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

这个项目的目标是解决复杂几何和统一弦理论交界处的一些问题，这些问题的解决对于进一步的发展是至关重要的。一条潜在的共同线索是超对称性，这是不久前在物理学中出现的。然而，它在几何和分析领域的重要性，如超黎曼曲面理论、超模空间和非线性偏微分方程组理论，直到最近才得到更充分的认识，我们的理解仍然很不完整。超对称性导致某些上同调约束。从分析的角度来看，能够实现这些约束的偏微分方程组本身就具有相当大的兴趣。它们提出了许多新的挑战，对该理论的未来发展应该是非常有价值的。在过去，同样的偏微分方程，如果它受到几何或物理的高度限制，从非常不同的数学应用中出现的情况并不少见。我们可以从这些新的方程式中期待同样的结果，它们的进展将具有广泛的价值。该研究项目还汇集了数学和物理的几个领域的思想和技术，它应该为学生和年轻的博士后研究人员提供一个很好的培训基础。更具体地说，由超对称引起的上同调约束是厄米度规的Kahler条件的推广，但可能与之显著不同。因此，它们一方面通向非Kahler几何，另一方面通向可能比复杂的Monge-Ampere方程或Kahler-Ricci流复杂得多的偏微分方程组。新的困难来自这样一个事实，即方程是系统，或者它们可能涉及曲率张量的更高次方，或者它们的长期行为对初始数据的依赖可能更加微妙。这个项目的一个主要目标是从异常流动开始，为这类方程发展一个普遍的理论。这些流是由PI和合作者引入的，目的是实现上同调约束，并在提供Kahler几何基本结果的新证明方面显示了它们的力量，例如Fu-Yau定理和Calabi猜想的Yau解。另一个目标是发展一种混合上同调，它可以帮助从超模空间到模空间的非全纯投影中提取全纯散射幅度。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 goal of this project is to address some problems at the interface of complex geometry and unified string theories, whose solution is essential for further progress. An underlying common thread is supersymmetry, which appeared in physics a while ago. However, its importance in areas of geometry and analysis such as the theory of super Riemann surfaces, supermoduli space, and the theory of non-linear partial differential equations, has only recently been more fully appreciated, and our understanding is still very incomplete. Supersymmetry results in certain cohomological constraints. The partial differential equations which can implement these constraints are of considerable interest in their own right from the point of view of analysis. They pose many new challenges which should be very valuable for the future development of the theory. It has not been uncommon in the past for the same partial differential equation, if it is highly constrained by either geometry or physics, to emerge from very different applications of mathematics. We can expect the same from these new equations, and progress on them to be of wide value. The research project also brings together ideas and techniques from several areas of mathematics and physics, and it should provide an excellent training ground for students and young postdoctoral researchers.More specifically, the cohomological constraints arising from supersymmetry are generalizations of, but may be markedly different from, the Kahler condition of Hermitian metrics. As such, they lead on one side to non-Kahler geometry, and on the other side, to partial differential equations which can be much more complicated than the complex Monge-Ampere equation or the Kahler-Ricci flow. New difficulties arise from the facts that the equations are systems, or they may involve higher powers of the curvature tensor, or the dependence of their long-time behavior on the initial data may be more delicate. A major goal of this project is to develop a general theory for such equations, beginning with Anomaly flows. These are flows introduced by the PI and collaborators with the precise goal of implementing cohomological constraints, and which have shown their power in providing new proofs of fundamental results in Kahler geometry such as the Fu-Yau theorem and Yau's solution of the Calabi conjecture. Another goal is the development of a hybrid cohomology which can help extract holomorphic scattering amplitudes from non-holomorphic projections from supermoduli space to moduli space.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.