Classification of Lagrangian fibrations

超卡勒流形出现在高能理论物理的许多不同领域，例如空间参数化杨-米尔斯瞬子、磁单极子和希格斯场。在其代数几何表现中，超卡勒流形表现为全纯辛流形。全纯辛流形上的纤维被称为拉格朗日纤维，因为相对于全纯辛结构纤维必须是拉格朗日纤维。此外，通用纤维必须是阿贝尔品种，拉格朗日纤维可以看作是椭圆K3表面的高维类似物。希钦系统是一个著名的非紧凑型例子。在这个项目中，P.I.将研究（紧致）拉格朗日颤动的分类。目标是在某些特殊情况下对拉格朗日纤维进行分类：当纤维是曲线的雅可比矩阵或主要极化时，当纤维在四维（阿贝尔曲面作为纤维）中局部等平凡时。P.I.也将探索拉格朗日纤维的新结构，并解决变形类的有限性问题。这个项目将发展一些已知的在理论物理中出现的特殊类型的几何，特别是在量子场论和弦理论中。主要的焦点将放在超卡勒几何上，这是一类允许里奇平坦度量的空间，即真空中爱因斯坦方程的解。P.I.还将研究广义复几何，这是数学家们发明的，用于连接复杂几何和辛几何，IIA型和IIB型弦理论的基础，并有助于解开弦理论家预测的镜像对称现象。简而言之，这些都是创新的新型几何，主要是为了响应理论物理学家的要求而发展起来的，但也有固有的数学兴趣。P.I.将通过组织和参与跨学科会议和研讨会积极促进数学家和物理学家之间的思想交流，并参与这些主题的研究生培训。

Hyperkahler manifolds arise in many different areas of high energy theoretical physics, for example as spaces parametrizing Yang-Mills instantons, magnetic monopoles, and Higgs fields. In their algebro-geometric manifestation, hyperkahler manifolds appear as holomorphic symplectic manifolds. Fibrations on holomorphic symplectic manifolds are known as Lagrangian fibrations, since the fibers must be Lagrangian with respect to the holomorphic symplectic structure. In addition, the generic fibres must be abelian varieties, and Lagrangian fibrations may be viewed as higher-dimensional analogues of elliptic K3 surfaces. The Hitchin system is a celebrated non-compact example. In this project the P.I. will study the classification of (compact) Lagrangian fibrations. The goals are to classify Lagrangian fibrations in some particular cases: when the fibers are Jacobians of curves or principally polarized, when the fibration is locally isotrivial, in dimension four (abelian surfaces as fibers). The P.I. will also explore new constructions of Lagrangian fibrations, and address the question of finiteness of deformation classes.This project will develop some particular kinds of geometry that are known to arise in theoretical physics, particularly in quantum field theory and string theory. The main focus will be on hyperkahler geometry, a class of spaces admitting Ricci-flat metrics, i.e., solutions to Einstein's equations in a vacuum. The P.I. will also study generalized complex geometry, which was invented by mathematicians to bridge complex and symplectic geometry, on which type IIA and IIB string theory are based, and to help unravel the Mirror Symmetry phenomenon predicted by string theorists. In short, these are all innovative new kinds of geometry developed largely in response to the demands of theoretical physicists, but also of inherent mathematical interest. The P.I. will actively promote the exchange of ideas between mathematicians and physicists by organizing and participating in cross-disciplinary conferences and workshops, and is also involved in the training of graduate students in these topics.