An Enumerative Approach to Heegaard Floer

你如何区分三角形和六边形？简单：事实上，这两个“不变量”是等价的，因为它们通过方程X=V-E+F相关联，其中F计算由边勾勒出的二维表面或“面”的数量，X是一个有几百年历史的不变量，称为欧拉特征。一个n边形的欧拉特征线X=n-n+1=1，而一个足球的表面的欧拉特征线X=60-90+32=2。事实上，任何二维球体都有X=2，不管它是如何“绗缝”的。“一个裹着百吉饼的被子的欧拉特征为0，而一个G孔的椒盐卷饼的表面的欧拉特征为X=2-2G。另一个不变量，“基本群”，最近成为头条新闻，在佩雷尔曼证明庞加莱猜想和阿戈尔的突破奖研究之间。基本组跟踪将字符串缠绕在对象周围的可能方法。绳子只是从足球上福尔斯下来，但它可以在百吉饼上缠绕无数次，或者无可救药地缠在椒盐卷饼上。数学家在几何和拓扑学方面的许多工作都涉及类似的“不变量”。上面的缝合曲面被称为2-流形，因为它是由二维的部分构建的。我研究了一种称为Heegaard Floer homology（HF）的3-流形不变量，它是为了帮助计算一个4-流形不变量而发明的，该不变量可以区分在4-流形上进行微积分的不同自洽方法。虽然HF已经有近20年的历史了，但在许多情况下，HF应该告诉我们关于这些三维流形的什么仍然是一个谜。然而，最近的一些照片可能会提供解开这个谜的钥匙。Boyer、Gordon和沃森已经证明了HF为零的3-流形--“L-空间”--和基本群中的所有包装都可以被赋予“左序”（LO）的3-流形之间的二分法--这是一种特殊的群序，相对于附加的弦包装来说是自洽的。与此同时，Juhasz用co-oriented taught foliation（CTF）在L-空间和3-流形之间建立了一个二分法--一种从平面（像一张纸）和具有负欧拉特征的2-流形中雕刻3-流形的方法。Nemethi最近证明了一个定理，该定理将L-空间与代数奇点联系起来--由多项式之间的关系控制的特殊挤压点。受这些观察的启发，我自己的研究努力超越这种二分法，而是使用HF作为区分和计数LO、CTF和某些其他结构的工具。这种方法以前从未被采用过，但我已经开发出了新的工具，并开始产生效果。这些努力中的任何一项的成功都可能使几何学家和拓扑学家能够回答的问题的类型发生革命性的变化。事实上，令人惊讶的是，数学家对我们所居住的世界的三维几何和拓扑仍然知之甚少，尽管错综复杂的物体的行为，特别是在分子水平上，影响着我们的日常生活。例如，打结的DNA的拓扑结构影响拓扑异构酶可以制备用于转录的DNA的速度，这是开发中的某些癌症治疗的重要因素。帝国理工学院的多萝西·巴克博士研究了诸如此类的问题，她的许多学生使用Heegaard Floer同源性来推进我们对结和缠结的三维拓扑结构的理解。谁知道HF的未来会有哪些应用？

How do you distinguish a triangle from a hexagon? Easy: you count the edges E or vertices V. In fact, these two "invariants" are equivalent, since they are related by the equation X=V-E+F, where F counts the number of two dimensional surfaces or "faces" outlined by edges, and X is a centuries-old invariant called the Euler characteristic. An n-gon has Euler characteristic X=n-n+1=1, while the surface of a football has Euler characteristic X=60-90+32=2. In fact, any 2-dimensional sphere has X=2, no matter how it is "quilted." A quilt wrapping a bagel has Euler characteristic 0, and the surface of a G-holed pretzel has X=2-2G. Another invariant, the "fundamental group," has made recent headlines, between Perelman's proof of the Poincare conjecture and Agol's Breakthrough Prize research. The fundamental group keeps track of possible ways to wrap string around an object. The string just falls off a football, but it can wrap countably many times around a bagel or get hopelessly tangled on a pretzel.A lot of the work of mathematicians in geometry and topology deals with "invariants" analogous to these. The quilted surfaces above are called 2-manifolds, since built from 2-dimensional pieces. I study an invariant for 3-manifolds called Heegaard Floer homology (HF), which was invented to help calculate a 4-manifold invariant that distinguishes different self-consistent ways of doing calculus on 4-manifolds. Although HF is nearly 2 decades old, in many cases it remains a mystery what HF should tell us about these 3-manifolds.There are some recent conjectures, however, which might provide keys to unlock this mystery. Boyer, Gordon, and Watson have conjectured a dichotomy between 3-manifolds with vanishing HF--"L-spaces"--and 3-manifolds for which all the wrappings in the fundamental group can be given an "left order" (LO)--a special ordering on the group which is self-consistent with respect to the addition of additional string wraps. Juhasz, meanwhile, has conjectured a dichotomy between L-spaces and 3-manifolds with a co-oriented taut foliation (CTF)--a certain way of sculpting a 3-manifold out of planes (like sheets of paper) and 2-manifolds with negative Euler characteristic. Nemethi has recently proven a theorem relating L-spaces to algebraic singularities--special squashed up points governed by relationships among polynomials.Inspired by these observations, my own research strives to move beyond such dichotomies and instead use HF as a tool to distinguish and count LOs, CTFs, and certain other structures. This approach has never been taken before, but I have already developed new tools which have begun to deliver results. Success in any one of these endeavours could revolutionize the types of questions geometers and topologists are able to answer.In fact, it is astonishing how little mathematicians still know about the 3-dimensional geometry and topology of the world we inhabit, even though the behaviour of intricately tangled objects, particularly at the molecular level, impacts our daily lives. For example, the topology of knotted DNA influences the speed with which topoisomerase enzymes can prepare DNA for transcription, an important factor for certain cancer treatments in development. Dr Dorothy Buck at Imperial College studies questions such as these, and many of her students use Heegaard Floer homology to advance our understanding of the 3-dimensional topology of knots and tangles. Who knows what applications of HF the future might hold?