John Francis - Moduli of stratifications and factorization homology
The Ran space Ran(X) is the space of finite subsets of X, topologized so that points can collide. Ran spaces have been studied in diverse works from Borsuk–Ulam and Bott, to Beilinson–Drinfeld, Gaitsgory–Lurie and others. The alpha form of factorization homology takes as input a manifold or variety X together with a suitable algebraic coefficient system A, and it outputs the sheaf homology of Ran(X) with coefficients defined by A. Factorization homology simultaneously generalizes singular homology, Hochschild homology, and conformal blocks or observables in conformal field theory. This alpha form of factorization homology has applications to the study of mapping spaces in algebraic topology, bundles on algebraic curves, and perturbative quantum field theory. There is also a beta form of factorization homology, where one replaces the Ran space with a moduli space of stratifications, designed to overcome certain strict limitations of the alpha form. The key notion is that of a constructible bundle: in terms of its functor of points, a K-point of this moduli space is a constructible bundle over K. One application is to proving the Cobordism Hypothesis, after Baez–Dolan, Costello, Hopkins–Lurie, and Lurie. This is joint work with David Ayala.