Mathematical Sciences: Geometric Topology

伯曼将研究结理论，特别关注辫理论在结和链分类中的应用。第一个项目是与Menasco合作，旨在通过辫子理论来解决链接问题。如果成功，对主要定理的研究，可以被描述为“具有稳定性的马尔可夫定理”，将通过显示存在一系列降低复杂性的动作来建立结问题算法解决方案的存在性，这些动作将一个结的任意闭合辫子代表变成一个具有最小复杂性的辫子。此外，该算法的实现还需要识别某些降低复杂性的操作何时适用。第二个项目是瓦西里耶夫结不变量的研究。一个目标是估计i阶瓦西里耶夫不变量的数目，并可能证明这个数目无限制地增长。第三个问题涉及约翰·穆迪最近对局陈述的忠实性的反例及其与琼斯陈述的相应问题的相关性。伯曼希望确定主席团代表的核心。最后一个项目涉及理解所有琼斯不变量的集合是否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)群体在树上的行为；（ii）使用代数几何计算代数曲面的Donaldson不变量；(3)计算由长圆柱形管连接的两片4流形的Donaldson不变量。第一个项目，与R. Skora合作，关注r树上群的自由行动的分类，特别是哪些群自由行动的问题。他们一直在研究具有合并或hnn扩展的特定类型的非平凡自由产物群。他们建议将这项研究扩展到更普遍的汞合金。项目（ii）和（iii）涉及光滑4流形的Donaldson不变量。项目（ii）探讨与代数几何的联系。Morgan和R. Friedman已经对一般类型椭圆曲面的不变量进行了部分计算，并提出了对其计算的扩展，特别是对椭圆曲面的计算。项目（iii）探讨了一些纯解析的微分几何问题，特别是无限圆柱体上有限能量的ASD连接，以提供4流形中嵌入的黎曼曲面与唐纳森不变量值之间的联系。这些拓扑学项目与数学的其他领域密切相关，例如代数、代数几何，甚至数学物理。进展可以采取许多不同的形式，注意到有关各方的多样性令人鼓舞。

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.