The Purdue Topology Seminar is held Wednesdays 2:30pm3:20pm at LWSN B155 unless otherwise noted.
Primary contact is Peter Patzt (ppatzt at purdue dot edu)
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. 

March 7: 
Eric Ramos (University of Michigan) 

Title: Representation stability in the configuration spaces of graphs


Abstract: A graph is a 1dimensional CW complex. Configuration spaces of graphs have
recently risen to popularity due to their deep connections with robotics
and motion planning. In this talk we will begin by giving an overview of
the state of the art in the study of graph configuration spaces. Following
this, we will discuss what the tools of representation stability, and other
related techniques from asymptotic algebra, can tell us about these spaces. A
particular focus will be placed on recent work of the speaker, as well as
the speaker and Lütgehetmann, on how it is perhaps more correct to think
about these spaces by fixing the number of points being configured, and
allowing the graph to vary in some natural way. 


March 21: 
Bernardo Villarreal (IUPUI) 

Title: Classifying spaces for commutativity


Abstract: In this talk I will define the space BcomG arising from commuting tuples in G originally presented by A. Adem, F. Cohen and E. Torres. This space sits inside the classifying space BG and I will focus on describing the space BcomG for G=SU(2), U(2) and O(2), via its integral and mod 2 cohomology ring together with its Steenrod algebra. If time permits, for the Lie groups above, I’ll describe the homotopy type of the homotopy fiber of the inclusion BcomG into BG, denoted EcomG. This is joint work with O. Antolín and S. Gritschacher. 

March 28: 
John WiltshireGordon (University of Wisconsin) 

Title: Configuration space in a product


Abstract: Write Conf(n,X) for the space of injections {1,…,n} —–> X. For
example, the space Conf(n, R) is homotopy equivalent to a discrete space
with cardinality n!. In contrast, the space Conf(n, R x R) seems much more
interesting and complicated. In this talk, we explain how to compute the
homology of Conf(n, R x R) using only information about configurations in
R. The technique generalizes to general products as well. In making the
calculation, we give a handson demonstration of new software for
pruning chain complexes of presheaves on a poset. 

April 6 (Friday): 
Manuel Krannich (University of Copenhagen) 

Title: Homological stability of topological moduli spaces


Abstract: Since the seventies, many families of topological moduli spaces have been proven to stabilize homologically, including moduli spaces of Riemann surfaces (Harer), unordered configuration spaces (McDuff, Segal), and moduli spaces of higherdimensional manifolds (Galatius, RandalWilliams). From the perspective of homotopy theory, a common structure these examples share is that of an E_2algebra, or at least of a module over such an algebra. In this talk, I will introduce a framework which provides a uniform treatment of classical and new (twisted) homological stability results from this perspective. If time permits, I will also discuss how these results imply representation stability for related moduli spaces. 

April 11: 
Nate Harman (University of Chicago) 

Title: Interpolating categories and representation stability


Abstract: We discuss certain algebraic families of categories that
“interpolate” categories of representations of families of groups and
algebras. The focus will be on how these families of categories give rise
to stable sequences of representations, giving new proofs of several known
stability results as well as some new ones. 

April 18: 
Paul VanKoughnett (Northwestern University) 

Title: Localizations of Etheory


Abstract: Chromatic homotopy theory uses the theory of formal groups from
algebraic geometry to construct new topological invariants. The tightest
link between the two worlds is Morava Etheory, a homotopical avatar of the
space of deformations of a formal group of fixed height. We study what
happens when Etheory undergoes chromatic localization, forcing the height
of this formal group to decrease. We give modular descriptions of the
resulting objects, and applications to the study of power operations in
homotopy theory. 

