Mathematical and Computational Linguistics Project N.1 Persistent Homology of Syntactic Parameters

In recent years, a new approach to data analysis has been developed, based on topological methods. The basic idea is to understand structures in a (large) set of data in a (high-dimensional) space, by associating to it a simplicial topological space and studying its topology. The starting point is a set of data with a proximity parameter (such as a distance function). The simplicial complex (Vietoris-Rips complex) is constructed by taking the set of data as the vertex set and assigning a k-dimensional face (k-simplex) to a k + 1-tuple of data {x0, . . . , xk} iff the distances satisfy d(xi, xj) ≤ for all 0 ≥ i, j ≤ k. Other versions of simplicial complexes associated to sets of data are described in [1]. General introductions to topological data analysis can be found in [1], [2], [3], [6]. Software packages are available at [5].