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

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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.