Katz and J. Tate.
11.24.09:
Charles Rezk (
UIUC).
Koszul resolutions for algebras of power operations.
This will be a Tuesday talk, at 4pm in 2-132. Morava E-theory (the complex oriented cohomology theories
associated to deformations of formal groups) are structured commutative
ring spectra, and so support a well-behaved theory of power operations.
We describe what is know about this theory, and we prove a conjecture of
Ando, Hopkins, and Strickland, that the ring of power operations for
such theories is Koszul.
Costenoble and Waner showed that grouplike equivariant E_infty-
spaces model equivariant infinite loop spaces. Shimakawa gave an
equivariant analog of Gamma-spaces to model equivariant infinite loop
spaces.
We describe equivariant Gamma spaces as defined by Shimakawa. We show
that the categories of equivariant E_infty-spaces and equivariant
Gamma-spaces are Quillen equivalent with appropriate model categories.
Following Segal's work, we give a construction of equivariant Gamma-
spaces (and hence of equivariant infinite loop spaces) from symmetric
monoidal G-categories for finite group G.
The (stable) chromatic spectral sequence has had a significant impact on
our understanding of the stable homotopy groups of the spheres. I will
talk about preliminary attempts to construct an unstable version. I will
try to describe a filtration of the stable chromatic spectral sequence
induced by the Hopf rings for the odd spheres. There are natural
questions that arise in the unstable world (e.g. an unstable version of
the Morava stabilizer algebra) and a chromatic interpretation of the Hopf
invariant.
Lurie's theorem allows the functorial construction of
E_infty ring spectra associated to certain p-divisible groups.
In this talk I will discuss three situations in which we can apply
this and attempts to understand the computational results. The first
is joint work with Behrens on the relationship between the moduli of
elliptic curves and certain moduli of abelian surfaces with complex
multiplication. The second is joint work with Hill on Shimura curves
that parametrize "false elliptic curves", and in particular trying to
obtain computations of the homotopy of the associated spectra without
niceties such as q-expansions and Weierstrass equations. The third is
on using Zink's work on displays to produce E_infty ring spectra
from purely algebraic input data, in the form of invertible matrices
over Witt rings.
11.02.09:
Hans-Werner Henn (
Strasbourg).
The rationalization of the K(2)-local sphere and Picard groups
at chromatic level 2 for p=3.
In this talk I will present joint work with various subsets of Goerss,
Karamanov, Mahowald and Rezk. In earlier work with Goerss, Mahowald and Rezk we
constructed a resolution of the K(2)-local sphere at the prime 3 in terms of
well understood homotopy fixed point spectra which are closely related to the
spectrum TMF of topological modular forms. In this talk I intend to describe
consequences of the existence of this resolution, both for the calculation of
the rational homotopy groups of the K(2)-local sphere and for the Picard groups
of E(2)-local and K(2)-local spectra at p=3.
Grothendieck's anabelian conjectures say that hyperbolic
algebraic curves over number fields should be K(pi,1)'s in algebraic
geometry. It follows that conjecturally the rational points on such a
curve are the sections of etale pi_1 of the structure map. These
conjectures are analogous to equivalences between fixed points and
homotopy fixed points of Galois actions on related topological spaces.
We use cohomological obstructions of Jordan Ellenberg coming from
nilpotent approximations to the curve to study the sections of etale
pi_1 of the structure map. We will relate Ellenberg's obstructions to
Massey products, and explicitly compute mod 2 versions of the first
and second for P^1-{0,1,infty} over Q. Over R, we show the first
obstruction alone determines the connected components of real points
of the curve from those of the Jacobian.
The playing field of profinite homotopy theory is provided by the
homotopy categories of profinite spaces and profinite spectra. A
motivating application is the connection to algebraic geometry. For
example the etale fundamental group and continuous etale
cohomology of a scheme can be defined in a unified way using a
profinite etale realization functor. We will discuss this functor
and use it to define etale topological cobordism. But it turned out
that profinite structures might be useful in other areas such as
Lubin-Tate spectra. If time permits we will discuss this idea in
progress as well.
This will be a Tuesday talk, at 4pm in 2-132.
09.28.09:
Mike Hill (
Harvard).
Equivariant computations and RO(G)-graded spectral sequences.
This talk will motivate and develop a bordism category consisting of
singular manifolds. Applications will be discussed having relevance
to 'stable' characteristic classes of families of smooth manifolds and
Gromov-Witten theory.
The 6-connected cover of Spin(n), known as the group String(n), has fascinating
connections with both abstract homotopy theory (through String Bordism and TMF)
and with quantum field theory (through the 2D SUSY non-linear sigma model). A
better geometric understanding of String geometry has the potential to offer new
interactions between these fields. Unfortunately all previous models of
String(n) are infinite dimensional, making a thorough geometric understanding
elusive. In this talk we will construct a finite dimensional model of String(n)
as a higher categorical version of a group (known as a 2-group). In the process,
we will "categorify" the classical notions of group cohomology and derived
functor. In particular we will categorify Segal's topological group cohomology,
thereby obtaining a classification of extensions of topological 2-groups.
Please send mailing list requests and questions to Inna Zakharevich.
Other useful links:
