Applications of rational homotopy theory

当一个人希望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想象成这个空间中的路径是非常有用的，空间的几何形状携带着系统固有的一些信息。知道空间有多复杂——例如，它有洞吗？——是很重要的。大型数据集（例如人类基因组）或高维数据集也是这种空间数学思想的例子，发现其中的任何几何结构，或者知道它们有多复杂，都具有直接的实际意义。代数拓扑使用代数对空间的一些底层几何结构进行编码，使空间（及其孔）可以像数字一样进行操作。这阐明了它们的一些内在属性，这些属性通常很难以其他方式看到。理性同伦是代数拓扑中的一种专长，它代表了编码信息量和操作方便性之间的一种极好的折衷。在这个项目中，我们将使用有理同伦的工具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”（构建空间所需的最小简单块数），空间的“Toral Rank”（它拥有的最简单对称类型的最大维数），并研究空间的“椭圆”逼近，其中两个经典代数空间度量都是有限的。我们还将使用新兴的“持久同调”工具来研究大型数据集的几何，并尝试使用理性同伦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.