RUI: Polygonal Knot Theory, Controlled Topology and Topology of Homology Manifolds

摘要奖：DMS-0104111主要研究人员：Heather M.Johnston多边形结理论研究三个空间中嵌入的多边形的同构类，其中边的数目和长度在整个同构中是固定的。PI和合著者发现了多边形的第一个例子，这些多边形拓扑不打结，但嵌入的保序类不是平凡的。国际和平研究所和学生合作者将调查一些问题，比如等边多边形是否有这样的例子。外科理论研究给定同伦类型内的流形结构的集合。同调流形的Bryant-Ferry-Mio-Weinberger外科精确序列已被PI用来证明直到S余边法，流形的许多几何性质也适用于同调流形。Novikov猜想和与之相关的粗略Novikov猜想是我们理解外科理论的关键。PI已经发展了一些技巧来攻击异常非一致可压缩空间的粗略Novikov猜想。受控有界外科理论将被用来进一步研究同调流形和粗Novikov猜想。到目前为止，拓扑学家已经研究了行为良好的空间，如流形。然而，奇异空间在分析、代数几何和物理等学科中出现的频率越来越高。在这个项目中研究的不可分解的同调流形是特殊的，它们与欧几里得邻域没有任何点。也许有一天，这些奇怪的空间会解释弦理论所预言的宇宙的额外维度。粗几何和John Roe的粗同调是将空间的大尺度行为从局部信息中分离出来的方法。粗几何和拓扑的不变量仅取决于空间的大尺度行为。更好地理解这些物体及其不变量将有助于拓扑学家对现代物理学中出现的不同类型的奇异空间进行分类。在多边形结理论中，研究了一类不同类型的奇异空间，这是对经典研究结的一种新的扭转。这一新理论的模型是，通过通用的柔性接头(可能是橡胶管)将棍子端到端连接起来，形成一个闭合环。PI已经制作了第一个卡住(无法解开)的配置的例子，但仅仅是因为它们是由棍子制成的。如果同样的构型是由弦组成的，它们就可以被解开。弦的纽结理论已经被应用于蛋白质和DNA分子的研究。在本文所讨论的模型中，对于少量原子，每个键都可以用一根棒子表示，这是一种比过去使用的弦更适合小分子的模型。

AbstractAward: DMS-0104111Principal Investigator: Heather M. JohnstonPolygonal knot theory studies the isotopy classes of embeddedpolygons in three space where the number and length of edges arefixed throughout the isotopy. The PI and co-author have foundthe first examples of polygons which are topologically unknotted,but for which the isotopy class of embeddings is nontrivial. ThePI and student collaborators will investigate questions such aswhether or not there are any such examples for equilateralpolygons. Surgery theory studies the set of manifold structureswithin a given homotopy type. The Bryant-Ferry-Mio-Weinbergersurgery exact sequence for homology manifolds has been used bythe PI to prove that up to s-cobordism many of the geometricproperties of manifolds also hold for homology manifolds. TheNovikov conjecture and related coarse Novikov conjecture are keysto our understanding of surgery theory. The PI has developedsome techniques for attacking the coarse Novikov conjecture forunusual non uniformly contractible spaces. Controlled and boundedsurgery theory will be used to further investigate homologymanifolds and the coarse Novikov conjecture.Up to now, topologists have studied well-behaved spaces such asmanifolds. Yet singular spaces arise more and more frequently insubjects such as analysis, algebraic geometry and physics. Thenon-resolvable homology manifolds studied in this project are sosingular that they have no points whatsoever with Euclideanneighborhoods. Perhaps these strange spaces will someday accountfor the extra dimensions of the universe predicted by stringtheory. Coarse geometry, and the coarse homology of John Roe areways of separating the large scale behavior of spaces from thelocal information. The invariants of coarse geometry andtopology depend only on the large scale behavior of thespace. Better understanding of these objects and their invariantswill help topologists to classify the different types of singularspaces, which appear throughout modern physics. In polygonal knottheory, a new twist on the classical study of knots, a differenttype of singular spaces is studied. This new theory is modeled bysticks joined end to end by universally flexible joints (rubbertubing perhaps) to form a closed loop. The PI has produced thefirst examples of configurations which are stuck (cannot beunraveled), but only because they are made of sticks. If thesame configurations were made of string they could be unraveled.Knot theory of strings has been applied to the study of proteinand DNA molecules. For a small number of atoms each bond can berepresented by a stick in the model discussed here, This is aricher and more appropriate model for small molecules than thestring which has been used in the past.