Gromov-Witten Invariants and Isoparametric submanifolds

项目编号：dms -0071372项目负责人：刘晓波本项目主要研究两个方面的问题。辛几何中的Virasoro猜想断言紧辛流形的Gromov-Witten不变量的生成函数必须被Virasoro代数中的微分算子序列湮灭。首席研究员将研究第一属维拉索罗的推测，或许还有更普遍的约束是否适用于所有投射品种。甚至有可能由零属不变量显式地计算出一类初等Gromov-Witten不变量的生成函数，尽管这比Virasoro猜想所暗示的要多。第二个要讨论的领域是子流形几何，特别是等参子流形的性质，包括无限维子流形。我们知道一些很好的等参子流形结构，但是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基于辛几何的，辛几何以二维表面的面积作为其基本测量。辛流形的gromov - witten不变量可以被描述为一组数，它们计算满足某些约束的曲面的数量，例如与流形的特定子集相交。这种枚举不变量是代数几何中一个古老的问题，在代数几何中，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被发现与生成函数有关，这些函数将gromov - witten不变量族集合成可管理的形式。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不变量与完全不同问题的数学联系起来。在将子流形作为弯曲物体的研究中，出现了一种不同的几何，我们使用几种弯曲的测量来捕捉直观的感觉，即三维空间中的非球面，如果其半径小，则比其半径大时更弯曲。等参子流形是相对于更大空间具有特别简单曲率特性的嵌入对象，这种特性是由这个局部条件经常给整个子流形施加强对称特性所驱动的。

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.