Geometry and Topology of the Heisenberg Groups

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

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.