Project Grant 2303508

The National Science Foundation (NSF) awarded a $152,862 Project Grant under its Mathematical and Physical Sciences program (CFDA 47.049) to Rutgers, The State University of New Jersey - Newark Division. The grant will fund research focused on addressing open problems in geometric analysis and exploring their applications in fields such as geometry, topology, and mathematical physics. Key objectives include developing novel approaches in the regularity theory for linear and fully nonlinear partial differential equations on complex manifolds, with an emphasis on complex Monge-Ampère equations and associated Kähler metrics. The research aims to advance techniques like auxiliary differential equations and establish uniform Poincaré and Sobolev inequalities, as well as heat kernel estimates, to analyze the compactness of Kähler metric families and investigate the asymptotic behavior of metrics near singularities. The project will also explore the parabolic approach in high-dimensional supergravity to discover new solutions to coupled systems, thereby deepening the understanding of the underlying space. The grant period runs from August 15, 2023 to July 31, 2026.