Contact Geometry and Einstein Manifolds

摘要奖：DMS-9970904主要研究员：Charles P.Boyer和Krzysztof Galicki教授Boyer和Galicki提议研究几何和拓扑学中的几个项目。所有项目的目标是研究黎曼几何中的基本问题，主要有两个焦点，联系几何和爱因斯坦度量学。这里提出的问题是基于主要研究者的工作，该工作探索了Sasakian-Einstein空间的接触几何与两种特殊的Kaehler几何之间的基本关系。这些是q阶乘Fano簇(具有正数量曲率的Kaehler-Einstein度量)和Calabi-Yau流形(Kaehler Ricci-Flat度量)。我们最近的工作在研究许多经典问题方面取得了重大进展，并导致了全新技术的发展。最值得注意的是，我们能够在任何大于5的奇数维上构造具有任意第二Betti数的紧局部不可约正标量曲率爱因斯坦空间。我们现在所处的点上，这个提议中描述的几个分类问题和猜想似乎是我们可以达到的。此外，我们正在研究的爱因斯坦几何类型最近在超保角形场理论和弦理论的背景下成为理论物理学的兴趣，并赋予我们的工作跨学科的特点。数学是我们现代技术的基础，它的许多理解和发展必须先于技术进步。然而，我们对特殊类型几何的研究与现代理论物理中的一些重要问题密切相关，应该为理解它们提供重要的数学基础。

AbstractAward: DMS-9970904Principal Investigator: Charles P. Boyer and Krzysztof GalickiProfessors Boyer and Galicki propose to investigate severalprojects in geometry and topology. The objective of all theprojects is to investigate fundamental questions in RiemannianGeometry with the two main focal points, Contact Geometry andEinstein Metrics. The questions and problems proposed here arebased on the principal investigators' work which exploited afundamental relationship between contact geometry ofSasakian-Einstein spaces and two particular kinds of Kaehlergeometry. These are the Q-factorial Fano varieties(Kaehler-Einstein metrics with positive scalar curvature) and theCalabi-Yau manifolds (Kaehler Ricci-flat metrics). Our recentwork has led to a significant progress in studying many classicalproblems as well as resulted in the development of completely newtechniques. Most notably, we were able to construct compactlocally irreducible positive scalar curvature Einstein spaceswith arbitrary second Betti number in any odd dimension greaterthan 5. We now are at the point where several classificationproblems and conjectures described in this proposal appear to bewithin our reach. Furthermore, the type of Einstein geometry thatwe are studying has recently become of interest in TheoreticalPhysics in the context of superconformal field theory and stringtheory, and gives our work an interdisciplinary flavor.Mathematics is the foundation upon which our modern technology isbuilt, and much of its understanding and development must preceedtechnological progress. Nevertheless, our research into aparticular type of geometry is closely linked to some importantproblems in modern Theoretical Physics and should provide animportant mathematical basis for their understanding.