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NSF卓越研究计划支持的该项目中，首席研究员（PI）将数学分析一类来自几何和物理学的方程式。应用程序包括在几何问题中的最佳形状的存在和表征，这对科学家和工程师在理解宇宙方面和最佳设计对现实世界中重要的物体和工具的最佳设计有帮助。物理应用包括描述对能量关键状态，这些状态对于理解量化相关能量的问题，例如某些Chern-Simons-Higgs模型中的Euler流量和冷凝水的涡流。该项目中所研究的方程式的一个特殊性是，它们验证了强烈的相互作用和量化现象，这些现象是通过模拟现实生活问题的许多部分微分方程所享受的。该研究的目的是开发可用于建立这种方程式的存在机制的方法，以验证量化现象和强烈相互作用的现象。 PI将指导学生研究并组织几何分析项目主题的高级研讨会。该项目还具有一个旨在增加STEM学科中代表性不足的组的组成部分。为此，PI将驾驶一座博士学位的桥梁。主要任务是通过博士学位增加妇女和少数民族的数量。霍华德大学和美国境内数学学位。这项研究的主要目标涉及Q狂热类型的非紧凑几何变异问题。它们一方面是：非线性偏微分方程描述了Riemannian指标与处方Q展界型数量之一的形式相形的形式，另一方面：非线性偏微分方程的系统描述了Chern-Simons理论中的平均值和TODA问题。这些方程是作为能量功能的Euler-Lagrange方程而出现的，对于某些Moser-trudinger型不平等至关重要。该项目的重点是共振案例，这是当相关的Euler-Lagrange功能的某些非压缩流量线的积累点（相关变化问题的无穷大属性的所谓的真实临界点）发生时。该项目将使用Abbas Bahri无限的关键点的工具调查存在机制。 PI将通过为这种类型的问题开发Morse和代数拓扑论点来建立新的存在结果。确切地说，他将为Q狂欢类型方程的存在建立完整的理论和莫尔斯理论。此外，与霍华德大学的艺术与科学研究生院合作，PI将基于这些主题和其他相关的共同不变的变异问题在几何分析中组织互动研讨会，以招募和培训研究生进行研究。该研究项目的教育和宣传部分将使PI能够揭示不同级别和不同背景的学生如何使用数学来模拟和解决可行的现实世界问题，以激励学生使用数学来实现重要性的科学挑战，从而通过数学奖励进行数学的兴趣，以反映NSF的宣传，并提高了他们的兴趣。优点和更广泛的影响审查标准。

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