Mathematical Sciences: Geometric Topology

伯曼将研究纽结理论， 辫子理论在分类中的应用 结和链接。 第一个项目，与梅纳斯科，旨在 算法解决方案的链接问题，通过理论 辫子 如果成功的话，主要定理的工作，这可能是 被描述为“稳定化的马尔可夫定理”，将 建立一个算法解决方案的存在的结 问题，表明有一系列的复杂性降低， 移动采取任意闭合辫子代表一个 结到一个具有最小复杂性的节点。 执行 此外，该算法将需要识别何时确定 降低复杂性的举措是适用的。 第二个项目是 Vassiliev的纽结不变量的研究 一个目标 估计i阶Vassiliev不变量的个数， 也许可以证明这个数字的增长是无限的。 第三 约翰·穆迪最近的一个反例， 及其与相应的 关于琼斯的提问 伯曼希望能找出 Burau表示的核心。 最终项目 关于了解所有琼斯的集合是否 不变量可以忠实于结和链接类型。 Morgan打算研究拓扑学的三个相关领域： (i)树上群的作用;（ii）唐纳森的计算 代数曲面的不变量，使用代数几何;以及 (iii)四维流形的唐纳森不变量的计算 是由两块由一根长圆柱形管连接而成的。 的 第一个项目，R。斯科拉，涉及免费的分类 R-树上的群的作用，特别是， 哪些群体可以自由行动 他们一直在研究 是某些类型的非平凡的免费产品， 或HNN扩展。 他们建议将这项研究扩展到更多 一般汞合金。 项目（ii）和（iii）涉及唐纳森的 光滑四维流形的不变量 项目（二）探索 与代数几何的联系 Morgan和R.弗里德曼有 部分计算这些不变量的椭圆曲面的 一般类型，并建议扩展他们的计算，特别是 椭圆曲面的那些。 项目（三）探讨了一些纯粹的 解析的微分几何问题，特别是ASD 连接的有限能量无限圆柱体，为了 在四维流形中提供嵌入的黎曼曲面之间的链接 和唐纳森不变量的值。 这些是拓扑学中的项目， 数学的其他领域，例如，代数，代数几何， 甚至数学物理学。 要取得进展， 不同的形式，令人鼓舞的是， 感兴趣的各方。

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.