Geometry and Topology of the Heisenberg Groups

海森堡群第一次出现在赫尔曼·魏尔的证明中，他证明了薛定谔和海森堡的量子力学方法在数学上是等价的。然而，海森堡群的应用范围远远超出了量子力学。海森堡小组在数学的许多领域发挥着重要作用，在数学生物学中也在视觉皮层数学模型的发展中发挥着重要作用。虽然海森堡群已经被研究了几十年，但最近的发展，导致在纯数学和应用数学的新领域的应用，提供了新的视角。这个研究项目将推进当代数学这个非常活跃的领域。该项目将为研究生和博士后提供充分的机会，在这个重要的数学领域进行培训，桥梁不同的数学领域。四名研究生和两名博士后将积极从事与项目相关的课题研究。这项研究将导致在格罗莫夫先生的开创性工作中介绍的计划的发展，它将由几个独立但相关的任务组成。大部分任务将作为PI与他的研究生，博士后助理和来自美国和欧洲的其他研究人员的联合项目进行。该项目将调查以下问题。(1)发展了Heisenberg群的Lipschitz同伦群理论。与经典同伦群不同，Lipschitz同伦群提供了对Holder、Lipschitz和Sobolev映射到Heisenberg群的几何更深入的了解。(2)刻画具有Lipschitz扩展性质的Heisenberg群对。Lipschitz可拓性的研究是非常困难的，因为经典的拓扑方法是失败的，需要更多的定量和解析的方法。(3)求解映射到Heisenberg群的Whitney可拓问题。虽然惠特尼扩展定理已被证明是分析中最具影响力的结果之一，但海森堡群提供了一个迄今尚未研究的非线性约束。(4)发展Holder连续映射的微分学。这个主题最近和独立发现的许多研究人员在不同的数学领域,和π的目标是找到一种统一的方法将导致某种结果的简化格罗莫夫和海森堡集团将链接的方法开发与集成在凸几何刚度问题。(5)构造M. Gromov关于Holder映射到Heisenberg群的猜想的一个反例。数值算例显示了意想不到的结果，但它们必须得到严格的验证。(6)基于Heisenberg群的不可整流性，找到有界长度畸变映射的新表征。本课题还研究了凸函数的逼近、Sobolev空间中极大函数的有界性、雅可比矩阵变符号的同胚、具有正雅可比矩阵的Sobolev映射的连续性。

The Heisenberg group appeared for the first time in Herman Weyl's proof that the Schrodinger and the Heisenberg approaches to quantum mechanics are mathematically equivalent. However, the scope of applications of the Heisenberg groups goes far beyond quantum mechanics. The Heisenberg groups play an important role in many areas of mathematics and also in the mathematical biology in the development of a mathematical model of the visual cortex. Although the Heisenberg groups have been studied for several decades now, recent developments, leading to applications in new areas of pure and applied mathematics, provide new perspectives. This research project will advance this very active area of contemporary mathematics. The project will provide ample opportunity for graduate students and postdoctoral associates to be trained in this important area that bridges different fields of mathematics. Four graduate students and two postdoctoral associates will actively work on topics related to the project. The research will lead to a development of the program introduced in a seminal work of M. Gromov and it will consist of several independent, but related tasks. Most of the tasks will be carried out as a joint projects of the PI with his graduate students, postdoctoral associates and other researchers from the US and Europe. The project will investigate the following problems. (1) Develop the theory of Lipschitz homotopy groups of the Heisenberg groups. Unlike the classical homotopy groups, the Lipschitz homotopy groups provide a deeper insight into the geometry of Holder, Lipschitz, and Sobolev mappings into the Heisenberg group. (2) Characterize the pairs of the Heisenberg groups that have the Lipschitz extension property. The Lipschitz extension property is very difficult to study, because the methods of the classical topology fail and a more quantitative and analytic methods are necessary. (3) Solve the problem of the Whitney extension for mappings into the Heisenberg group. While the Whitney extension theorem has proven to be one of the most influential results in analysis, the Heisenberg group provides a non-linear constraint that has not been investigated so far. (4) Develop the differential calculus of Holder continuous maps. This subject has recently and independently been discovered by many researchers working in different areas of mathematics and the goal of the PI is to find a unified approach which will lead to a simplification of some results of Gromov and will link the methods developed for the Heisenberg group with the geometric rigidity problems in convex integration. (5) Construct a counterexample to a conjecture of M. Gromov about Holder mappings into the Heisenberg group. Numerical examples show unexpected results, but they have to be verified rigorously. (6) Find a new characterization of mappings of bounded length distortion, a study motivated by unrectifiability of the Heisenberg group. Other topics under study in the project are: approximation of convex functions, boundedness of maximal functions in Sobolev spaces, homeomorphisms whose Jacobian changes sign, and continuity of Sobolev mappings with positive Jacobian.