Excellence in Research: Morse theory and Algebraic Topological Methods for Q-curvature type equations

卓越研究：Q 曲率型方程的莫尔斯理论和代数拓扑方法

• 批准号: 2000164
• 负责人: Cheikh Ndiaye
• 金额: $ 44.8万
• 依托单位: Howard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000164&HistoricalAwards=false
• 关键词: Excellence; Research; Morse; theory; Algebraic

In this project supported by NSF's Excellence in Research program, the principal investigator (PI) will mathematically analyze a class of equations that arise from geometry and physics. The applications include the existence and characterization of optimal shapes in geometric problems that are helpful for scientists and engineers in understanding the universe and for optimal design of important objects and tools in the real world. The physics applications include describing energy critical states which are important for the understanding of the problems where an associated energy is quantized, such as, vortices of Euler flows and condensates in some Chern-Simons-Higgs models. One particularity of the equations under study in this project is that they verify the phenomena of strong interaction and quantization, which are enjoyed by many partial differential equations modeling real life problems. The aim of the research is to develop methods that can be used to establish existence mechanisms for such equations that verify the phenomena of quantization and strong interaction. The PI will mentor student research and organize Senior Seminar in Geometric Analysis project topics. The project also has a component that seeks to increase the number of underrepresented groups in STEM disciplines. To this end, the PI will pilot a Bridge to Ph.D. program with the main mission being to increase the number of women and minorities with Ph.D. degrees in Mathematics at Howard University and within the United States. The main goal of this research deals with non-compact geometric variational problems of Q-curvature type. They are on one hand: nonlinear partial differential equations describing the conformal deformation of a Riemannian metric to one of prescribed Q-curvature type quantity, and on the other hand: systems of nonlinear partial differential equations describing the Mean Field and Toda problems from Chern-Simons Theory. These equations arise as Euler-Lagrange equation of energy functionals which are critical with respect to some Moser-Trudinger type inequalities. The focus of the project is on the resonant cases which are when accumulations points of some non-compact flow lines of a pseudo-gradient of the associated Euler-Lagrange functional, the so-called true critical points at infinity of the associated variation problem, occur. The project will investigate existence mechanism using the tools of critical points at infinity of Abbas Bahri. The PI will establish new existence results by developing Morse and algebraic topological arguments for this type of problems. Precisely he will establish a full Degree Theory and Morse Theory for existence for Q-curvature type equations. Moreover, in collaboration with Howard University's Graduate School of Arts and Sciences, the PI will organize an interactive seminar in geometric analysis based on these topics and other related conformally invariant variational problems to recruit and train graduate students to do research. The educational and outreach component of this research project will allow the PI to expose students of different levels and diverse backgrounds how mathematics can be used to model and solve viable real-world problems, to motivate students to use mathematics to undertake scientific challenges of importance, and to increase their interest in pursuing career in mathematics.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.

在这个由美国国家科学基金会卓越研究项目支持的项目中，首席研究员（PI）将对一类由几何和物理产生的方程进行数学分析。其应用包括几何问题中最优形状的存在和特征，这有助于科学家和工程师理解宇宙以及现实世界中重要物体和工具的最佳设计。物理应用包括描述能量临界状态，这对于理解相关能量被量子化的问题很重要，例如欧拉流的涡流和一些陈-西蒙斯-希格斯模型中的冷凝物。本项目所研究的方程的一个特点是它们验证了强相互作用和量子化现象，这是许多模拟现实生活问题的偏微分方程所享有的。本研究的目的是开发可用于建立验证量子化和强相互作用现象的此类方程的存在机制的方法。PI将指导学生的研究，并组织几何分析项目主题的高级研讨会。该项目还有一个组成部分，旨在增加STEM学科中代表性不足的群体的数量。为此，PI将试行一个通往博士学位的桥梁项目，其主要任务是增加在霍华德大学和美国国内获得数学博士学位的女性和少数族裔的人数。本研究的主要目标是处理q曲率型的非紧几何变分问题。它们一方面是描述黎曼度规向规定的q曲率型量之一的保形变形的非线性偏微分方程，另一方面是描述chen - simons理论中的平均场和Toda问题的非线性偏微分方程组。这些方程是能量泛函的欧拉-拉格朗日方程，对于某些Moser-Trudinger型不等式是至关重要的。该项目的重点是共振情况，即当相关欧拉-拉格朗日泛函的伪梯度的一些非紧致流线的累积点，即所谓的相关变分问题的无穷远处的真临界点出现时。该项目将利用Abbas Bahri的无穷远处临界点工具来研究存在机制。PI将通过对这类问题的莫尔斯和代数拓扑论证建立新的存在性结果。准确地说，他将建立q曲率型方程的全度理论和莫尔斯存在理论。此外，PI将与霍华德大学艺术与科学研究生院合作，组织一个基于这些主题和其他相关的共形不变变分问题的几何分析互动研讨会，以招募和培训研究生进行研究。该研究项目的教育和推广部分将允许PI向不同水平和不同背景的学生展示如何使用数学来建模和解决可行的现实世界问题，激励学生使用数学来承担重要的科学挑战，并增加他们追求数学事业的兴趣。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

First explicit constrained Willmore minimizers of non-rectangular conformal class

非矩形共形类的第一个显式约束 Willmore 最小化器

• DOI: 10.1016/j.aim.2021.107804
• 发表时间: 2021
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: Heller, Lynn;Ndiaye, Cheikh Birahim
• 通讯作者: Ndiaye, Cheikh Birahim

Isothermic constrained Willmore tori in 3-space

3 空间中的等温约束 Willmore 环面

• DOI: 10.1007/s10455-021-09778-1
• 发表时间: 2021
• 期刊: Annals of Global Analysis and Geometry
• 影响因子: 0.7
• 作者: Heller, Lynn;Heller, Sebastian;Ndiaye, Cheikh Birahim
• 通讯作者: Ndiaye, Cheikh Birahim

Stability properties of 2-lobed Delaunay tori in the 3-sphere

3 球体中 2 瓣 Delaunay 环面的稳定性特性

• DOI: 10.1016/j.difgeo.2021.101805
• 发表时间: 2021
• 期刊: Differential Geometry and its Applications
• 影响因子: 0.5
• 作者: Heller, Lynn;Heller, Sebastian;Ndiaye, Cheikh Birahim
• 通讯作者: Ndiaye, Cheikh Birahim

Optimal control for the infinity obstacle problem

无限远障碍问题的最优控制

• DOI: 10.1090/proc/15455
• 发表时间: 2021
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Mawi, Henok;Ndiaye, Cheikh Birahim
• 通讯作者: Ndiaye, Cheikh Birahim

Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian

• DOI: 10.3934/dcds.2022085
• 发表时间: 2021-07
• 期刊: Discrete and Continuous Dynamical Systems
• 影响因子: 1.1
• 作者: Martin Gebhard Mayer;C. B. Ndiaye