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

在这个由美国国家科学基金会的卓越研究计划支持的项目中，首席研究员(PI)将对一类从几何和物理中产生的方程进行数学分析。这些应用包括几何问题中最优形状的存在和表征，这有助于科学家和工程师理解宇宙，并有助于现实世界中重要物体和工具的优化设计。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是很重要的，例如一些Chern-Simons-Higgs模型中的欧拉流和凝聚体的涡旋。这个项目中研究的方程的一个特殊性是它们验证了强相互作用和量子化的现象，这是许多模拟现实生活问题的偏微分方程组所享有的。这项研究的目的是开发可用于建立此类方程的存在机制的方法，以验证量子化和强相互作用现象。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。该项目还有一个组成部分，旨在增加STEM学科中任职人数不足的群体的数量。为此，PI将试行一个通往博士学位的桥梁项目，主要任务是增加霍华德大学和美国境内拥有数学博士学位的女性和少数族裔的数量。本研究的主要目的是研究q-曲率型非紧几何变分问题。它们一方面是：描述黎曼度量到给定q-曲率类型量的共形变形的非线性偏微分方程组，另一方面是描述来自Chern-Simons理论的平均场和Toda问题的非线性偏微分方程组。这些方程是以能量泛函的欧拉-拉格朗日方程的形式出现的，它对于某些Moser-Trudinger型不等式是临界的。该项目的重点是当相关的欧拉-拉格朗日泛函的伪梯度的一些非紧致流线的聚点出现时的共振情况，即所谓的相关变分问题的无穷远处的真临界点。该项目将使用阿巴斯·巴赫里无穷远的临界点工具来研究存在机制。PI将通过发展这类问题的Morse拓扑论和代数拓扑论建立新的存在性结果。确切地说，他将建立q-曲率型方程的满度理论和Morse理论。此外，国际和平研究所将与霍华德大学文理研究生院合作，举办基于这些主题的几何分析和其他相关共形不变分问题的互动研讨会，以招募和培训研究生从事研究。这项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模拟和解决可行的现实世界问题，激励学生使用数学来应对重要的科学挑战，并提高他们在数学领域追求职业生涯的兴趣。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影响审查标准进行评估，被认为值得支持。

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.