Harmonic Maps, Geometric Rigidity, and Non-Abelian Hodge Theory

In everyday use, maps are representations of Earth's characteristics, providing a straightforward and efficient way to convey information about sizes, shapes, and distances between places, illustrating the spatial arrangement of elements on Earth. Earth is an example of a geometric space where angles and distances can be measured. Another example is Euclidean spaces used to model our physical world. Additionally, non-Euclidean or Riemannian spaces have numerous applications, notably in cosmology, allowing scientists to describe the large-scale structure, curvature, and topology of the universe, and in robotics, providing engineers with a mathematical framework to model the range of motion of robotic systems. Just as cartographers construct maps to reveal spatial information about regions on Earth, mathematicians construct maps between geometric spaces to uncover interesting features and analyze their geometry. In this research project, the principal investigator (PI) will focus on harmonic maps, a special type of maps that minimize a specific measure of energy between geometric spaces. By analyzing harmonic maps, the PI aims to uncover essential properties of various geometric spaces, contributing to a better understanding of the natural world. The mathematical theory of harmonic maps has practical applications in medicine (e.g., medical imaging) and computer science (e.g., computer vision), with potential to further scientific progress and societal welfare. Moreover, the project includes an educational aspect, offering guidance and support to graduate students, post-docs, and early-career mathematicians, especially those underrepresented in STEM fields. Harmonic map theory holds significant interest for mathematicians and physicists, playing a vital role in geometric analysis by serving as analytical objects that incorporate geometric, topological, and algebraic information of a given space. The techniques involving harmonic maps have been successfully applied in various mathematical contexts, yielding important results in rigidity problems, Hodge theory, and Teichmüller theory. The principal investigator (PI) will continue developing the theory of harmonic maps, motivated by its potential applications in different mathematical fields. The primary focus of the proposal revolves around three key areas: (1) Non-Abelian Hodge Theory, aiming to develop non-abelian Hodge theory over smooth quasi-project varieties and connecting topological data of Kahler manifolds to their holomorphic structure; (2) Geometric Rigidity, exploring a geometric approach to study the rigidity of lattices in semisimple Lie groups, particularly non-uniform lattices associated with non-compact manifolds and studying representations of lattices in isometry groups of smooth and singular spaces; (3) Regularity of Harmonic Maps, concentrating on problems related to the regularity theory of harmonic maps into singular targets, with the goal of better understanding singular sets and their potential applications. The research seeks to advance the theory of harmonic maps, unveiling new insights with practical implications in various mathematical fields. 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.