Mathematical Sciences: Studies on 3- and 4-Manifolds

9402988摩根教授摩根继续研究光滑4-流形M的Donaldson多项式不变量，包括寻找显式计算。一个是爆破公式，它用M的不变量来表示M与复射影平面的连通和的不变量。这样的公式对于理解不变量之间的一般关系应该是必不可少的。该项目的另一部分，继续与加州理工大学的Tomasz Mrowka开始的工作，研究被3环面分成两部分的4流形的不变量。这些不变量应该通过一个广义乘积公式与这两个部分的不变量相关联。以更定性的方式，摩根教授打算寻找可以定义这些不变量的其他背景，希望其他观点将阐明这一理论，并解释目前正在获得的一些令人惊讶的简单和连贯的数值结果。比尔曼教授的项目以3-空间中的结和环理论为中心。她从两个角度来看待这个问题。第一个是与W.Menasco的联合工作(现已进入第8个年头，大部分前期工作已经完成)。它的目标是通过辫子理论为链接问题提供算法解决方案。她的第二个项目与第一个项目密切相关，涉及三维空间中节点和链接的瓦西里耶夫-康采维奇不变量。这些数值不变量包括琼斯多项式及其推广中包含的所有信息，可能还有更多。一个中心问题是，瓦西里耶夫不变量是否比琼斯不变量包含更多的信息，特别是，它们是否检测方向。当不变量的阶数达到无穷大时，Vassiliev代数的维度的渐近性也涉及到Bman教授正在研究的组合学中的一些非常有趣的问题。摩根教授的研究涉及四维人流形的不变量，而比尔曼教授的研究集中在计算机和节点上。每一种都是通过使其易于进行数值或代数计算来驯服几何复杂性的努力。例如，考虑一下后者。虽然结是最常见的日常对象之一(每个渔民都知道)，但它们的分类在3-流形拓扑中是一个深刻而困难的问题。粗略地说，给出两个结，其中一个希望能够(在计算机的帮助下)决定其中一个是否可以扭曲和变形，保持其两端固定而不切断绳子)，直到它看起来像另一个。准确地描述一种做到这一点的方法是比尔曼教授(与纽约州立大学水牛城分校的W.Menasco)共同研究的一个悬而未决的问题。同一问题的另一个方面涉及到大量的可计算不变量集合(由V.Vassiliev发现)，它们给出了结点问题的部分答案。关于代数不变量，中心问题是瓦西里耶夫不变量到底遗漏了什么信息。虽然这项工作的动机是希望了解潜在的数学结构，但人们预计，对所问问题的答案将适用于任何出现结(或4-流形)的地方，即生物、化学和物理。***

9402988 Morgan Professor Morgan continues to study Donaldson polynomial invariants of smooth 4-manifolds M. This includes searching for explicit computations. One is a blow-up formula that would express the invariants for the connected sum of M with a complex projective plane in terms of those for M. Such a formula should prove essential for understanding general relations among the invariants. Another part of the project, continuing work begun with Tomasz Mrowka of Cal Tech, studies the invariants of a 4-manifold split into two pieces by a 3-torus. These invariants should be related by a generalized product formula to the invariants of the two pieces. In a more qualitative vein, Professor Morgan intends to look for other contexts in which these invariants may be defined, hoping that other points of view will illuminate the theory and will explain some of the surprisingly simple and coherent numerical results now being obtained. Professor Birman's project centers on the theory of knots and links in 3-space. She is approaching this problem from two points of view. The first is joint work with W. Menasco (now in its 8th year, with much of the preliminary work completed). Its goal is an algorithmic solution to the link problem, via the theory of braids. Her second project is closely related to the first, and involves Vassiliev-Kontsevich invariants of knots and links in 3-space. These numerical invariants include all the information which is contained in the Jones polynomial and its generalizations, and possibly more. A central question is whether Vassiliev invariants contain more information than Jones invariants, and, in particular, whether they detect orientation. The asymptotics of the dimension of the Vassiliev algebra, as the order of the invariants goes to infinity, also involves some very interesting problems in combinatorics, which Professor Birman is studying. Professor Morgan's research concerns invariants of four- dimensional man ifolds, while Professor Birman's centers on computers and knots. Each is an effort to tame geometric complexity by rendering it susceptible to numerical or algebraic computation. Consider the latter, for example. While knots are among the most familiar of everyday objects (as every fisherman knows), their classification turns out to be a deep and difficult problem in 3-manifold topology. Roughly speaking, given two knots, one would like to be able to decide (with the help of a computer) if one can be twisted and deformed, keeping its ends fixed and without cutting the string) until it looks like the other. Precisely describing a method for doing this is an unsolved problem on which Professor Birman is working (with W. Menasco, of SUNY Buffalo). A different aspect of the same question concerns a vast collection of computable invariants (discoverd by V. Vassiliev) which give a partial answer to the knot question. With regard to the algebraic invariants, the central question is exactly what information is 'missed' by the Vassiliev invariants. While the motivation for this work is a wish to understand the underlying mathematical structure, one would expect that answers to questions such as the ones being asked would have applications wherever knots (or 4-manifolds) occur, i.e. to biology, chemistry, and physics. ***