The PI will work on a range of research projects related to the role of the Brauer group in algebraic and formal geometry, with three key components: the interaction between moduli theory and the period-index problem for Brauer classes, the comparison between the cohomological and Azumaya Brauer groups, and the role of the formal Brauer group in lifting Fourier-Mukai equivalences from positive characteristic. First, as the PI has shown in his earlier research, viewing elements of the Brauer group as isomorphism classes of algebraic stacks gives rise to topological and geometric rigidifications of questions in classical algebra. Using these structures, one can relate these classical questions to modern questions about local-to-global principles and to the geometry of certain moduli spaces closely related to moduli spaces of vector bundles. The PI and his graduate students will continue to investigate these connections and push into new territory. Second, the PI has a list of proposed targets for testing the difference between the cohomological and Azumaya Brauer groups of algebraic spaces. A good understanding of these targets will involve computing their algebraic K-theory and relating it to the K-theory of resolutions of singularities, and the PI expects to work on these examples jointly with his graduate students. Finally, the PI will study Fourier-Mukai equivalences in positive characteristic, with an initial focus on K3 surfaces. One important tool will be crystalline and p-adic forms of the Mukai Hodge structure, and he conjectures that the formal Brauer group will play a role in positive characteristic analogous to that played by the transcendental lattice in Mukai's classical theory. As a special case, the PI and his graduate students will investigate the HKR isomorphism in positive characteristic and its connection with novel characteristic p deformations of moduli spaces of vector bundles of rank p on K3 surfaces coming from the tangent space to the formal Brauer group.
Originally devoted to the study of solutions of polynomial equations, algebraic geometry has vastly expanded its mandate in the last century to encompass a large range of interactions between algebra and geometry and a raft of applications in government and industry. The research component of this project will advance our understanding of an algebro-geometric object with both cryptographic and theoretical significance. The grant will fund work by the PI and the training of his graduate students. In addition to performing research, the PI will extend high school mathematical outreach efforts throughout the Pacific Northwest, with a focus on underserved populations in urban centers and outlying rural areas. He will give lectures and organize targeted online advertising through social networks and search engines in an effort to find and foster unusual mathematical talent, including in untraditional places and through untraditional means. This grant will also support a conference for a cohort of new PhDs in algebraic geometry, with a focus on broadening the interests and mathematical connections within this cohort. In an era of constrained resources and highly focused PhD training, it is essential for the future of the field that we continue to encourage a broad view of the community of algebraic geometers and promote mutual understanding among its members.
