Notre Dame Topology Seminar

Fall 2010 – Spring 2011 Topology Seminar
Felix Klein Seminar (on geometric things)
Notre Dame Department of Mathematics

Questions? Contact the organizer, Qayum Khan

Fall 2011 Schedule

Fall talks occur on Tuesdays during 12:30–13:30 (Eastern Time) in 125 Hayes-Healy Center.

October 11: John Francis (Northwestern U); Factorization homology of topological manifolds
~~October 18~~: NO SEMINAR; Notre Dame Fall Break
~~November 22~~: NO SEMINAR; Thanksgiving Break
November 29: Seunghun Hong (Pennsylvania State U); A Lie-algebraic approach to the local index theorem on a flag variety
December 6: Mark Powell (Indiana U); A second order algebraic knot concordance group
~~December 13~~: NO SEMINAR; Notre Dame Final Exam Week

Spring 2012 Schedule

Spring talks occur on Thursdays during 15:10–16:10 (Eastern Time) in 125 Hayes-Healy Center.

February 9: Daniel Berwick-Evans (U California-Berkeley); Supersymmetric field theories and cohomology
~~March 15~~: NO SEMINAR; Notre Dame Spring Break
~~May 10~~: NO SEMINAR; Notre Dame Final Exam Week

Abstracts of Invited Talks

October 11, 2011: John Francis

Factorization homology of topological manifolds

We describe an axiomatic characterization of the factorization homology (a.k.a. topological chiral homology) of topological manifolds, in a sense analogous to (and generalizing) the Eilenberg-Steenrod axioms for usual homology. This point of view provides a new proof of the nonabelian Poincare duality of Salvatore and Lurie, that factorization homology with coefficients in an n-fold loop space is homotopy equivalent to a space of compactly supported maps. In joint work with David Ayala and Hiro Tanaka, this method of proof generalizes to manifolds with boundary and to stratified manifolds. Time permitting, we will survey some other calculations of factorization homology.

November 29, 2011: Seunghun Hong

A Lie-algebraic approach to the local index theorem on a flag variety

Let G be a compact Lie group and let T be a maximal torus in G (more generally we could consider any connected closed subgroup). Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on G/T to the Weyl character formula. In this talk we shall explain how to prove the local index theorem on G/T using Lie algebra methods. Our method follows in outline the proof of the local index theorem due to Berline and Vergne. But our use of Kostant?s cubic Dirac operator in place of the riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.

December 6, 2011: Mark Powell

A second order algebraic knot concordance group

A knot in the three sphere is said to be slice if it bounds an embedded disc in the four ball. The group of knots under connected sum modulo slice knots is called the knot concordance group. Cochran, Orr and Teichner defined a geometric filtration of the concordance group related to Whitney towers and gropes. Their obstruction theory at each level depends on choices of the way in which the obstructions at lower levels vanish. I'll define an algebraic obstruction group of chain complexes which captures the first two COT stages in a single obstruction, and, if time permits, indicate how this can be extended to define an n-th order group.

February 9, 2012: Daniel Berwick-Evans

Supersymmetric field theories and cohomology

Stolz and Teichner have proposed an analogy connecting cohomology theories with supersymmetric field theories. Their three main examples include de Rham cohomology in dimension 0|1, K-theory in dimension 1|1, and---conjecturally---topological modular forms in dimension 2|1. In this talk I will report on some recent progress in understanding the analogy in other super dimensions: the three examples above appear to be very special.