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空间中的一条路径是非常有用的，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，它有任何洞吗？- 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，都具有直接的实际重要性。代数拓扑使用代数对空间的一些基本几何结构进行编码，使空间（及其洞）能够像数字一样被操纵。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专业，代表了编码信息量和易于操作之间的一个很好的折衷。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。我们将估计一个空间的“Lusternik-Schnirelmann范畴”（建立空间所需的最小简单块数），空间的“Toral秩”（它所拥有的最简单对称类型的最大维数），并研究“椭圆”空间的逼近，其中两个经典的空间代数测度都是有限的。我们还将使用“持久同源性”的新兴工具来研究大型数据集的几何形状，并尝试使用有理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.