Purdue University

Department of Mathematics
Topology Seminar


Spring 2013
Time and place: Thursdays 11:00-12:00 in BRNG 1254
Organizer: Ralph Kaufmann. Contact organizer.


02/14/13 Ralph Kaufmann. Berry Phases, Monopoles and Chern classes.
     Abstract: We discuss how characteristic classes come up when discussing a family of Hamiltonians over a given base space and relate this to well known constructions in physics. We will also give some examples from well known mathematics. If time permits, we will discuss how this is related to topological stability in certain materials.

02/21/13 + 02/28/13 Andrei Gabrielov: Combinatorial computation of the first Pontryagin class. (After Gabrielov, Gelfand and Losik, 1975).
    Abstract: When Pontryagin in 1940-ies introduced characteristic classes of smooth manifolds, he asked whether those classes (or, rather, cycles representing their duals) could be computed for simplicial manifolds in a purely combinatorial way. I will present the results of my work with I. Gelfand and M. Losik (1975)
on combinatorial computation of the first Pontryagin class. These results were presented by R. MacPherson at the Feb 1977 Bourbaki Seminar. At the Gelfand's memorial conference (Rutgers, 2009) MacPherson referred to this work as an unfinished attempt to define "combinatorial curvature."
The best reference to these results is Notes on "A combinatorial formula for P_1(X)" by D. Stone (Advances in Mathematics, 1979).

04/11/13 Ralph Kaufmann. Universal operations in Feynman categories and relations to other theories.
    Abstract: First we will briefly recall the definition of a Feynman category. We will then give the relations to colored operads and patterns which were defined by Getzler. Then we will discuss newer examples and the relation of Feynman categories to Lavwere theories and crossed simplicial groups.
Finally, we discuss how universal operations  natural arise from Feynman categories by taking colimits in a cocompletion.This includes the pre-Lie operations for operads, the Lie admissible operations for di-operads, the Kontsevich Soibelman minimal operad for operads with A_\infty multiplication.

04/18/13 Yu Tsumura.  A 3-2-1 topological quantum field theory extending the Reshetikhin-Turaev TQFT.
    Abstract: Reshetikhin and Turaev constructed a topological quantum field theory as a functor from the category of cobordisms to the category of vector spaces using a modular category as an input. In the talk, I will explain their construction and extend it to manifolds with corners. This extension is described in terms of 2-categories.

04/22/13 Noah Snyder (IU). The Space of Fusion Categories.
    Abstract. Fusion categories are quantum analogues of finite groups.  They play key roles in the study of topological quantum field theory, von Neumann subfactors, and condensed matter physics.  There's a topological space (more precisely a homotopy 3-type) whose points are fusion categories.  The goal of this talk is to explain what this space is and why it's interesting.  In particular I'll touch on Etingof-Nikshych-Ostrik's classification of G-extensions of fusion categories, joint work with Pinhas Grossman describing certain connected components coming from exceptional subfactors, joint work in progress with Chris Douglas and Chris Schommer-Pries giving an explicit construction of an O(3) action on this space, and joint work with Grossman and David Jordan on a fibration of closely relatedspaces.  Rather than dealing with any of these in depth, the emphasis of the talk will be on the big picture that a good way to study fusion categories is to look at the space of all of them.

04/25/13 Sasha Voronov (U of Minnesota). Quantum master equation and deformation theory.
Classical deformation theory is based on the Classical Master Equation (CME), a.k.a. the Maurer-Cartan Equation: dS + 1/2 [S,S] = 0. Physicists have been using a quantized CME, called the Quantum Master Equation (QME), a.k.a. the Batalin-Vilkovisky (BV) Master Equation: dS + h \DeltaS + 1/2 {S,S} = 0. The CME is defined in a dg Lie algebra g, whereas the QME is defined in a space V [[h]] of formal power series with values in a differential graded (dg) BV algebra V. One can anticipate a generalization of classical deformation theory arising from the QME or quantum deformation theory.

There are a few papers which may be viewed as making first steps in abstract quantum deformation theory: Quantum Backgrounds and QFT by Jae-Suk Park, Terilla, and Tradler; Modular Operads and Batalin-Vilkovisky Geometry by Barannikov; Smoothness Theorem for Differential BV Algebras by Terilla; and Quantizing Deformation Theory by Terilla.

Further steps in quantum deformation theory will be discussed in the talk.