For more information, contact Peter Scott at pscott@umich.edu.

If you want to check the list of seminars held in previous terms, click on the appropriate link below.

Fall 2002, the year 2000-1.

Schedule of Talks

Abstracts

January 10

I will discuss a new categorical formalism which enables a rigorous approach to both closed conformal field theory and boundary conformal field theory (D-brane categories). If time permits, I may discuss some related topics, including D-brane cohomology, modular functors via 2-vector spaces, and elliptic cohomology.

January 17

We describe Delzant's extension of the notion of the Scott complexity of a group to a complexity of homomorphisms of groups. This leads to a quick proof of the coherence of 3-manifold groups as well as Sela's acylindrical accessibility theorem.

January 31

Thurston's Geometrization theorem implies that an acylindrical hyperbolic manifold M admits a unique hyperbolic metric whose convex core has totally geodesic boundary. We show that this is the most efficient hyperbolic metric on M, in the sense that it is the hyperbolic metric whose convex core has least possible volume. The result above follows from an extension of work of Besson, Courtois and Gallot into the setting of Alexandrov spaces with lower bounds on curvature. This extension also has implications for volumes of hyperbolic cone manifolds.

February 7

The spinor bundle S(E) associated to an even dimensional real vector bundle E with spin structure has (at least) two roles in life: from a homotopy theory point of view it represents the K-theory Euler class of E; from a geometric/analytic point of view, the Dirac operator acts on the sections of the spinor bundle S(TX) associated to the tangent bundle of a spin manifold X. Analogously, it is believed that the spinor bundle or Fock space bundle $F(E)\to LX$ over the free loop space LX associated to an even dimensional vector bundle $E\to X$ with `string structure' plays a similar dual role: it should represent the Euler class of E in $tmf^*(X)$, and there should be a `Dirac-Witten operator' acting on the sections of F(TX), whose S^1-equivariant index is the Witten genus of X.

The main result of this joint work with Peter Teichner is that the spinor bundle F(E) can be equipped with additional structures we call `conformal connection' and `fusion'. We speculate that vector bundles over LX equipped with these two structures represent elements in $tmf^*(X)$.

March 14

Let N be a closed, connected, orientable 3-manifold with cyclic pi_1, and let K be a tame knot in N. The properties of the set of boundary slopes of K play a crucial role in the current approach of Culler, Dunfield, Jaco, and Shalen to proving the Poincare conjecture. The following theorem is potentially relevant:

Theorem. If pi_1(N) is odd cyclic and the exterior of K is irreducible, then at least one of the following conditions holds: (1) for every framing of the knot K, there exists a boundary slope S such that 1 < |S| \leq \infty; or (2) K is an iterated cable knot such that the exterior of the initial knot is a solid torus.

I'll discuss briefly the potential application, including related results. The main argument in the proof, based on an argument of Dunfield (in the case when K is hyperbolic), is of interest since it brings to bear some deep results in the theory of character varieties of hyperbolic 3-manifolds. In particular, the proof uses the Representation-Volume Rigidity theorem for finite-volume hyperbolic manifolds, and a theorem about a birational equivalence between (1) a certain curve C of characters in the PSL_2(C)-character variety of a 2-cusped hyperbolic manifold M and (2) its image under a certain rational map into the variety of one of the cusps of M.

March 21

Ahlfors' conjectured that the limit set of a finitely generated Kleinian group has Lebesgue measure zero in the Riemann sphere or is the full Riemann sphere. We verify this conjecture for Kleinian groups in the closure of the geometrically finite locus.

March 28

Marden's conjecture states that a complete hyperbolic 3-manifold with finitely generated fundamental group is tame, i.e. the interior of a compact 3-manifold. Given a manifold with torus boundary and hyperbolic interior (a hyperbolic knot complement), an exceptional Dehn filling is one which is reducible, or has finite fundamental group, or non-word-hyperbolic fundamental group. Assuming Marden's conjecture, we show that there are only finitely many 1-cusped hyperbolic knot complements which have > 8 exceptional Dehn fillings. It is conjectured that there are only finitely many with > 6 exceptional fillings, and an explicit list of such manifolds is conjectured to be the only examples. We will also show that for a non-compact hyperbolic manifold M with b_1(M)>2, the volume of M is >= 2pi*v_3/sqrt(3), where v_3 is the volume of a regular ideal tetrahedron in H^3. The common thread of these two theorems is a result of Anderson, Canary, Culler and Shalen which gives an improved Margulis lemma for tame free groups.

April 4

In joint work with Swarup, we obtained an algebraic analog of the JSJ decomposition of 3-manifolds. This led us to a better understanding of the topological result, and the purpose of this talk is to discuss our approach to the existence and uniqueness of the JSJ decomposition of a 3-manifold, and to Johannson's Deformation Theorem.

This page last updated20 .