New Perspectives on Four-Dimensional Geometry

四维对象的数学概念允许通过考虑其点(编码为三个空间坐标)以及任何其他相关的量，例如其温度、压力或电导率，来更全面地研究存在于我们的物理世界中的三维对象。这种同时处理空间坐标和其他逐点参数的方法在现代科学中既基本又普遍，从爱因斯坦广义相对论中的时空和粒子物理学中的D膜，到用于医学成像和诊断、工业机器人、城市交通流量、金融市场和无线通信等领域的数学模型。这个项目的主要目标是促进对作为抽象数学实体的四维对象的几何理解，这种方法足够通用，可以应用到任何使用四维模型的领域。特别是，这个项目将分析某些四维形状的刚性或延展性，在某些自然曲率假设下如何变化，以及如何有效地检测这些曲率属性。即将进行的研究的一个决定性特征是使用最近在传统上与几何无关的数学领域开发的尖端技术，为一个经典学科带来新的视角，试图颠覆性地解决它的一些最重要的开放问题。该项目还将通过宣传该领域的最新研究进展，部分支持纽约城市大学几何分析研讨会及其在城市大学研究生方案中的教育使命；以及在纽约市布朗克斯区的西语裔服务机构--纽约州立大学雷曼学院开展的几项公共外联活动。用更专业的术语来说，这个项目将探索凸代数几何和半定规划等新兴领域在几何分析和黎曼几何中的各种应用，例如通过研究具有截面曲率界限的四维流形的曲率算子的半代数集作为异面体阴影和异面体的极限。这种关于四维流形的曲率算子的实代数几何观点有望以截面曲率界限的显式代数(多项式)刻画、秩刚性结果和曲率估计的最优化的形式产生几个全局结果。此外，该项目试图利用前面提到的技术和它们的基本群的三分割法来检测四维流形中对正曲率的新的拓扑障碍。其他主题包括四维爱因斯坦流形、利玛窦孤子和利玛奇流。大多数研究将与博士生、职业生涯早期的数学家和其他研究人员合作进行，部分支持该领域新专家的培训。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The mathematical concept of a four-dimensional object allows a more comprehensive study of three-dimensional objects that exist in our physical world by considering its points (encoded as three space coordinates) together with any other relevant quantity, for example its temperature, pressure, or electric conductivity. This simultaneous treatment of space coordinates and other pointwise parameters is both essential and ubiquitous in modern science, from spacetimes in Einstein's theory of General Relativity and D-branes in Particle Physics, to mathematical models used in medical imaging and diagnosis, industrial robotics, urban traffic flows, financial markets, and wireless communications, among many others. The main goal of this project is to advance the geometric understanding of four-dimensional objects as abstract mathematical entities, an approach that is general enough to allow applications to any field that makes use of four-dimensional models. In particular, this project will analyze how rigid or malleable certain four-dimensional shapes are, how that changes under certain natural curvature assumptions, and how to efficiently detect these curvature properties. A defining characteristic of the research to be conducted is the use of cutting-edge techniques recently developed in areas of mathematics not traditionally associated with geometry, bringing new perspectives to a classical subject, in a disruptive attempt to solve some of its most important open questions. This project will also partially support the CUNY Geometric Analysis seminar, and its educational mission in the CUNY graduate program, through the communication of latest research advances in the field; as well as several public outreach activities at CUNY Lehman College, a Hispanic Serving Institution in the Bronx borough of New York City. In more technical terms, this project will pursue various applications of the emerging fields of Convex Algebraic Geometry and Semidefinite Programming to Geometric Analysis and Riemannian Geometry, for instance through the study of semialgebraic sets of curvature operators of four-manifolds with sectional curvature bounds as spectrahedral shadows and limits of spectrahedra. This real algebro-geometric viewpoint on curvature operators of four-manifolds is expected to have several global consequences in the form of explicit algebraic (polynomial) characterizations of sectional curvature bounds, rank rigidity results, and optimization of curvature estimates. Furthermore, the project seeks to detect new topological obstructions to positive curvature in four-manifolds using the aforementioned techniques and trisections of their fundamental groups. Other topics covered include four-dimensional Einstein manifolds, Ricci solitons, and Ricci flow. Most of the research will be conducted in collaboration with doctoral students, early-career mathematicians, and other researchers, partially supporting the training of new specialists in the field.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.