Alternating links and cobordism

数学中的结是空间中的一个闭合环--这与一根绳子上的结是一样的，只是我们规定绳子的两端在打结后应该连接在一起。这个研究项目将使用21世纪世纪的数学来解决结理论中一百多年来一直无法解决的问题。数学结理论始于19世纪世纪，试图根据开尔文勋爵的以太打结涡旋理论来组成一个元素表。以太理论被证明是不正确的，但它开始了一个丰富的数学研究，现在有许多重要的应用。特别是DNA分子表现出打结行为和数学性质的结所涉及的重要生物学意义。为了复制，打结的DNA需要通过一系列交叉变化而解开。一个交叉的变化或股通道是当一个股的结被切断，另一股通过削减，然后修复。泰特是第一个研究这些交叉变化，也是在19世纪。他定义了一个衡量结的复杂性的方法，称为解结数，它计算需要多少交叉变化才能完全解开结。直到今天，计算这些数仍然是一个非常困难的问题;事实上，没有已知的算法来确定一个纽结是否有等于1的解结数。纽结也被用于对三维和四维空间（或流形）进行数学描述，例如我们生活的宇宙。另一个众所周知的困难和重要的问题是决定一个结是否是切片;认为时间是第四维，切片结是时空中二维球体的快照。(This问题只有50岁左右）。在20世纪世纪的最后二十年里，唐纳森、维滕和其他人开创的新技术改变了数学家对四维几何和拓扑的理解。这种新的数学规范理论是从理论物理学的量子场论中衍生出来的。在过去的十年里，由于Ozsvath和Szabo的数学规范理论的新版本在纽结理论和3-欧文斯将联合收割机规范理论结果的唐纳森与新理论的奥兹瓦特和萨博攻击unknotting数和切片问题的一个主要类别的结称为交替结（这些包括众所周知的结，如奶奶结和礁石结或方结）这类结已知在打结的DNA中普遍存在。部分目标是找到一个完整的解决方案，以解开号码一和切片识别问题，这些结。生物学家感兴趣的进一步交叉变化信息也将被发现，以及对这类非常熟悉的结的神秘数学性质的新见解。

A knot in mathematics is a closed loop in space - this is the same thing as a knot in a piece of string except we stipulate that the ends of the string should be joined together after tying the knot. This research programme will use 21st century mathematics to solve problems in knot theory that have defied solution for over a hundred years.Mathematical knot theory began in the 19th century as an attempt to compose a table of elements based on Lord Kelvin's theory of knotted vortices in the aether. The aether theory proved incorrect but started a rich mathematical study that now has many important applications. In particular DNA molecules exhibit knotting behaviour and the mathematical properties of the knots involved have important biological implications. In order to replicate, knotted DNA needs to become unknotted by a sequence of crossing changes . A crossing change or strand passage is when one strand of the knot is cut, and another strand passes through the cut which is then repaired.Edinburgh physicist P. G. Tait was the first to study these crossing changes, also in the 19th century. He defined a measure of complexity of a knot called the unknotting number, which counts how many crossing changes are needed to completely undo the knot. Computing these numbers is a notoriously difficult problem to this day; in fact there is no known algorithm for deciding if a knot has unknotting number equal to one.Knots are also used in giving mathematical descriptions of 3 and 4 dimensional spaces (or manifolds) such as the universe we live in. Another notoriously difficult and important problem is to decide if a knot is slice; thinking of time as the fourth dimension, a slice knot is a snapshot of a two-dimensional sphere in spacetime. (This problem is only around fifty years old.)In the last two decades of the 20th century new techniques pioneered by Donaldson, Witten and others transformed mathematicians' understanding of 4 dimensional geometry and topology. This new mathematical gauge theory was derived from the quantum field theories of theoretical physics.In the last ten years, a new version of mathematical gauge theory due to Ozsvath and Szabo has made major progress on problems in knot theory and 3-dimensional topology.Owens will combine gauge theory results of Donaldson with the new theory of Ozsvath and Szabo to attack the unknotting number and slice problems for a major class of knots known as alternating knots (these include well-known knots such as the granny knot and reef or square knot) This class of knots is known to be prevalent in knotted DNA. Part of the goal is to find a complete solution to the unknotting number one and slice recognition problems for these knots. Further crossing change information of interest to biologists will also be discovered as well as new insights into the mysterious mathematical nature of this very familiar class of knots.