Submanifolds and Metrics in Contact Geometry

流形上的接触结构是两个多世纪前惠更斯、汉密尔顿和雅可比在几何光学方面的工作中诞生的自然物体。它们被许多数学家研究过，似乎涉及数学和物理的不同领域，但直到最近几十年它们才进入数学的前景。这是由于接触拓扑学的显著突破，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这个扩展到更高的维度，在那里这个探索可能同样具有启发性。首席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和黎曼几何的相互作用，他将通过与一大群研究生合作和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：低维接触几何和拓扑的相互作用，高维接触流形的性质和结构，以及接触几何和更熟悉的黎曼几何之间的联系。在低维中，主要的激励问题是确定哪三个流形允许紧接触结构。目前对这个问题已经有了相当多的了解，但对双曲同调球的了解却很少。这个问题将使用各种技术来研究，从凸面到全纯曲线和黎曼几何。此外，还将研究接触结构的各种性质之间的相互作用。低维接触几何已知很多，而高维接触几何所知甚少。首席研究员将研究高维接触流形的结构和性质。这方面的出发点将是研究各向同性和接触流形的接触子流形。这样的考虑已经在低维中产生了丰富的信息，预计在高维中也会有类似的成果。在过去的几年里，在接触几何和黎曼几何之间出现了一些有趣而微妙的联系。首席研究员将进一步探索这一点，希望找到与黎曼几何拓扑相关的经典结果的接触几何类似物。

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.