Applications of rational homotopy theory

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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.