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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来自202年美国救援计划法案（公法117-2）。该项目将利用与应用数学相关的方法来回答几何学中的基本问题，即多维对象的曲率如何约束其全局形状，以及在各种内在或外在条件下几何配置的刚性或延展性如何。这一议程的预期发现应该适用于数学模型，用于预测我们物理世界中各种大小部分的稳定性和最佳形状：从细胞膜，液滴和飞机机翼，到地球大气层不同层之间的界面，黑洞的事件视界，甚至整个宇宙。教学工作将使研究生和本科生参与发现过程。后者将得到CUNY Lehman College创建的3D打印和可视化实验室的支持，该实验室将成为纽约市布朗克斯区任何公立高等教育机构中的第一个，从而实现基于体验式学习的新形式的探究式教学。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，以吸引更多的学生学习数学，并帮助解决目前缺乏多样性和具有STEM资格的工人整体短缺的问题。该项目的调查路线可以分为两个主要类别，涉及凸代数几何或分叉理论在几何分析中的新应用。在第一类中，将通过联合收割机结合最近开发的凸优化工具（如半定规划）和经典的局部到全局方法（包括Chern-Weil理论、扭曲狄拉克算子的指数理论和Bochner技术）的策略来寻求闭流形上曲率条件的新拓扑障碍。特别地，曲率算子的谱面阴影上的多项式的极值将被用来界定某些具有非负或非正截面曲率的流形或特殊的完整流形的特征数。这些界限有望为有关球面积中正弯曲度规存在性的霍普夫问题、非负或非正曲率中欧拉特征线的符号以及关于具有正里奇曲率的弦流形的维滕属的斯托尔兹猜想提供新的线索。在第二类中，分歧理论的全局结果将用于分析关于对称性，稳定性，刚性和最小和常数平均曲率超曲面的多重性，爱因斯坦度量以及在共形或复杂几何中出现的其他偏微分方程的解，例如Yamabe问题及其许多变体。这种分叉理论的方法提供了几个优势，补充现有的变分方法，包括一个更好的控制拓扑结构和规则性的解决方案produced.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。