Gromov-Witten Invariants and Isoparametric submanifolds

Gromov-Witten 不变量和等参子流形

• 批准号: 0071372
• 负责人: Xiaobo Liu
• 金额: $ 7.52万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071372&HistoricalAwards=false
• 关键词: Gromov; Witten; Invariants; Isoparametric; submanifolds

AbstractAward: DMS-0071372Principal Investigator: Xiaobo LiuThis research program addresses problems in two areas. TheVirasoro conjecture in symplectic geometry asserts that thegenerating function of the Gromov-Witten invariants for a compactsymplectic manifold must be annihilated by a sequence ofdifferential operators in the Virasoro algebra. The principalinvestigator will study whether the genus-one Virasoro conjectureand perhaps more general constraints hold for all projectivevarieties. It is even possible that the generating function ofgenus-one primary Gromov-Witten invariants can be computedexplicitly from the genus-zero invariants, although this is morethan the Virasoro conjecture implies. The second area to beaddressed is submanifold geometry, especially the properties ofisoparametric submanifolds, including infinite dimensionalsubmanifolds. We know some good constructions of isoparametricsubmanifolds, but important aspects of the classification theoryare in early stages of development.Geometric descriptions of classical or quantum mechanical systemsare based in symplectic geometry, which takes as its basicmeasurement the area of two-dimensional surfaces. TheGromov-Witten invariants of a symplectic manifold can bedescribed as a family of number that count the number of surfacesof a nice sort that satisfy certain constraints, such asintersecting specified subsets of the manifold. Enumerativeinvariants of this kind are an ancient concern in algebraicgeometry, where we count the number of solutions to a system ofequations, and the subject was recently found to be connected tohigh energy physical theory, where the partition functions ofcertain quantum systems on a manifold have been found to berelated to generating functions that collect families ofGromov-Witten invariants into a manageable form. The Virasoroconjecture proposed by a group of physicists and mathematiciansasserts that these generating functions satisfy a family ofconstraints that simplify computations and connect theGromov-Witten invariants to the mathematics of completelydifferent problems. A different kind of geometry arises in thestudy of submanifolds as curved objects, where we use severalmeasurements of bending to capture the intuitive sense that asphere in three-space is more curved if its radius is small thanif its radius is large. An isoparametric submanifold is animbedded object which has particularly simple curvatureproperties with respect to the larger space, and the character ofthis specialty is driven by the fact that this local conditionfrequently imposes strong symmetry properties the wholesubmanifold.

摘要：DMS-0071372主要研究者：刘晓波这项研究计划解决了两个领域的问题。 辛几何中的Virasoro猜想指出，紧辛流形上的Gromov-Witten不变量的生成函数必被Virasoro代数中的一列微分算子零化。 首席研究员将研究第一属Virasoro限制以及可能更普遍的限制是否适用于所有投影变量。 甚至可能的是，生成函数的genus-one主要Gromov-Witten不变量可以计算从genus-zero不变量，虽然这是超过Virasoro猜想暗示。 第二个领域是子流形几何，特别是非参数子流形的性质，包括无穷维子流形。 我们知道等参子流形的一些好的构造，但分类理论的重要方面还处于发展的早期阶段。经典或量子力学系统的几何描述是以辛几何为基础的，辛几何以二维表面的面积为基本度量。 辛流形的Gromov-Witten不变量可以被描述为一个数族，它计算满足某些约束条件（如流形的相交子集）的一类曲面的个数。 这种类型的枚举不变量是代数几何中一个古老的问题，我们计算方程系统的解的数量，最近发现这个主题与高能物理理论有关，其中流形上某些量子系统的配分函数与生成函数有关，生成函数将Gromov-Witten不变量的家族收集到一个可管理的形式中。 由一群物理学家和数学家提出的Virasoro猜想断言，这些生成函数满足一系列约束，这些约束简化了计算，并将Gromov-Witten不变量与完全不同的数学问题联系起来。 在研究作为弯曲物体的子流形时，出现了一种不同的几何学，我们使用几种弯曲的测量来捕捉三维空间中的非球面在其半径较小时比半径较大时更弯曲的直观感觉。 一个等参子流形是相对于较大空间具有特别简单的曲率性质的活动物体，而这种特殊性的特征是由这样一个事实驱动的，即这种局部条件经常赋予整体子流形强的对称性。