January 10: 
Jeremy Miller (Purdue University) 

Title: High dimensional cohomology of congruence subgroups 

Abstract: The level p congruence subgroup of SL_n(Z) is defined to be the subgroup of
matrices congruent to the identity matrix mod p. These groups have trivial
cohomology in high enough degrees. In the 1970s, Lee and Szczarba gave a
conjectural description of the top cohomology groups of these congruence
subgroups. In joint work in progress with Patzt and Putman, we show that this
conjecture is false and that these congruence subgroups have extra exotic
cohomology classes in their top degree cohomology coming from the first homology
group of the associated modular curve. I will also discuss join work in progress
with Patzt and Nagpal on a stability pattern in the high dimensional cohomology
of congruence subgroups. 

January 24: 
Jeff Smith 

Title: Compactly generated stable homotopy theories 

Abstract: In this talk I will follow one idea from Morita theory from
computing the center of matrix rings to the modern definition of Ktheory.
Along the way we compute the topological Hochschild cohomology of some
endomorphism spectra and give examples of spectra all having the same
SpanierWhitehead dual. 

January 31: 
Ryan Spitler (Purdue University) 

Title: Profinite Completions and Representations of Groups 

Abstract: The profinite completion of a group $\Gamma$, $\widehat{\Gamma}$, encodes all of the information of the finite quotients of $\Gamma$. A residually finite group $\Gamma$ is called profinitely rigid if for any other residually finite group $\Delta$, $\widehat{\Gamma} \cong \widehat{\Delta}$ implies $\Gamma \cong \Delta$. I will discuss some ways that $\widehat{\Gamma}$ can be used to understand linear representations of $\Gamma$ and applications to questions related to profinite rigidity. In particular, I will explain the role this plays in forthcoming work with Bridson, McReynolds, and Reid which establishes the profinite rigidity of the fundamental group a certain finite volume hyperbolic 3orbifold. 


February 16 (Friday): 
Andy Putman (Notre Dame University) 

Title: The Johnson filtration is finitely generated 

Abstract: A recent breakthrough of ErshovHe shows that the Johnson kernel
subgroup of the mapping class group is finitely generated for g at
least 12. In joint work with Ershov and Church, I have extended this
to show that every term of the lower central series of the Torelli
group is finitely generated once the genus is sufficiently large. A
byproduct of our work is a proof that the Johnson kernel is finitely
generated for g at least 4 which is remarkably simple (so simple, in
fact, that I will be able to give it in nearly complete detail in this
talk). 

February 21: 
Corbett Redden (Long Island University) 

Title: Differential Equivariant Cohomology 

Abstract: Suppose G is a compact Lie group acting on a smooth manifold M. The "differential quotient stack" assigns to any test manifold X the groupoid of principal Gbundles on X with connection and equivariant map to M. I will explain how the differential cohomology groups (Deligne cohomology) of this stack provide a natural home for equivariant ChernWeil theory. I will also explain, from joint work with Byungdo Park, how equivariant S^1gerbe connections are classified by degree 3 classes. 

February 28: 
Jeremy Hahn (Harvard University) 

Title: Even spaces and variants of periodic complex bordism


Abstract: I will describe a classical construction of MUP, the periodic complex
bordism spectrum, due to Victor Snaith. Joint work with Allen Yuan reveals
that the multiplicative properties of this construction are surprisingly
subtle. Inspired by work of Gepner and Snaith, I will also discuss
potential analogues of the construction in exotic homotopy theories. Some
of this involves work in progress with Dylan Wilson. 








