|Ken Bromberg, Caltech||10:30-11:30 am||151 Sloan||The topology of deformation spaces of hyperbolic 3-manifolds|
|Justin Roberts, UCSD||1:30-2:30 pm||151 Sloan||Asymptotics of 6j symbols|
|Allen Knutson, UC Berkeley||2:45-3:45 pm||151 Sloan||Singularities of differentiable maps from equivariant cohomology and algebraic geometry|
|Benson Farb, U Chicago||4-5 pm||151 Sloan||The isometry groups of finite-volume Riemannian manifolds and their universal covers|
Ken Bromberg. Title: The topology of deformation spaces of hyperbolic 3-manifolds
Abstract: Recently Brock, Canary and Minsky have announced the completion of Minsky's program to prove the Ending Lamination Conjecture. This result gives a complete classification of all complete hyperbolic structures on a 3-manifold with incrompressible boundary. However this classifaction is not a parameterization for the map to space of classifying objects in not continuous. In this talk we will give an overview of what is known and what is conjectured about the topology of these deformation spaces.
Justin Roberts. Title: Asymptotics of 6j symbols
Abstract: A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j-symbol, which is a purely algebraic object, but it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the 6j-symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. I will explain some of the geometric ingredients of my 1999 proof of their formula. For quantum topologists, the problem of the asymptotic behaviour of the quantum 6j-symbols, which ought to be related to spherical tetrahedra, is an important part of the general programme of relating quantum invariants with classical topology. I will describe the recent proof of such an asymptotic formula by Yuka Taylor and Chris Woodward, as well as the peculiar scissors congruences arising from symmetries of 6j-symbols.
Allen Knutson. Title: Singularities of differentiable maps from equivariant cohomology and algebraic geometry
Abstract: Thom asked: given a smooth map f: M -> N between complex manifolds, generic in its homotopy class, how can one compute the class [Mr] of the locus where the differential drops rank (say, to r)? There turns out to be a universal formula in the Chern classes of TM and f*TN, the Thom-Porteous formula. Recently several authors independently pointed out a universal construction explaining such a formula, and the role of Borel's equivariant cohomology construction in reducing it to an algebraic geometry question. I'll explain this in detail, and sketch some more advanced questions for which the algebraic geometry question admits a nice, combinatorial, answer. This work is joint with Ezra Miller and Mark Shimozono.
Benson Farb. Title: The isometry groups of finite-volume Riemannian manifolds and their universal covers
Abstract: Shmuel Weinberger and I have recently classified (up to finite index) the isometry groups of all finite-volume Riemannian manifolds and their universal covers. I will explain this result and some of its corollaries. This talk will be in the same general direction as, but have little overlap with, my colloquium talk (on Thursday the 29th).