Contact Geometry and Einstein Manifolds

AbstractAward：DMS-9970904首席研究员：Charles P. Boyer和Krzysztof Galicki Boyer和Galicki教授提议研究几何和拓扑学中的几个项目。所有这些项目的目标是调查黎曼几何的两个主要焦点，接触几何和爱因斯坦的基本问题。 这里提出的问题和难题是基于主要研究者的工作，他们利用了Sasakian-Einstein空间的接触几何和两种特殊的Kaehler几何之间的基本关系。这些是Q-阶乘Fano簇（具有正标量曲率的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，我们能够构造紧的局部不可约的正标量曲率爱因斯坦空间与任意第二贝蒂数在任何奇数维大于5。我们现在正处在这样一个点上，在这个建议中描述的几个分类问题和解决办法似乎是我们力所能及的。此外，我们正在研究的爱因斯坦几何类型最近在超共形场论和弦论的背景下引起了理论物理学的兴趣，并使我们的工作具有了跨学科的味道。数学是我们现代技术的基础，它的理解和发展必须先于技术进步。 然而，我们对特定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，并应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.