Problems in Complex Geometry, Partial Differential Equations, and Mathematical Physics

对自然界所有力的统一理论的探索，通过弦理论引出了新的方程，这些方程需要一个具有挑战性的、不熟悉的几何。本研究致力于发展一个理论的几何偏微分方程，可以解决这样的方程。这样的理论需要在偏微分方程、几何学和高能物理学之间的界面领域取得进展，并在某一天达到对物理定律最基本水平的理解。本科生和研究生的培训是该项目的一个组成部分。对于下一代学者来说，尽快引入这些困难的问题尤为重要。这些新方程的一个特征是结合了上同调约束的曲率条件。因此，它们可以被视为定义“规范度量”。PI将努力开发几何流方法来找到这样的度量，并考虑张量和旋量场的流。从偏微分方程理论的观点来看，这些几何流动本身就很有趣。特别感兴趣的是，以确定表现的超对称性和对偶性，预计特别是要链接方面的复杂几何方面的辛几何。一个混合上同调理论也将发展与超弦微扰理论的可能应用。这个奖项反映了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 search for a unified theory of all forces of nature has led through string theories to new equations requiring a challenging and unfamiliar geometry. This research is devoted to the development of a theory of geometric partial differential equations which can address such equations. Such a theory is needed progress is to be made in this area of interface between partial differential equations, geometry, and high energy physics, and arrive some day at an understanding of physical laws at their most fundamental level. The training of both undergraduate and graduate students is an integral component of the project. It is particularly important for the next generation of scholars to be introduced to these difficult issues as soon as possible.A characteristic feature of these new equations is a curvature condition combined with a cohomological constraint. Thus they can be viewed as defining “canonical metrics”. The PI will strive to develop geometric flow methods to find such metrics, and to consider flows of both tensor and spinor fields. These geometric flows are interesting in their own right, from the point of view of the theory of partial differential equations. Of particular interest is to determine the manifestations of supersymmetry and duality, which are expected in particular to link aspects of complex geometry to aspects of symplectic geometry. A hybrid cohomology theory will also be developed with possible applications to superstring perturbation theory.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.