Floer homology and low-dimensional topology

数学结被想象为简单但打结的封闭绳索，自12世纪末以来一直被科学家研究。早期的动机之一是利用这些物体作为理解原子和物质性质的一种方式。多年来，与自然科学的联系日益密切，如今，DNA拓扑学是一个活跃的跨学科研究领域。从纽结理论的早期开始，纽结的分类问题就发挥了核心和激励作用。基本的问题似乎很简单：人们能说出两个给定的结在拓扑上是否等价吗？解决这个问题的方法使拓扑学家研究了各种类型的纽结，并试图刻画它们的众多不变量。此外，几代数学家构造了越来越复杂的纽结不变量，旨在捕捉纽结的基本性质，如亏格、切片亏格、解结数等。从数学的角度来看，纽结的许多用途之一是它们与称为三维流形的对象密切相关，而三维流形的分类一直是60多年来数学中的一个中心问题。事实上，这两者(非T和3-流形)之间的关系的例子是由Dehn运算和沿纽结的循环分支覆盖等构造提供的。我们的研究涉及纽结和这样的3-流形的不变量，我们感兴趣的是从流形上的椭圆型偏微分方程得到的不变量。特别是，对Floer同调理论的各种形式的研究帮助证明了低维拓扑领域的各种猜想，申请人的研究方案概述了可以用这些强大的不变量来解决的各种问题。

Mathematical knots, visualised as a simple but knotted closed rope, have been studied by scientists since the end of the XIXth century. One of the early motivations was to use these objects as a way to understand properties of atoms and matter. The ties to natural sciences have grown over the years and nowadays, for example, the study of DNA topology is an active, interdisciplinary area of research.From early days of Knot theory, the classification problem for knots has played a central and motivating role. The basic problem seems quite simple: can one tell if two given knots are topologically equivalent or not. Approaches to this problem have led topologists to study various classes of knots and to try to characterize their numerous invariants. Moreover, generations of mathematicians have constructed increasingly complex knot invariants which aim to capture essential properties of knots such as genus, slice genus, unknotting number and so on.One of the many uses of knots, from a mathematical perspective, is that they are intimately related to objects called 3-manifolds, whose classification has been a central problem in Mathematics for more than 60 years. Indeed examples of the relationship between the two (nonts and 3-manifolds) are provided by constructions like Dehn surgery and cyclic branched covers along knots.  Our research is concerned with invariants of knots and such 3-manifolds, the invariants we are interested in being obtained from elliptic partial differential equations on manifolds. In particular, the study of Floer homology theory in its various flavours has helped to prove various conjectures in the area of low-dimensional topology and the applicant's research proposal outlines various problems which may be approached with these powerful invariants.