Submanifolds and Metrics in Contact Geometry

流形上的接触结构是两个多世纪前惠更斯、汉密尔顿和雅可比关于几何光学的工作中诞生的自然对象。它们被许多数学家研究过，似乎触及了数学和物理的不同领域，但只是在过去的几十年里，它们才进入数学的前台。这是由于接触拓扑学的重大突破，产生了丰富而美丽的理论和许多应用。研究子集及其与这些结构的相互作用有助于理解三维空间，并导致了深刻的进步。首席研究员现在将把它扩展到更高的维度，在那里这种探索很可能同样具有启发性。主要研究员还将继续研究低维空间上接触结构的性质及其与拓扑学和黎曼几何的相互作用，并通过与大量研究生合作以及组织会议和研讨会来培养下一代研究人员。该奖项支持的研究将集中在围绕三个广泛主题的问题上：接触几何和拓扑在低维的相互作用，接触流形在高维的性质和结构，以及接触几何和更熟悉的黎曼几何之间的联系。在低维的主要动机问题是确定哪些三流形承认一个紧密接触的结构。目前相当多的是知道这个问题，但很少有人知道它的双曲同调球。这个问题将研究使用各种技术，从凸曲面，全纯曲线和黎曼几何。此外，将研究了解接触结构可能具有的各种属性之间的相互作用。虽然在低维接触几何知道很多，但在更高的维度上却知之甚少。主要研究者将研究高维接触流形的构造和性质。这将是起点的研究各向同性和接触子流形的接触流形。这样的考虑导致了低维信息的丰富，预计在更高的维度也会同样富有成效。在过去的几年里，接触几何和黎曼几何之间有一些有趣而微妙的联系。主要研究者将进一步探索这一点，希望找到接触几何类似物的经典结果有关拓扑黎曼几何。

Contact structures on manifolds are natural objects, born over two centuries ago, in the work of Huygens, Hamilton, and Jacobi, on geometric optics. They have been studied by many mathematicians and seem to touch on diverse areas of mathematics and physics, but only in the last few decades have they moved into the foreground of mathematics. This is due to the remarkable breakthroughs in contact topology, resulting in a rich and beautiful theory with many applications. Studying subsets and their interactions with such structures was instrumental in the understanding of three-dimensional spaces, and it led to profound progress. The Principal Investigator will now extend this to higher dimensions, where this exploration is likely to prove equally illuminating. The Principal Investigator will also continue to study properties of contact structures on low-dimensional spaces and their interaction with topology and Riemannian geometry, and he will train the next generation of researchers by working with a large group of graduate students and organizing conferences and seminars.The research supported by this award will focus on problems centered around three broad topics: the interactions of contact geometry and topology in low dimensions, properties and constructions of contact manifolds in higher dimensions, and connections between contact geometry and the more familiar Riemannian geometry. In low dimensions the main motivating question is to determine which three-manifolds admit a tight contact structure. Currently quite a bit is known about this question, but very little is known about it for hyperbolic homology spheres. This problem will be studied using a variety of techniques, from convex surfaces, to holomorphic curves and Riemannian geometry. In addition, understanding interactions between various properties a contact structure can have will be studied. While much is known about contact geometry in low dimensions, there is very little known in higher dimensions. The principal investigator will study constructions and properties of high-dimensional contact manifolds. The starting point for this will be the study of isotropic and contact submanifolds of contact manifolds. Such considerations have led to a wealth of information in low-dimensions and it is expected to be similarly fruitful in higher dimensions as well. In the past few years there have been some interesting and subtle connections between contact geometry and Riemannian geometry. The Principal Investigator will explore this further hoping to find contact geometric analogs of classical results relating topology to Riemannian geometry.