Applications of rational homotopy theory

当一个人想要定量地研究任何一种系统时，考虑系统的所有可能的构型是有益的，数学家称之为构型的‘空间’。把系统在时间上的演化看作是这个空间中的一条路径是非常有用的，而空间的几何形状携带着一些系统固有的信息。了解空间可能有多复杂--例如它有洞吗？--具有一定的重要性。大量的数据集(例如人类基因组)，或者高维的数据集，也是这种空间数学概念的例子，发现其中的任何几何结构，或者知道它们有多复杂，可能具有直接的实际重要性。代数拓扑学使用代数对空间的一些基本几何结构进行编码，使空间(及其空洞)能够像数字一样被处理。这阐明了它们的一些内在属性，这些属性通常很难用其他方式看到。有理同伦是代数拓扑学中的一个特长，它代表了编码信息量和操作简易性之间的极好折衷。在这个项目中，我们将使用有理同伦的工具来继续我们对空间复杂性的研究。我们将估计空间的‘Lusternik-Schnirelmann范畴’(建立空间所需的最小简单片数)，空间的‘托拉尔秩’(它所具有的最简单对称类型的最大维度)，并研究用‘椭圆’来逼近空间，其中空间的两个经典代数度量都是有限的。我们还将使用新兴的“持久同伦”工具来研究大型数据集的几何形状，并试图使用有理同伦中的技术来改进这些方法。

When one wishes to study any kind of system quantitatively, it is profitable to consider all possible configurations of the system, called the 'space' of configurations by mathematicians. It is very useful to think of the evolution of the system in time as a path in this space, and the geometry of the space carries some information which is intrinsic to the system. Knowing how complicated the space can be - e.g. does it have any holes? - is of some importance.  Large data sets (e.g. the human genome), or data sets in high dimensions, are also examples of this mathematical idea of space, and discovering any geometric structure in these, or knowing how complicated they are, can be of immediate practical importance. Algebraic topology encodes some of the underlying geometric structure of spaces using algebra, enabling spaces (and their holes) to be manipulated as if they were numbers. This elucidates some of their intrinsic properties, which are often hard to see any other way. Rational homotopy, a specialty within algebraic topology, represents an excellent compromise between the amount of information encoded and the ease of manipulation. In this project, we will use the tools of rational homotopy to continue our study of the complexity of spaces. We will estimate the 'Lusternik-Schnirelmann category' of a space (the minimum number of simple pieces needed to build the space), the 'Toral Rank' of space (the maximum dimension of the simplest type of symmetry it possesses), and study the approximation of spaces by 'elliptic' ones, for which two classical algebraic measures of spaces are both finite. We will also investigate the geometry of large data sets using the emerging tool of 'persistent homology', and attempt to improve these methods using techniques from rational homotopy.