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

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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.