CAREER: Stability, Kahler Geometry, and the Hele-Shaw Flow

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Mathematics can be used to understand a vast range of different things. The research in this project is motivated by the study of shapes, in particular one can ask the question: what does it mean for an object to have a "best shape"? Surprisingly it turns out that aspects of this question are related to a seemingly unrelated piece of fluid mechanics called the Hele-Shaw flow. The project aims at building, extending and exploiting this deep connection with the expectation of revealing new insights into both shape and fluid flows, which is likely to have impact in both pure and applied mathematics. The project includes an educational program that provides specialized training in this area aimed at the next generation of research mathematicians, and a component that explores how undergraduate-level mathematics can be communicated though the medium of sequential art.The research project combines algebraic geometry, complex geometry and free boundary problems using algebraic, analytic and computational techniques. First, the investigator studies a new notion of K-stable map which extends and clarifies the Yau-Tian-Donaldson conjecture connecting algebraic stability with canonical Kahler metrics. Second, the investigator builds on and extend the his previous success in studying problems in complex geometry through the Hele-Shaw flow, giving new insight into what kinds of behavior solutions to such equations can display. The educational activities aimed at training early-career mathematicians include a combined two-week graduate-school and research workshop, and activities aimed at undergraduates include an annual undergraduate research symposium, and summer research opportunities related to this research.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.