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.
Abstract: 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.
