Ricci Curvature and Torsion

点击翻译按钮获取中文摘要

Geometry plays a fundamental role in different areas of mathematics, physics, and other scientific disciplines. A central and deeply studied concept in our modern approach to geometry is that of `curvature.’ Uncovering the modern definition of curvature played a key role in the development of Einstein’s theory of relativity over a century ago. On the other hand, the idea of `torsion,’ is a less understood, though no less important, part of geometry. This project will support fundamental research into geometric structures which naturally incorporate torsion, aiming at new applications in mathematics and physics. In addition to supporting research this award will support various training initiatives for undergraduates and graduate students. This includes mentorship of undergraduates to prepare them for graduate study, and professional training and guidance for graduate students. As UC Irvine is nationally recognized as having a diverse student body, this training will support goals of inclusive excellence across at all higher educational levels.The Ricci curvature, Einstein metrics, and Ricci flow play a central, essential, role in our understanding of geometry and analysis, deeply influencing our understanding of a diverse array of subjects such as heat kernels, Brownian motion, partial differential equations, low-dimensional topology, complex geometry, algebraic geometry, and more. Taking inspiration from a variety of new developments in topology, Riemannian geometry, mathematical physics, and complex geometry, a natural extension of Ricci curvature has emerged which incorporates torsion. Attendant to this are natural generalizations of Einstein metrics and Ricci flow. The research project aims at uncovering fundamental geometric and analytic aspects of the generalized Ricci curvature, generalized Einstein metrics, and generalized Ricci flow, ultimately building towards uniformization and classification results with further applications to differential geometry, PDE, complex manifolds, and mathematical physics. The award will aid the training of students at the undergraduate and graduate levels, through outreach activities, by incorporating them in research projects, and through professional training programs.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.