Geometry and Topology of the Heisenberg Groups

海森堡群首次出现在赫尔曼·韦尔的证明中，证明了量子力学的薛定谔方法和海森伯格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。海森伯格小组在许多数学领域和数学生物学中扮演着重要的角色，在视觉皮质的数学模型的发展中。尽管海森伯格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。这项研究项目将推动当代数学这一非常活跃的领域的发展。该项目将为研究生和博士后助理提供充分的机会，在这个连接不同数学领域的重要领域进行培训。四名研究生和两名博士后助理将积极研究与该项目相关的主题。这项研究将导致M.Gromov一项开创性工作中介绍的计划的发展，该计划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执行。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Soblev映射到Heisenberg群的几何的更深层次的了解。(2)刻画具有Lipschitz扩张性质的Heisenberg群对。由于经典拓扑学的方法失效了，需要一种更定量的分析方法来研究Lipschitz延拓性质，这是一个非常困难的问题。(3)解决了映射到Heisenberg群的Whitney扩张问题。虽然惠特尼延拓定理已被证明是分析中最有影响力的结果之一，但海森伯格群提供了一个迄今尚未被研究的非线性约束。(4)推广了Holder连续映射的微积分。这个问题是最近由许多在不同数学领域工作的研究人员独立发现的，PI的目标是找到一个统一的方法，它将导致Gromov的一些结果的简化，并将为Heisenberg群发展的方法与凸积分中的几何刚性问题联系起来。(5)对M.Gromov关于Holder映射到Heisenberg群的猜想构造一个反例。数值例子显示了意想不到的结果，但它们必须得到严格的验证。(6)研究了Heisenberg群的不可纠性，得到了有界长度偏差映射的一个新刻画。该项目中的其他研究课题包括：凸函数的逼近，极大函数在Sobolev空间中的有界性，Jacobian变号的同胚，以及Sobolev映射具有正Jacobian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.