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

为了寻找自然力的统一理论，通过弦理论得出了新的方程，需要一种具有挑战性的和不熟悉的几何学。这项研究致力于发展一种能够处理此类方程的几何偏微分方程组理论。这样的理论需要在偏微分方程式、几何和高能物理之间的接口领域取得进展，并有一天在最基本的层面上理解物理规律。对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，尽快了解这些困难的问题尤为重要。这些新方程的一个特征是曲率条件和上同调约束相结合。因此，它们可以被视为定义了“规范指标”。PI将努力开发几何流动方法来寻找这样的度量，并考虑张量场和旋量场的流动。从偏微分方程组理论的角度来看，这些几何流动本身就很有趣。特别令人感兴趣的是确定超对称性和对偶性的表现形式，它们特别被期望将复杂几何的各个方面与辛几何的各个方面联系起来。混合上同调理论也将被开发，并可能应用于超弦微扰理论。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.