Algebro-Geometric Applications of Factorization Homology

Topology is the study of shapes, more precisely of topological spaces, up to continuous deformations such as stretching, twisting, and bending, but not tearing or gluing. For example, to a topologist, a mug is equivalent to a doughnut. One of the main ways to study topological spaces is to attach to them algebraic invariants, values that remain unchanged under continuous deformations. Betti numbers, and their more refined version, (co)homology, are among the oldest such invariants. Roughly speaking, the k-th Betti number counts the number of k-dimensional holes in a given space. For example, all Betti numbers of the plane are zeros. On the other hand, the first Betti number of a circle is onethe number of one-dimensional holes. Algebraic geometry, on the other hand, is the study of algebraic varieties, solutions to systems of polynomial equations, which are far more rigid than topological spaces. The solutions one considers live in general number systems called rings, which include the familiar rational, real, and complex numbers but also discrete ones such as finite fieldsnumber systems with only finitely many numbers. An important example is mod p modular arithmetic, where one considers integers up to multiples of a fixed prime p. One of the main triumphs of 20th-century mathematics is the development of scheme theory and associated Betti cohomology theories which allow one to put geometry on the set of solutions and to count the number of holes in them, even when working over finite fields where the sets of solutions are finite and discrete. In fact, Weil conjectures (now theorems) give the precise sense in which the topology of the set of complex solutions, its Betti numbers in particular, governs the number of solutions over finite fields. Applying topological methods to study algebro-geometric objects has been the theme of many mathematical breakthroughs during the last couple of decades. Factorization homology, the research proposals main focus, is a beautiful marriage of ideas coming from topology, algebraic geometry, representation theory, and physics. Unlike the additive nature of usual cohomology theories (number of points/holes adds up), factorization homology yields multiplicative invariants. This special feature has been exploited to attack problems in many different areas of mathematics with great successes. This research plan proposes to extend and generalize factorization homology in the context of algebraic geometry and apply it to problems regarding configuration spaces, Hilbert schemes of points, cohomology of moduli spaces of bundles, and knot invariants. These problems lie at an interesting crossroad of algebraic geometry, representation theory, and topology. Factorization homology has not been used to attack these problems, and the methods proposed, if successful, will infuse new techniques to the studies of these subjects.

Given a space X, the configuration spaces of X are geometric objects representing all possible ways "particles" can be placed on X. These configuration spaces play a central role in many areas of mathematics, as they lie in the interface of topology, algebraic geometry, and representation theory, etc. As the number of particles increases, the configuration space becomes more and more complicated. One way to measure the complexity is via counting the number of holes in various dimensions, or mathematically speaking, via the homology groups. It is known that the number of holes in a fixed dimension stabilizes when the number of particles is large enough, a phenomenon known as homological stability. One of the main achievements of the project is a result that exhibits new stability phenomena that involve holes of different dimensions. These so-called higher stability results are much more subtle and have so far been established in very few examples.