Mathematical Sciences: Studies on 3- and 4-Manifolds

9402988 Morgan教授继续研究光滑4流形m的Donaldson多项式不变量，包括寻找显式计算。一个是用M的不变量表示复射影平面上M的连通和的不变量的扩展公式，这样的公式对于理解不变量之间的一般关系是必不可少的。该项目的另一部分是由加州理工学院的Tomasz Mrowka继续进行的工作，研究被3环面分成两部分的4流形的不变量。这些不变量应该用一个广义的乘积公式与这两部分的不变量联系起来。从更定性的角度来看，摩根教授打算寻找可以定义这些不变量的其他背景，希望其他观点能够阐明这一理论，并解释目前获得的一些令人惊讶的简单而连贯的数值结果。伯曼教授的研究项目集中在三维空间中的结和链理论。她从两个角度来看待这个问题。第一个是与W. Menasco的合作（现在是第8个年头，大部分初步工作已经完成）。它的目标是通过辫的理论，通过算法解决链接问题。她的第二个项目与第一个项目密切相关，涉及三维空间中结点和链接的Vassiliev-Kontsevich不变量。这些数值不变量包含了琼斯多项式及其推广中包含的所有信息，甚至更多。一个核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。瓦西里耶夫代数维数的渐近性，当不变量的阶趋于无穷时，也涉及到组合学中一些非常有趣的问题，伯曼教授正在研究。摩根教授的研究涉及四维人折叠的不变量，而伯曼教授的研究则集中在计算机和结上。每一种方法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。以后者为例。虽然结是最熟悉的日常物品之一（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，给定两个结，其中一个希望能够（在计算机的帮助下）决定是否可以扭曲和变形其中一个，保持其末端固定，而不切断绳子)，直到它看起来像另一个。精确地描述这样做的方法是伯曼教授（与纽约州立大学布法罗分校的W. Menasco）正在研究的一个尚未解决的问题。同一问题的另一个方面涉及大量可计算不变量（由V. Vassiliev发现），这些不变量给出了结问题的部分答案。关于代数不变量，核心问题是瓦西里耶夫不变量究竟“遗漏”了什么信息。虽然这项工作的动机是希望理解潜在的数学结构，但人们会期望对这些问题的答案可以应用于任何出现结（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. ***