CAREER: Conformal Geometry and Monge-Ampere Type Equations

保形几何的研究一直是微分几何的一个基本课题。它与偏微分方程和数学物理有着紧密的联系。在本研究计划中，PI将进行共形不变量的研究，并研究共形不变量在几何不等式中的重要作用。PI还计划将该理论推向柯西-黎曼几何领域，这是微分几何的另一个重要领域。该奖项还包括对不同学术水平学生的教育活动的支持。PI将组织第一年的几何分析研究生迷你课程，为本科生创建研究计划，并扩大MADGS研讨会，以增加研究生的参与，特别是那些来自代表性不足的群体。这些活动旨在为早期的职业研究人员提供机会，并鼓励他们的合作。本研究项目的一个方向是了解共形不变量以及它们如何控制非紧流形端点的渐近行为，从而影响几何不等式，包括等周不等式。另一个方向是研究共形不变量和Monge-Ampere方程之间的关系，Monge-Ampere方程是数学和物理中自然出现的最重要的方程之一。PI将继续她的工作，建设完全非线性泛函，使他们阐明了几何和分析功能的方程。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.