LEAP-MPS: Two Conjectures in Mathematical Relativity

这个项目将使用几何方法来解决广义相对论数学中的两个重要问题：彭罗斯不等式猜想和霍罗威茨-迈尔斯猜想。 广义相对论是一种引力的几何理论，其中引力可以用时空的曲率来解释。 也许广义相对论最吸引人的预测是黑洞的存在--在时空中极其密集和巨大的区域，引力极其强大。 最近LIGO对黑洞合并的引力波探测，以及EHT合作的M87超大质量黑洞的“图片”，证实了这一预测。在该项目的一部分中，PI将考虑具有角动量的彭罗斯不等式猜想，该猜想旨在澄清旋转黑洞的总能量，视界面积和角动量之间的关系。 PI还将研究Horowitz Myers猜想，该猜想将建立在反德西特/共形场论（AdS/CFT）对应的基础上，这是现代物理学中与量子场论和超引力理论相关的突破。 该项目将为本科生提供机会，从事广义相对论数学方面的研究项目，重点是从代表性不足的群体中招收学生。 克拉克大学举办的外展活动将鼓励当地高中生探索STEM职业。Khuri，H. Kunduri和S.- T. Yau，PI研究了负弯曲流形上的奇异调和映射和黎曼流形上的反曲率流，以证明广义相对论中涉及（总或准局部）能量，角动量，视界面积和旋转黑洞电荷的几个几何不等式。这些结果为解决具有角动量的Penrose不等式猜想提供了基础。另一方面，与P. K. Hung和M. Khuri，PI证明了一个正能量定理的渐近双曲初始数据集与环面无穷大和光滑的非空边界使用新的水平集技术的时空调和函数。 PI将修改这项技术，以处理完整的渐近双曲流形与环面无穷大，这是霍洛维茨-迈尔斯猜想的设置。这一奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

This project will use geometric methods to address two important conjectures in the mathematics of general relativity: the Penrose inequality conjecture, and the Horowitz-Myers conjecture. General relativity is a geometric theory of gravitation in which gravity can be explained by the curvature of spacetime. Perhaps the most fascinating prediction of general relativity is the existence of black holes – extremely dense and massive regions of spacetime where gravity is extremely powerful. The recent gravitational wave detection of black hole mergers by LIGO, and the `picture' of the M87 supermassive black hole by the EHT collaboration, confirm this prediction. In one part of the project, the PI will consider the Penrose inequality conjecture with angular momentum, which aims to clarify the relationship between the total energy, horizon area, and angular momentum of a rotating black hole. The PI will also work on the Horowitz-Myers conjecture, which would build on the anti-de Sitter/conformal field theory (AdS/CFT) correspondence, a breakthrough in modern physics relating quantum field theories and supergravity theory. The project will provide opportunities for undergraduate students to work on research projects in the mathematics of general relativity, with an emphasis on the recruitment of students from underrepresented groups. Outreach activities hosted at Clark University will encourage local high school students to explore careers in STEM.In recent joint work with M. Khuri, H. Kunduri, and S.-T. Yau, the PI has studied singular harmonic maps to negatively curved manifolds and inverse curvature flows on Riemannian manifolds to prove several geometric inequalities involving (total or quasi-local) energy, angular momentum, horizon area, and charges of rotating black holes in general relativity. These results provide the foundation to address the Penrose inequality conjecture with angular momentum. On the other hand, in collaboration with P.-K. Hung and M. Khuri, the PI proved a positive energy theorem for asymptotically hyperboloidal initial data sets with toroidal infinity and smooth non-empty boundary using the new level set technique of spacetime harmonic functions. The PI will modify this technique to deal with complete asymptotically hyperbolic manifolds with toroidal infinity, which is the setting of the Horowitz-Myers conjecture.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.