The Principal Investigator, Dr. Christina Sormani, brought together a team of sixteen postdocs and doctoral students to analyze sequences of curved geometric spaces with applications to general relativity.   Some of these postdocs began working with Dr. Sormani in the Summer of 2017 at the Fields Institute in Toronto, while others only came on board recently when Dr. Sormani was visiting the Simons Center for Geometry and Physics and the Institute for Advanced Study during her sabbatical in Fall 2018-Spring 2019.   Located at a variety of universities around the world, the team came together through a series of workshops.  The funding has successfully supported not only the research and travel of the sixteen junior mathematicians on the team but also trained over two hundred additional mathematicians who participated in the 2020 Virtual Workshop on Ricci and Scalar Curvature.

The team has focused on both the direct applications of intrinsic flat convergence to sequences of curved spaces that arise in general relativity and the development of new mathematical techniques that can be applied to better understand these curved spaces.   A few decades ago, Schoen and Yau proved that when the mass of an isolated gravitational system in space is zero, then the space has zero curvature and is flat.  However it is known that even with a tiny amount of mass, space can be strongly curved and have deep gravity wells.   Nevertheless it is conjectured that the curved space will be close to flat Euclidean space in the intrinsic flat sense.  A related question proposed by Gromov is whether a curved torus with almost nonnegative scalar curvature is close to a flat torus in the same sense.   This grant has partially funded research on these questions by team members: Dr. Allen, Ms. Hernandez, Dr. Parise, Dr. Payne, Dr. Wang, Dr. Cabrera-Pacheco, Dr. Ketterer, Dr. Perales, and Dr. Bryden.  Their work completed jointly with Professors Sormani, Sakovich, and Khuri has resulted in five papers posted in the arxiv some of which have already been accepted for publication.   

Key development of theoretical techniques was completed by Dr. Kazaras, Dr. Allen, Dr. Basilio, Dr. Bryden, Dr. Kazaras, Dr. Perales, Dr. Adelstein, Dr. Taskent, Dr. Taylor-Bruce, and Dr. Mouille, resulting in nine papers that are posted on the arxiv some of which are already published.  It akes a few years for a theoretical paper in Geometric Analysis to be carefully checked before being accepted for publication.   One of these papers (by Dr. Basilo, Dr. Kazaras, and Prof. Sormani) provides the first example of a sequence of Riemannian manifolds of positive scalar curvature which converges in the intrinsic flat sense to a space with no geodesics: the sphere with the restricted distance from Euclidean space.   Another pair of papers (by Prof. Sormani with Dr. Allen and by Dr. Allen with Dr. Bryden) explore the relationships between various notions of convergence in Geometric Analysis.   

Perhaps the most important paper resulting from this project is joint work of Prof. Sormani with postdocs, Dr. Allen and Dr. Perales.   This paper develops a new technique for proving sequences of manifolds converge in the intrinsic flat sense.   This new technique is applied to prove that if a sequence of spaces has distances bounded below by the limit space’s distance and volumes converging to the limit space’s volume, then the sequence of manifolds converges in the intrinsic flat sense.   This theorem is directly applied to Gromov’s Torus Conjecture in a subsequent paper by Dr. Perales, Dr. Ketterer, and Dr. Cabrera-Pacheco.  It is extended in work of Dr. Allen and Dr. Perales to manifolds with boundary and applied to solve a special case of the Positive Mass Conjecure mentioned above.   It is expected that this paper will have many exciting applications in the years to come.

This project has welcomed diverse mathematicians from around the world to participate.  Of the sixteen members of this research team of postdocs and doctoral students: three are women, one is Black, and three are Hispanic. Some are employed at universities in the United States, Mexico, and Canada, and others in Europe.  The undergraduate research team that contributed to one of the papers on this project included four black students and six women students out of twelve, some of whom are now pursuing doctorates and others applying to doctoral programs.   The participants in the Virtual Workshop on Ricci and Scalar Curvature, were located in the United States, Mexico, Canada, Argentina, Brasil, Colombia, Uruguay, Spain, Italy, Switzerland, France, the United Kingdom, Germany, the Netherlands, Sweden, Russia, Turkey, India, China, Japan, Egypt, and Morocco.   Junior participants were funded with stipends to pay for the equipment and childcare they needed to participate online.   This workshop enabled  mathematicians to work together even when the pandemic caused the cancellation of nearly all the other conferences in Geometric Analysis worldwide.

Last Modified: 12/28/2020
Modified by: Christina A Sormani