Birman will investigate knot theory, with a particular focus on the applications of the theory of braids to the classification of knots and links. The first project, with Menasco, aims at an algorithmic solution to the link problem via the theory of braids. If successful, work on the main theorem, which may be described as "Markov's Theorem with stabilization," will establish the existence of an algorithmic solution to the knot problem by showing that there is a series of complexity-reducing moves which take an arbitrary closed braid representative of a knot to one which has minimum complexity. The implementation of the algorithm will require, in addition, recognizing when certain complexity-reducing moves are applicable. The second project is an investigation of the knot invariants of Vassiliev. One goal is to estimate the number of Vassiliev invariants of order i, and perhaps prove that this number grows without bound. The third concerns John Moody's recent counterexample to the faithfulness of Burau representations and its relevance to the corresponding question for the Jones representations. Birman hopes to identify the kernel of the Burau representation. The final project concerns understanding whether the collection of all Jones invariants can be faithful on knot and link types. Morgan intends to work on three related areas of topology: (i) actions of groups on trees; (ii) computation of Donaldson invariants for algebraic surfaces, using algebraic geometry; and (iii) computations of Donaldson invariants of 4-manifolds which are made of two pieces joined by a long cylindrical tube. The first project, with R. Skora, concerns the classification of free actions of groups on R-trees, and in particular, the question of which groups act freely. They have been studying groups which are certain types of non-trivial free products with amalgamation or HNN-extensions. They propose to expand this study to more general amalgams. Projects (ii) and (iii) concern Donaldson's invariants for smooth 4-manifolds. Project (ii) explores connections with algebraic geometry. Morgan and R. Friedman have partially computed these invariants for elliptic surfaces of general type and propose to extend their computations, especially those for elliptic surfaces. Project (iii) explores some purely analytic, differential geometric questions, particularly, the ASD connections of finite energy on infinite cylinders, in order to provide a link between embedded riemann surfaces in a 4-manifold and the values of the Donaldson invariant. These are projects in topology which interact heavily with other areas of mathematics, e.g., algebra, algebraic geometry, and even mathematical physics. Progress could take many different forms, and it is encouraging to note the variety of interested parties.
