This $450,287 Project Grant was awarded by the National Science Foundation (NSF) Division of Mathematical Sciences under the Mathematical and Physical Sciences program (CFDA 47.049). The grant supports research at The Johns Hopkins University to advance the theory of harmonic maps, which have practical applications in areas like medical imaging and computer vision. Specifically, the research focuses on three key areas: (1) developing non-abelian Hodge theory to connect topological data of Kähler manifolds to their holomorphic structure, (2) exploring a geometric approach to study the rigidity of lattices in semisimple Lie groups, and (3) investigating the regularity theory of harmonic maps into singular targets to better understand their singular sets and potential applications. This work aims to uncover new insights with implications across mathematical fields while also providing guidance and support to graduate students, postdocs, and early-career mathematicians, especially those underrepresented in STEM.