Title 1: Gevrey series in quantum topology

Abstract 1: A power series is Gevrey if the n-th coefficient is bounded by 
n! C^n. Gevrey series are typically factorially divergent. We aim to show 
that the LMO invariant of a 3-manifold is a Gevrey series. To achieve 
this, we need to replace the LMO graph-valued invariant by a power series 
invariant. Weight systems do exactly that. However weight systems are not 
unique (there is one per simple Lie algebra, and even more exotic ones are 
known). It would be nice to replace the LMO invariant Z by a single power 
series |Z| such that (a) |Z| is Gevrey, (b) the image of Z under any 
weight system from Lie algebras is Gevrey and (c) |Z|=1 iff Z=1. Can we 
achive all three tasks simoultaneously? And what happens if we replace the 
LMO invariant by the Kontsevich integral? These questions have a simple 
and positive answer. Interested? Come and listen. This is joint work with 
T.T.Q. Le.

Title 2: Resurgent functions in quantum topology

Abstract 2: A resurgent function (due to Ecalle) is a holomorphic function 
originally  convergent in a small disk, with endless analytic continuation 
in the complex plane, minus a countable set of singularities. Examples of 
resurgent functions are meromorphic functions, and algebraic functions, or 
more generally, solutions to regular singular differential equations.
The position and shape of singularities are important invariants of 
resurgent functions. Ecalle proved that formal solutions of differential 
or difference equations (linear or not) have unique resurgent solutions, 
which in addition determine the differential equation.
What does this have to do with quantum topology? In the talk we will 
formulate a general resurgence conjecture for power series associated to 
knotted objects, and we will show how this conjecture implies the 
existence of asymptotic expansions of quantum invariants (to all orders 
and with exponentially small corrections included). In particular, the 
position of the singularities of these resurgent functions are expected to 
be determined by the i vol + CS invariant of parabolic SL(2,C) representations
of the knotted objects. As a nontrivial test case, we will give a proof of
our resurgence conjecture for the simplest knot: the trefoil, and for a
simple 3-manifold: the Poincare homology sphere. Time permitting, we will 
indicate a proof of our resurgence conjecture for the 4_1 knot.
This is joint work with Ovidiu Costin.