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 Semester 2011 Schedule

12:30–13:30 (Eastern Time) Tuesdays @ 125 Hayes-Healy Center

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

Spring Semester 2012 Schedule

15:10–16:10 (Eastern Time) Thursdays @ 125 Hayes-Healy Center

09 February: Daniel Berwick-Evans (U California-Berkeley); Supersymmetric field theories and cohomology
23 February: Sam Gunningham (Northwestern U); Spin Hurwitz numbers and TQFT
~~15 March~~: NO SEMINAR; Notre Dame Spring Break
29 March: Christopher Schommer-Pries (Massachusetts I Technology); On the unicity of the homotopy theory of higher categories, I
continued - 16:30–17:30 Thursday 29 March @ 117 HAYE: Christopher Schommer-Pries (Massachusetts I Technology); On the unicity of the homotopy theory of higher categories, II
~~05 April~~: NO SEMINAR; Easter Break
26 April: Tam Nguyen-Phan (U Chicago); Finite volume, negatively curved manifolds
~~10 May~~: NO SEMINAR; Notre Dame Final Exam Week

Abstracts of Invited Talks

11 October 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.

29 November 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.

06 December 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.

09 February 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.

23 February 2012: Sam Gunningham

Spin Hurwitz numbers and TQFT

Spin Hurwitz numbers count ramified covers of surfaces with a sign according to the Atiyah invariant of the covering surface. I will describe how to compute these numbers using ideas from topological quantum field theories (TQFT), homotopy theory, and spin representation theory of the symmetric group.

29 March 2012: Christopher Schommer-Pries

On the unicity of the homotopy theory of higher categories

We will discuss joint work with Clark Barwick, in which we propose four axioms that a quasicategory should satisfy to be considered a reasonable homotopy theory of (∞,n)-categories. This axiomatization requires that a homotopy theory of (∞,n)-categories, when equipped with a small amount of extra structure, satisfies a simple, yet surprising, universal property. We further prove that the space of such quasicategories is homotopy equivalent to the n-fold product RP^∞ × … × RP^∞. This generalizes a theorem of Toen when n=1, and it verifies two conjectures of Simpson. In particular, any two such quasicategories are equivalent. We also provide a large class of examples of models satisfying our axioms, including those of Joyal, Kan, Lurie, Simpson, and Rezk.

26 April 2012: Tam Nguyen-Phan

Finite volume, negatively curved manifolds

I will talk about the structure of noncompact, complete, finite volume, negatively curved manifolds. I will discuss how different curvature conditions control the topological properties of these manifolds.