Abstract:
A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's $_1\psi_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems $\tilde A_n$ respectively $\tilde A_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.

Abstract:
In [5], P. Lecomte conjectured the existence of a natural and conformally invariant quantization. In [7], we gave a proof of this theorem thanks to the theory of Cartan connections. In this paper, we give an explicit formula for the natural and conformally invariant quantization of trace-free symbols thanks to the method used in [7] and to tools already used in [8] in the projective setting. This formula is extremely similar to the one giving the natural and projectively invariant quantization in [8].

Abstract:
We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial.

Abstract:
We study the asymptotic expansion of the colored Jones polynomial (the Melvin-Morton expansion) using a recursion formula for the deframed universal weight system for the $sl(2)$ Lie algebra. Combined with the formula for the universal weight system for the Lie superalgebra $gl(1|1)$ (which corresponds to the Alexander-Conway knot polynomial) this formula gives a very short proof of the Melvin-Morton conjecture relating the colored Jones invariant and the Alexander-Conway polynomial of knots.

Abstract:
We use the Chern-Simons quantum field theory in order to prove a recently conjectured limitation on the 1/K expansion of the Jones polynomial of a knot and its relation to the Alexander polynomial. This limitation allows us to derive a surgery formula for the loop corrections to the contribution of the trivial connection to Witten's invariant. The 2-loop part of this formula coincides with Walker's surgery formula for Casson-Walker invariant. This proves a conjecture that Casson-Walker invariant is a 2-loop correction to the trivial connection contribution to Witten's invariant of a rational homology sphere. A contribution of the trivial connection to Witten's invariant of a manifold with nontrivial rational homology is calculated for the case of Seifert manifolds.

Abstract:
Turaev's shadow formula calculates the SU(2)-Reshetikhin-Turaev-Witten invariants using shadows, and its expression is somehow similar to a Euler characteristic. We give a short proof of this formula using skein theory. The formula applies to pairs (M,G) where M is a closed oriented 3-manifold and GcM is a (possibly empty) colored framed trivalent graph (for instance, a framed knot or link).

Abstract:
In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula involves a new class of generalized Littlewood-Richardson coefficients, all of which surprisingly seem to be non-negative. We conjecture that each of these coefficients count the number of sequences of semistandard Young tableaux which satisfy certain conditions. In this paper I give a proof of this conjecture in the special case where the quiver variety can be described by at most four vector bundles. I also prove that the general conjecture follows from a simple combinatorial statement for which substantial computer verification has been obtained.

Abstract:
We study a computational method of the hyperbolic Reidemeister torsion (also called in the literature the non-abelian Reidemeister torsion) induced by J. Porti for complete hyperbolic three-dimensional manifolds with cusps. The derivative of the twisted Alexander invariant for a hyperbolic knot exterior gives the hyperbolic torsion. We prove such a derivative formula of the twisted Alexander invariant for hyperbolic link exteriors like the Whitehead link exterior. We provide the framework for the derivative formula to work, which consists of assumptions on the topology of the manifold and on the representations involved in the definition of the twisted Alexander invariant, and prove derivative formula in that context. We also explore the symmetry properties (with sign) of the twisted Alexander invariant and prove that it is in fact a polynomial invariant, like the usual Alexander polynomial.

Abstract:
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.