Purdue Topology Seminar, Spring 2018

The Purdue Topology Seminar is held Wednesdays 2:30pm-3:20pm 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 K-theory. Along the way we compute the topological Hochschild cohomology of some endomorphism spectra and give examples of spectra all having the same Spanier-Whitehead 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 3-orbifold.
February 7: TBD
February 16 (Friday): Andy Putman (Notre Dame University)
Title: The Johnson filtration is finitely generated
Abstract: A recent breakthrough of Ershov-He 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 G-bundles 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 Chern-Weil theory. I will also explain, from joint work with Byungdo Park, how equivariant S^1-gerbe 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 1-dimensional 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 14: Spring Break
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 Wiltshire-Gordon (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 hands-on 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 higher-dimensional manifolds (Galatius, Randal-Williams). From the perspective of homotopy theory, a common structure these examples share is that of an E_2-algebra, 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 E-theory
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 E-theory, a homotopical avatar of the space of deformations of a formal group of fixed height. We study what happens when E-theory 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.
April 25: TBD