Principal Bundles and Higgs Bundles in Algebraic Geometry

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

The study of principal bundles began in the early XXth century by physicists as a formalism to describe electromagnetism. Later, this was extended to encompass strong and weak interactions, so that principal bundles became a basis for the so-called standard model - a physical theory describing three out of four fundamental interactions. In mathematics, principal bundles penetrate many areas: geometry, number theory, mathematical physics, and others. In 1950's Fields Medalist Jean-Pierre Serre recognized the importance of principal bundles in algebraic geometry. In his 1958 seminal paper he gave the first modern definition of a principal bundle and formulated a certain deep conjecture. This conjecture, as well as some remaining questions, are among the oldest unsolved foundational questions in mathematics. The first part of this project is aimed at proving some of these conjectures. The remaining parts of the projects are related to the so-called Higgs principal bundles, which can be thought of as mathematical incarnations of the Higgs boson -- a recently found elementary particle. These parts of the project belong to the famous Langlands program unifying number theory, algebraic geometry, harmonic analysis, and mathematical physics. This award will support continuing research in these areas. Advising students and giving talks at conferences are going to be part of the proposed activity. In more detail, the first project originated from the Grothendieck-Serre conjecture on principal bundles, which was settled recently for rings containing fields by Ivan Panin and the PI. The PI plans to extend the proof to the mixed characteristic case as well as to work on the purity conjecture for principal bundles. The purity conjecture is, in a sense, the next logical step after the Grothendieck-Serre conjecture. In the second project, the PI intends to construct and prove the local Langlands duality for Hitchin systems and to derive some cases of the global Langlands duality for Hitchin systems from the local case. The third project is devoted to counting motivic volumes of moduli stacks of principal Higgs bundles. The goal is to generalize the previous results of the PI and other people from the GL(n) case to the case of arbitrary reductive groups. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.