On universal Vassiliev invariants

Using properties of ordered exponentials and the definition of the Drinfeld associator as a monodromy operator for the Knizhnik-Zamolodchikov equations, we prove that the analytic and the combinatorial definitions of the universal Vassiliev invariants of links are equivalent.Supported by Fonds national suisse de la recherche scientifiqueURA 1436 du CNRS, associée à l'Ecole Normale Supérieure de Lyon et au laboratoire d'Annecyle-Vieux de Physique des Particules     

A construction of thec<1 modular invariant partition functions

Decomposition theorems for certain representations of Kac-Moody algebras which are needed for the construction of modular invariant unitary conformal models are proved. It is shown that allc<1 modular invariant models can then be recovered from gauged free fermionic models, including the exceptional cases.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY85-15857Supported by the Swiss National Science FoundationSupported in part by the American-Israeli Binational Science Foundation and the Israeli Academy of Sciences     

Quasi-quantum groups,knots, three-manifolds,and topological field theory

We show how to construct, starting from a quasi-Hopf algebra, or quasiquantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This happens for a finite-dimensional quasi-quantum group, whose definition involves a finite groupG, and a 3-cocycle , which was first studied by Dijkgraaf, Pasquier, and Roche. We treat this example in more detail, and argue that in this case the invariants agree with the partition function of the topological field theory of Dijkgraaf and Witten depending on the same dataG, .     

Mean curvature flow through singularities for surfaces of rotation

In this paper, we study generalized “viscosity” solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.