Mathematical Sciences: Representations of Three-Manifold Groups

Andrew Casson将研究3流形的几何化问题。他将尝试改进“强环面定理”，得出包含本质奇异环面的不可约3-流形要么包含1属的本质嵌入面，要么是Seifert纤维空间。他的方法与图基亚使用的方法有关。他还将尝试在3流形上寻找保证PSL2(C)中基本群表示存在的良好可用条件。特别是，他将寻求找到双曲或其他几何上有趣的表示的方法。robon Kirby将继续研究Witten的程序，通过SU(n)值连接的chen - simons不变量的“平均值”来定义框架3流形的不变量。他还将研究由嵌入2球上的“Gluck扭转”得到的光滑4流形的微分同构分类，或对具有一个双点的浸入2球的推广。Yakov Eliashberg将研究Legendrian结的不变量。这三个项目都涉及三维流形的不同方面，它们是由3个独立坐标在局部描述的自然几何物体。3流形的结构与数学的其他部分有关，因为这种流形是常方程或微分方程的解集。与物理理论的相关性也很明显。

Andrew Casson will study aspects of the geometrization problem for 3-manifolds. He will attempt to improve the "Strong torus theorem" to conclude that irreducible 3-manifolds which contain essential singular tori either contain essential embedded surfaces of genus one or are Seifert fibered spaces. His method is related to one used by Tukia. He will also attempt to find good usable conditions on a 3-manifold guaranteeing the existence of representations of the fundamental group in PSL2(C). In particular, he will seek methods of finding hyperbolic or other geometrically interesting representations. Robion Kirby will continue studying Witten's program for defining an invariant of framed 3-manifolds via an "average" of the Chern-Simons invariants for SU(n) valued connections. He will also work on the diffeomorphism classification of smooth 4-manifolds obtained by "Gluck twists" on imbedded 2-spheres, or a generalization to immersed 2-spheres with one double point. Yakov Eliashberg will study invariants of Legendrian knots. All three projects concern different aspects of 3-dimensional manifolds, which are natural geometric objects describable locally by 3 independent coordinates. The structure of 3-manifolds has relevance to other parts of mathematics inasmuch as such manifolds occur as solution sets of ordinary or differential equations. Relevance to physical theories is also evident.