Mathematical Sciences: Low-Dimensional Topology, Geometry and Thin-Positions

这个职业进阶奖支持在结理论、最小曲面理论和3流形理论领域中出现的问题的数学研究。这个项目的重点是通过薄位置的概念来探索低维拓扑和几何之间的边界区域。在结理论中，这个概念首先由D. Gabai提出，并用于解决长期存在的性质R猜想。鲁宾斯坦（H. Rubenstein）对最小曲面理论中薄位置的类比进行了广泛的研究；特别地，薄位置被证明是解决3球识别问题的关键。薄位置的结理论解释最近被用来给出鲁宾斯坦定理的另一种证明，证实了这两种观点之间的密切联系。在两个不同领域之间寻求这种联系将会非常富有成效。这个项目有两个目标；首先，深入了解最小曲面理论，特别是pl -最小曲面理论。其次，利用薄位置，利用结理论的见解来解决最小曲面理论中有趣的问题，反之亦然。＊＊＊

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. ***