Geometric Partial Differential Equations and Complex Geometry

Project: Research project

Project Details

Description

Overview
The PI proposes to use techniques from geometric analysis and nonlinear partial differential equations to investigate problems about the geometry of complex (and symplectic) manifolds. The first project is about understanding limits of Ricci-flat Calabi-Yau manifolds as the Kahler class degenerates. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. The second project concerns the long-time behavior of the Ricci flow on compact Kahler manifolds, in the most difficult case when collapsing occurs at infinite time. The third project is centered on Donaldson's program to extend Yau's solution of the Calabi Conjecture in Kahler geometry to symplectic four-manifolds. This would provide a new analytic tool to construct symplectic forms four-manifolds as solutions of a highly nonlinear PDE, and would have striking applications in symplectic topology
Intellectual Merit
The study of Calabi-Yau manifolds is a central topic in geometry, having ramifications in fields as diverse as algebraic geometry, number theory and theoretical physics. The proposed problems in particular will complete our understanding of the collapsing of such spaces which admits fibrations. The Ricci flow is the most well-known intrinsic geometric flow, which has been used in recent years to prove spectacular results, including the Poincare Conjecture. The proposed research aims to clarify the behavior of the flow on higher-dimensional compact Kahler manifolds, a key question in complex geometry. The proposed work on symplectic four-manifolds will deepen our understanding of the geometry of these spaces and will introduce new analytic tools in the field.
Broader Impacts
The PI will continue to organize a learning seminar at his institution, to introduce undergraduate and graduate students to these topics, and training them to give their own presentations. He will also advise one graduate student (with another one possibly joining this year), and will continue advising undergraduate students' honors theses and mentoring postdocs and visitors. The PI will also continue to organize conferences, workshops and summer schools, with a sizable portion of the speakers being early career mathematicians and members of underrepresented groups.
StatusActive
Effective start/end date9/1/198/31/22

Funding

  • National Science Foundation (DMS-1903147)

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