Complex and Symplectic Geometry of Complexifications

DMS-0204634 Richard Hindone可以联系到光滑流形上余切丛上的辛结构，以及其余切丛内区域上的实解析黎曼流形上的标准复结构。本项目将继续研究这些辛复结构的研究者的工作。一个广泛的目标是发现辛几何与原始流形上的光滑结构有多么密切的关系，以及复几何如何反映黎曼性质。另外，由于协调复结构的性质中包含了大量的辛信息，因此将辛几何与复几何联系起来也是很有趣的。余切丛是辛流形的最简单的例子，然而辛几何中的许多基本问题在这些情况下仍然是难以捉摸的。研究人员将对拉格朗日子流形的辛结构和保序类的唯一性进行研究。我们研究的复流形自然地出现在数学的许多分支中，特别是代数几何和表示论。它们的几何的重要方面仍有待理解，以及应该应用于数论的分析问题。辛几何起源于经典和量子力学的现代数学语言。它现在是纯数学研究的一个令人兴奋的领域，但继续受益于当前的物理思想。自八十年代以来，人们已经清楚地看到，从研究相容的复杂结构中也可以获得对辛几何的许多见解。幸运的是，复杂几何的研究长期以来一直是数学的主要焦点。这个项目将对辛几何和复几何进行研究，重点是两者之间的相互作用。具体的研究对象是作为辛流形的第一个例子出现的，也是作为在数学和理论物理的许多分支中频繁出现的复杂流形的例子而产生的。

DMS-0204634Richard HindOne can associate to a smooth manifold a symplectic structureon its cotangent bundle, and to a real-analytic Riemannian manifolda canonical complex structure on a domain inside its cotangent bundle.This project will continue work of the investigator studying thesesymplectic and complex structures. A broad aim is to discover howclosely the symplectic geometry relates to the smooth structure onthe original manifold, and how the complex geometry reflects theRiemannian properties. Also, since much symplectic infomation iscontained in properties of compatible complex structures, it is alsointeresting to relate the symplectic and complex geometry. Cotangentbundles are the simplest examples of symplectic manifolds, yet manyfundamental problems in symplectic geometry are still elusive inthese cases. The investigator will conduct research into theuniqueness of symplectic structures and of isotopy classes ofLagrangian submanifolds. The complex manifolds we study arisenaturally in many branches of mathematics, notably algebraic geometryand representation theory. Important aspects of their geometryremain to be understood, together with analytical problems whichshould have applications to number theory. Symplectic geometry originated as the modern mathematicallanguage of classical and quantum mechanics. It is now an excitingarea of pure mathematical research, but continues to benefit fromcurrent physical ideas. Since the eighties it has been clear thatmuch insight into symplectic geometry can also be gained fromstudying compatible complex structures. Fortunately the study ofcomplex geometry has long been a main focus of mathematics. Thisproject will conduct research into both symplectic and complexgeometry, focussing on interactions between the two. The specificobjects of study arise as the first examples of symplecticmanifolds, and as examples of complex manifolds occuring frequentlyin many branches of mathematics and theoretical physics.