L1

Quasifuchsian punctured-torus groups are the `simplest'
Kleinian groups with an interesting deformation theory. I will show that the convex core of the quotient of hyperbolic 3-space by such a group admits a decomposition into ideal tetrahedra which is canonical in two completely independent senses: one combinatorial, the other geometric. One upshot is a proof of the Bending Lamination Conjecture for such groups.

Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n

has a (generally branched) solution with leading order behaviour

proportional to

(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. Jointly with R.G. Halburd we consider the subclass of equations for which each possible leading order term of

this

form corresponds to a one-parameter family of solutions represented near

z_0

by a Laurent series in fractional powers of z-z_0. For this class of

equations we show that the only movable singularities that can be reached

by

analytic continuation along finite-length curves are of the algebraic type

just described. This work generalizes previous results of S. Shimomura.

The only other possible kind of movable singularity that might occur is an

accumulation point of algebraic singularities that can be reached by

analytic continuation along infinitely long paths ending at a finite point

in the complex plane. This behaviour cannot occur for constant coefficient

equations in the class considered. However, an example of R. A. Smith

shows

that such singularities do occur in solutions of a simple autonomous

second-order differential equation outside the class we consider here.

This is the first of two seminars this afternoon describing a generalization of Donaldson-Thomas invariants, joint work of Yinan Song and Dominic Joyce. We shall define invariants "counting" semistable coherent sheaves on a Calabi-Yau 3-fold. Our invariants are invariant under deformations of the complex structure of the underlying Calabi-Yau 3-fold, and have known transformation law under change of stability condition.

This first seminar constructs an auxiliary invariant "counting" stable pairs (s,E), where E is a Gieseker semistable coherent sheaf with fixed Hilbert polynomial and s : O(-n) --> E for n >> 0 is a morphism of sheaves, and (s,E) satisfies a stability condition. Using Behrend-Fantechi's approach to obstruction theories and virtual classes we prove this auxiliary invariant is unchanged under deformation of the underlying Calabi-Yau 3-fold.

As a small step towards an understanding of the relationship of the two fields in the title, I will present a uniformness result for embeddings of finitely generated, virtually free groups as cocompact, discrete subgroups in totally disconnected, locally compact groups.