9402988 Morgan Professor Morgan continues to study Donaldson polynomial invariants of smooth 4-manifolds M. This includes searching for explicit computations. One is a blow-up formula that would express the invariants for the connected sum of M with a complex projective plane in terms of those for M. Such a formula should prove essential for understanding general relations among the invariants. Another part of the project, continuing work begun with Tomasz Mrowka of Cal Tech, studies the invariants of a 4-manifold split into two pieces by a 3-torus. These invariants should be related by a generalized product formula to the invariants of the two pieces. In a more qualitative vein, Professor Morgan intends to look for other contexts in which these invariants may be defined, hoping that other points of view will illuminate the theory and will explain some of the surprisingly simple and coherent numerical results now being obtained. Professor Birman's project centers on the theory of knots and links in 3-space. She is approaching this problem from two points of view. The first is joint work with W. Menasco (now in its 8th year, with much of the preliminary work completed). Its goal is an algorithmic solution to the link problem, via the theory of braids. Her second project is closely related to the first, and involves Vassiliev-Kontsevich invariants of knots and links in 3-space. These numerical invariants include all the information which is contained in the Jones polynomial and its generalizations, and possibly more. A central question is whether Vassiliev invariants contain more information than Jones invariants, and, in particular, whether they detect orientation. The asymptotics of the dimension of the Vassiliev algebra, as the order of the invariants goes to infinity, also involves some very interesting problems in combinatorics, which Professor Birman is studying. Professor Morgan's research concerns invariants of four- dimensional man ifolds, while Professor Birman's centers on computers and knots. Each is an effort to tame geometric complexity by rendering it susceptible to numerical or algebraic computation. Consider the latter, for example. While knots are among the most familiar of everyday objects (as every fisherman knows), their classification turns out to be a deep and difficult problem in 3-manifold topology. Roughly speaking, given two knots, one would like to be able to decide (with the help of a computer) if one can be twisted and deformed, keeping its ends fixed and without cutting the string) until it looks like the other. Precisely describing a method for doing this is an unsolved problem on which Professor Birman is working (with W. Menasco, of SUNY Buffalo). A different aspect of the same question concerns a vast collection of computable invariants (discoverd by V. Vassiliev) which give a partial answer to the knot question. With regard to the algebraic invariants, the central question is exactly what information is 'missed' by the Vassiliev invariants. While the motivation for this work is a wish to understand the underlying mathematical structure, one would expect that answers to questions such as the ones being asked would have applications wherever knots (or 4-manifolds) occur, i.e. to biology, chemistry, and physics. ***
