CAREER: Curvature, Topology, and Geometric Partial Differential Equations, with new tools from Applied Mathematics

职业：曲率、拓扑和几何偏微分方程，以及应用数学的新工具

基本信息

批准号：
2142575
负责人：
Renato Ghini Bettiol
金额：
$ 50万
依托单位：
Research Foundation Of The City University Of New York (Lehman)
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2027-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2142575&HistoricalAwards=false
关键词：
CAREER Curvature Topology Geometric Partial

项目摘要

This award is funded in whole or in part under the American Rescue Plan Act of 202 (Public Law 117-2). This project will harness methods associated with applied mathematics to answer fundamental questions in geometry regarding how the curvature of a multi-dimensional object constrains its global shape, and how rigid or malleable geometric configurations can be under various intrinsic or extrinsic conditions. The anticipated findings of this agenda should be applicable to mathematical models used to predict the stability and optimal shape of a wide range of small and large parts of our physical world: from cell membranes, fluid droplets, and airplane wings, to the interface between different layers of the Earth's atmosphere, the event horizon of a black hole, and even the entire universe. Pedagogical efforts will engage graduate and undergraduate students in the discovery process. The latter will be supported by the creation of a 3D printing and visualization lab at CUNY Lehman College, which will be the first of its kind in any public institution of higher education in the Bronx borough of New York City, enabling new forms of inquiry-based instruction grounded on experiential learning. This facility will also be used to host events in partnership with CUNY Bronx Community College, to attract more students to Mathematics and help address the current lack of diversity and overall shortage of workers with STEM qualifications.The lines of investigation in this project can be separated in two main categories, involving novel applications of either convex algebraic geometry or bifurcation theory to geometric analysis. In the first category, new topological obstructions to curvature conditions on closed manifolds will be sought through strategies that combine recently developed convex optimization tools, such as semidefinite programming, and classical local-to-global methods, including Chern-Weil theory, Index theory for twisted Dirac operators, and the Bochner technique. In particular, extremal values of polynomials on spectrahedral shadows of curvature operators will be used to bound characteristic numbers of certain manifolds with nonnegative or nonpositive sectional curvature, or special holonomy. These bounds are expected to shed new light on the Hopf Questions about existence of positively curved metrics in products of spheres, and the sign of the Euler characteristic in nonnegative or nonpositive curvature, as well as on the Stolz conjecture on the Witten genus of string manifolds with positive Ricci curvature. In the second category, global results from bifurcation theory will be used to analyze issues regarding symmetry, stability, rigidity, and multiplicity of minimal and constant mean curvature hypersurfaces, Einstein metrics, and solutions to other partial differential equations that arise in conformal or complex geometry, such as the Yamabe problem and its many variants. This bifurcation-theoretic approach provides several advantages which complement existing variational methods, including a finer control on the topology and regularity of the solutions produced.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.

该奖项的全部或部分资金根据《美国救援计划法 202》（公法 117-2）提供。该项目将利用与应用数学相关的方法来回答几何中的基本问题，例如多维物体的曲率如何限制其整体形状，以及在各种内在或外在条件下刚性或可塑的几何配置如何。本议程的预期结果应适用于数学模型，用于预测我们物理世界的各种大大小小的部分的稳定性和最佳形状：从细胞膜、液滴和飞机机翼，到不同物体之间的界面地球大气层、黑洞的事件视界，甚至整个宇宙。教学工作将让研究生和本科生参与发现过程。后者将得到纽约市立大学雷曼学院 (CUNY Lehman College) 创建的 3D 打印和可视化实验室的支持，这将是纽约市布朗克斯区公立高等教育机构中的第一个此类实验室，从而实现新形式的探究——以体验式学习为基础的教学。该设施还将用于与纽约市立大学布朗克斯社区学院合作举办活动，以吸引更多学生学习数学，并帮助解决目前缺乏多样性和具有 STEM 资格的工作者总体短缺的问题。该项目的调查路线可以分开分为两个主要类别，涉及凸代数几何或分岔理论在几何分析中的新颖应用。在第一类中，将通过结合最近开发的凸优化工具（例如半定规划）和经典的局部到全局方法（包括 Chern-Weil 理论、指数理论）的策略来寻找闭流形上曲率条件的新拓扑障碍。扭曲的狄拉克算子和博赫纳技术。特别是，曲率算子的谱面阴影多项式的极值将用于限制具有非负或非正截面曲率或特殊完整的某些流形的特征数。这些界限有望为关于球体乘积中正曲率度量的存在性的霍普夫问题、非负或非正曲率中欧拉特征的符号以及关于弦流形维滕属的斯托尔兹猜想提供新的线索具有正 Ricci 曲率。在第二类中，分岔理论的全局结果将用于分析有关对称性、稳定性、刚性以及最小和恒定平均曲率超曲面的多重性、爱因斯坦度量以及共形或复杂几何中出现的其他偏微分方程的解的问题，例如 Yamabe 问题及其许多变体。这种分岔理论方法提供了多种优势，可以补充现有的变分方法，包括对所产生的解决方案的拓扑和规律性进行更精细的控制。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优势和更广泛的评估进行评估，被认为值得支持。影响审查标准。