Past Topology Seminar

In this talk, we introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover all homotopy groups of any dimensional spheres can be obtained from the geometric Massey products on certain links.

Bourdon's building is a negatively curved 2-complex built out of hyperbolic right-angled polygons. Its automorphism group is large (uncountable) and remarkably rich. We study, and mostly answer, the question of when there is a discrete subgroup of the automorphism group such that the quotient is a closed surface of genus g. This involves some fun elementary combinatorics, but quickly leads to open questions in group theory and number theory. This is joint work with David Futer.

The talk will begin with a brief account of the construction of string topology operations. I will point out some mysteries with the formulation of these operations, such as the role of (moduli) of surfaces, and pose some questions. The remainder of the talk will address these issues. In particular, I will sketch some ideas for a higher-dimensional version of string topology. For instance, (1) I will describe an E_{d+1} algebra structure on the (shifted) homology of the free mapping space H_*(Map(S^d,M^n)) and (2) I will outline how to obtain operations H_*(Map(P,M)) -> H_*(Map(Q,M)) indexes by a moduli space of zero-surgery data on a smooth d-manifold P with resulting surgered manifold Q.

The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group G as well as quotients spaces given by the associated space of representations.

These spaces of representations admit the structure of a simplicial space at the heart of the work here.

Features of geometric realizations will be developed.

What is the fundamental group or the first homology group of the associated space in case G is a finite, discrete group ?

This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.

TBA

Physical space-time is a manifold with a Lorentzianmetric, but the more mathematical treatments of the theory usually prefer toreplace the metric with a positive - i.e. Riemannian - one. The passage fromLorentzian to Riemannian metrics is called 'Wick rotation'. In my talk I shallgive a precise description of what is involved, and shall explain some of itsimplications for physics.

We define a chain complex B_*(C, M) (the "blob complex") associated to an n-category C and an n-manifold M. This is in some sense the derived category version of a TQFT. Various special cases of the blob complex are

familiar: (a) if M = S^1, then the blob complex is homotopy equivalent to the Hochschild complex of the 1-category C; (b) for * = 0, H_0 of the blob complex is the Hilbert space of the TQFT based on C; (c) if C is a commutative polynomial ring (viewed as an n-category), then the blob complex is homotopy equivalent to singular chains on the configuration (Dold-Thom) space of M. The blob complex enjoys various nice formal properties, including a higher dimensional generalization of the Deligne conjecture for Hochschild cohomology.

If time allows I will discuss applications to contact structures on 3-manifolds and Khovanov homology for links in the boundaries of 4-manifolds. This is joint work with Scott Morrison.

I shall describe the notion of finite decomposition complexity (FDC), introduced in joint work with Romain Tessera and Guoliang Yu on the Novikov and related conjectures. The talk will focus on the definition of FDC and examples of groups having FDC.