CAREER: Conformal Geometry and Monge-Ampere Type Equations

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The study of conformal geometry has always been a fundamental subject in differential geometry. It has a tight relationship to partial differential equations and mathematical physics. In this research project, the PI will conduct study on conformal invariants, and investigate important roles of conformal invariants in geometric inequalities. The PI also plans to push the theory to the field of Cauchy-Riemann geometry, another important field of differential geometry. The award also includes support for educational activities for students of different academic levels. The PI will organize the first-year graduate mini-courses in geometric analysis, create research programs for undergraduate students, and expand the MADGS workshop to increase the participation of graduate students, especially of those from underrepresented groups. Those activities aim to provide opportunities to early career researchers and encourage their collaborations.One direction of this research project is to understand conformal invariants and how they control the asymptotic behavior at the ends of noncompact manifolds, and thus affect geometric inequalities including the isoperimetric inequality. Another direction is to study the relationship between conformal invariants and Monge-Ampere equations, one of the most important equations that arises naturally in mathematics and physics. The PI will continue her work on the construction of fully nonlinear functionals so that they shed light on both geometric and analytic features of the equation. The methods will incorporate those from conformal geometry, harmonic analysis and partial differential equations.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.