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

RUI：多边形结理论、受控拓扑和同调流形拓扑

• 批准号: 0104111
• 负责人: Heather Johnston
• 金额: $ 9.36万
• 依托单位: Vassar College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104111&HistoricalAwards=false
• 关键词: RUI; Polygonal; Knot; Theory; Controlled

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.

摘要奖：DMS-0104111主要研究者：石楠M. Johnston多边形纽结理论研究了三个空间中边的数目和长度在整个合痕中是固定的嵌入多边形的合痕类。 PI和合著者已经发现了第一个例子的多边形是拓扑unknotted，但其中的isotopy类的嵌入是非平凡的。PI和学生合作者将调查问题，如是否有任何这样的例子等边多边形。手术理论研究的是给定同伦类型内的流形结构集。同调流形的Bryant-Ferry-Mio-Weinbergersurgery精确序列已经被PI用来证明直到s-协边主义，流形的许多几何性质也适用于同调流形。诺维科夫猜想及相关的粗糙诺维科夫猜想是理解外科理论的关键。 PI已经发展了一些技术来攻击不寻常的非一致可收缩空间的粗糙Novikov猜想。受控有界手术理论将被用来进一步研究同调流形和粗糙Novikov猜想。到目前为止，拓扑学家已经研究了流形等良好行为的空间。 然而奇异空间在分析、代数几何和物理等学科中出现得越来越频繁。 本项目中研究的不可分解同调流形是奇异的，它们没有任何具有欧氏邻域的点。 也许这些奇怪的空间有一天会解释弦论所预言的宇宙的额外维度。粗糙几何和约翰·罗伊的粗糙同调是将空间的大尺度行为与局部信息分开的方法。 粗糙几何和拓扑的不变量只依赖于空间的大尺度行为。更好地理解这些对象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。在多边形纽结理论中，对纽结的经典研究进行了新的扭曲，研究了一种不同类型的奇异空间。这个新理论是由万向柔性接头（也许是橡胶管）首尾相连形成一个闭环的操纵杆模型。PI已经制作了第一个被卡住（无法解开）的配置示例，但这只是因为它们是由棍子制成的。 如果同样的结构是由弦构成的，那么它们就可以被解开。弦的纽结理论已经被应用于蛋白质和DNA分子的研究。 对于少数原子，每个键都可以用这里讨论的模型中的一根棍子来表示。这是一个比过去所用的弦更适合小分子的模型。