Principal Investigator: Mark A. Stern
In collaboration with Anda Degeratu, the principal investigator will continue their study of harmonic spinors on manifolds with degenerating spin structures. Their primary goal is to use these degenerating spin structures to extend Witten's technique for proving the positive mass theorem to new classes of higher dimensional nonspin manifolds. They will also investigate the application of index theory to these degenerate spin structures in order to find new relations for the A hat genus on nonspin manifolds. The principal investigator will also continue his study of a geometric model of B fields and the higher p-form potentials arising in string theory, which is suitable for understanding how these fields affect the application of geometric analysis to string theory. He will then apply this improved understanding to various questions in string theory, including joint work with Savdeep Sethi on the analysis of gravitational anomalies of field theories arising from string theories on singular spaces.
At low speeds and masses, general relativity should reduce to Newton's theory of gravity. This correspondence is confused by the fact that there is no definition of mass in general relativity which allows one to define the mass density at a single point in space. One can only define the mass of large regions. Physicists have introduced several notions of mass to obviate this difficulty, the best known of which is the ADM mass. The positive mass conjecture posits that this complicated ADM mass is never negative. This is important because a negative mass universe seems physically unreasonable. Moreover, the truth of the positive mass conjecture has strong implications for the very stability of space. In lower dimensions (including 4) this conjecture was proved by Schoen and Yau. Recent work in string theory has led to the study of gravity in more than 4 spacetime dimensions and thus interest in higher dimensional positive mass theorems under investigation in this project. Many related questions arising in the gravity theories produced by string theory are not currently amenable to careful mathematical analysis because the correct mathematical translation of some of the important string theory concepts, such as higher p form potentials, remains unclear. One of the goals of this project is to provide and apply such a translation.
