# Project Details

### Description

Geometry and Analysis on Calabi-Yau and Hermitian Manifolds

The PI proposes to investigate several problems about the geometry of complex (and symplectic) manifolds using nonlinear partial differential equations. The first project is about understanding the ways in which Ricci-flat Calabi-Yau manifolds can degenerate in families. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. In the second project the PI will study the geometry of Hermitian manifolds using the Chern-Ricci flow, an extension of the K¨ahler-Ricci flow to all complex manifolds. This flow is intimately related to the complex structure of the manifold and will be

used to widen our understanding of non-K¨ahler compact complex surfaces. The third project is centered on Donaldson’s program to extend Yau’s solution of the Calabi Conjecture in K¨ahler geometry to symplectic four-manifolds. This would

provide a new and powerful analytic tool to construct symplectic forms on closed symplectic four-manifolds as solutions of a highly nonlinear PDE, and would allow to solve basic open questions in symplectic topology, such as: given a compact

almost-complex four-manifold, when are there compatible symplectic forms?

Intellectual merit of the project

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 classification of compact complex surfaces is to this day incomplete, and any new analytic tool that applies to all Hermitian manifolds (such as the Chern-Ricci flow) is extremely useful. The proposed work on symplectic fourmanifolds 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. Thanks to the rich interplay between geometry and physics, geometric aspects closely related to the proposed research have found applications in various areas of theoretical physics. The PI also proposes to support two graduate students at his institution, as well as mentoring one undergraduate student per year for a summer Research Experience for Undergraduates project at the PI’s institution.

The PI proposes to investigate several problems about the geometry of complex (and symplectic) manifolds using nonlinear partial differential equations. The first project is about understanding the ways in which Ricci-flat Calabi-Yau manifolds can degenerate in families. This is closely related to the theory of mirror symmetry, which was inspired by physical considerations. In the second project the PI will study the geometry of Hermitian manifolds using the Chern-Ricci flow, an extension of the K¨ahler-Ricci flow to all complex manifolds. This flow is intimately related to the complex structure of the manifold and will be

used to widen our understanding of non-K¨ahler compact complex surfaces. The third project is centered on Donaldson’s program to extend Yau’s solution of the Calabi Conjecture in K¨ahler geometry to symplectic four-manifolds. This would

provide a new and powerful analytic tool to construct symplectic forms on closed symplectic four-manifolds as solutions of a highly nonlinear PDE, and would allow to solve basic open questions in symplectic topology, such as: given a compact

almost-complex four-manifold, when are there compatible symplectic forms?

Intellectual merit of the project

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 classification of compact complex surfaces is to this day incomplete, and any new analytic tool that applies to all Hermitian manifolds (such as the Chern-Ricci flow) is extremely useful. The proposed work on symplectic fourmanifolds 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. Thanks to the rich interplay between geometry and physics, geometric aspects closely related to the proposed research have found applications in various areas of theoretical physics. The PI also proposes to support two graduate students at his institution, as well as mentoring one undergraduate student per year for a summer Research Experience for Undergraduates project at the PI’s institution.

Status | Finished |
---|---|

Effective start/end date | 9/15/13 → 8/31/16 |

### Funding

- National Science Foundation (DMS-1308988)