9409743 Thompson This career advancement award supports mathematical research on problems arising in the fields of knot theory, minimal surface theory and the theory of 3-manifolds. The emphasis of this project is to explore an area at the border between low-dimensional topology and geometry via the concept of thin position. In knot theory, this concept was first introduced by D. Gabai and used to solve the long-standing property R conjecture. The analog to thin position in minimal surface theory has been studied extensively the H. Rubenstein; in particular, thin position proved to be the key to the solution to the recognition problem for the 3-sphere. The knot theoretic interpretation of thin position has recently been used to give an alternate proof of Rubenstein's theorem, confirming close connections between the two points of view. Pursuit of such linkage between two different areas promises to be extremely fruitful. The goals of this project are twofold; first, to acquire a deep understanding of minimal surface theory, particularly PL-minimal surface theory. Second, to exploit thin position, with the intent of using insights from knot theory to solve interesting questions in minimal surface theory and vice-versa. ***
