Department of Mathematics Topology Seminar

01/24: James McClure:  A Convenient Category of C^{infty} Manifolds.
Abstract:  This is joint with Gerd Laures.  We show that by expanding the
category of manifolds with corners one obtains a category in which gluing is
canonical.

01/31: Ben Walter: Rational Hopf Invariants via coLie Coalgebras.
Abstract:   This is joint work with Dev Sinha.  Using colie coalgebras of
 graphs, we define natural, rational functionals on homotopy
 groups using cochain data.  Our work extends work of Sullivan,
 Chen and Hain, and overlaps with work of Boardman and Steer.
 Proofs are simple and the underlying geometric picture is
 compelling.

02/07: Matthew Ando (UIUC). A quasicategory approach to units of ring spectra and Thom spectra.
Abstract:   May-Quinn-Ray-Tornehave developed the obstruction theory for E_\infty
orientations of E_\infty Thom spectra. Initially we set out to develop
the A_\infty analogue. In both cases, the complications arise from the
fact that the Thom isomorphism arises from trivializing a homotopy
sheaf of module spectra. Quasicategories are designed to provide a natural
setting for such sheaves, and so we should have been merely pleased
rather than pleasantly surprised that in this setting the obstruction
theory simplifies a great deal.

02/21: Philip Hackney (Purdue): Coproducts, Smith Chains, and Hochschild Cohomology.
Abstract: The Gerstenhaber algebra structure on the Hochschild cohomology of an
algebra A is naturally defined at the cochain level. The cup product is
actually the dual of the cut coproduct on the Hochschild chains if A is
commutative. When A is a complete intersection, Larry Smith developed a
small resolution of A; we introduce a coproduct in this setting which
plays the role of the cut coproduct.

03/06: Anatoly Libgober (UIC). Higher genera in algebraic geometry.
Abstract:   I will discuss elliptic genus and Hirzebruch's chi_y genus generalizations
of Novikov's higher signatures. Applications to multiplicativity of chi_y
in for locally trivial fibrations and birational analogs of Novikov's conjecture
will be given.

03/27: Javier Zuniga (Purdue). The Master Equation of Open-closed String Theory.
Abstract: I will set up a Quantum Master Equation for open-closed string theory and construct a
solution using a compactified version of moduli space of bordered Riemann Surfaces.

04/03: Alastair Hamilton (UConn). Noncommutative geometry and
compactifications fo the moduli space of curves.
Abstract: There is a theorem, due to Kontsevich, which states that the homology
of the moduli space of curves can be expressed as the homology of a
certain Lie algebra. In this talk I will explain how the homology of a
certain compactification of the moduli space, introduced by Kontsevich
in his study of Witten's conjectures, can be expressed as the homology
of a certain differential graded Lie algebra by deforming Kontsevich's
original Lie algebra using a Lie bialgebra structure considered by many authors.

04/10: Jinhyun Park (Purdue). Hochschild and cyclic homologies as additive K-theories
This talk is introductory: we recall the Lie algebra homology, Leibniz homology
and discuss how they are related to cyclic homology and Hochschild homology.
Then we argue why cyclic homology and Hochschild homology are additive K-theories.
We will talk about the aim of the second talk as well.

04/17: Jinhyun Park (Purdue). Commutative differential graded algebras of additive Chow cycles
We first mention the relationship between Quillen K-theory and higher Chow groups.
Then, we discuss briefly the additive higher Chow cycles and what it aims to achieve
in connection with additive K-theory. We will define motivic analogues of Connes'
boundary map and the shuffle product whose traditional place of discussion
is Hochschild complex. Using them, we discuss how to give the structure of CDGA on
additive higher Chow groups.