The PI will engage in several projects at the border of algebraic geometry and algebraic topology. Three projects aim to use topological methods to understand the Brauer group, Azumaya algebras, and more generally torsors on schemes. (1) The PI will study the extent to which the foundational results of Jackowski, McClure, and Oliver on maps between classifying spaces of complex algebraic groups can be extended to finite approximations to these classifying spaces. Progress on this problem will enable the solution of a host of problems about when torsors for complex algebraic groups extend from the generic point of a scheme to the entire scheme. In low dimensions, early progress on this problem has been used by the PI and Ben Williams to settle an old question of Auslander and Goldman on the existence of Azumaya maximal orders in unramified division algebras, where it transpires that there are purely topological obstructions to the existence of these Azumaya maximal orders. (2) The PI will work toward computing the Chow groups and singular cohomology of the classifying spaces of special linear groups by various central subgroups. This has been done in special cases by Vezzosi and Vistoli. However, greater generality is needed for most applications. These Chow groups are fundamental objects in algebraic geometry, controlling the characteristic classes associated to certain torsors of fundamental importance in the study of the Brauer group. The computations will be directly useful to the first project, and to the following project. (3) The PI and Ben Williams previously formulated the topological period-index problem and established first results. They will continue this study, especially as it relates to the algebraic period-index conjecture. In particular, their results in low dimensions suggest a method for disproving the period-index conjecture, which would be a fundamental advance. Following this idea to its conclusion is the major aspiration of the first set of projects. A fourth project aims to continue to build a bridge between higher category theory and classical algebraic geometry, bringing the formidable techniques of the former to bear on various questions in the arithmetic of derived categories. For example, the PI is developing a toolbox using higher category theory that will allow a purely derived-category proof of Panin's computations of the K-theory of projective homogeneous spaces, once the existence of certain exceptional objects on the split forms of these spaces is known.
The PI proposes work in algebraic geometry and algebraic topology, two areas of modern mathematics. Algebraic geometry is an ancient subject with many connections to real-world problems. Its goal is to understand the geometry of solutions sets of polynomial equations, equations of central importance in various disciplines, such as theoretical physics, cryptography, and the modeling of dynamical systems like weather. Algebraic topology on the other hand developed more recently, in the 19th century, and aims to study a general notion of shape, less rigid than the idea of shape studied in geometry. It has found striking applications in the last decade, for instance to the analysis of large data sets that occur in computer vision and cancer research, frequently finding patterns that more traditional methods of data analysis fail to find. The proposal of the PI will bring the considerable machinery and insight of algebraic topology to bear on several questions in algebraic geometry which have been identified by the community as among the most important.
