University of Georgia
Mathematics Department Colloquium
Jingzhi Tie
University of California, Irvine
March 15 1999
Heisenberg Group and Laguerre Calculus
Abstract: The Heisenberg group is the simplest, non-commutative, nilpotent
Lie group. It arises in two fundamental but different settings in analysis.
On the one hand, it can be realized as the boundary of the unit ball
in several complex variables. On the other hand, there is its genesis
in the context of Quantum Mechanics. The Laguerre calculus is the
symbolic tensor calculus induced by the Laguerre function on
the Heisenberg group. In this talk, I will introduce the Heisenberg group
and its Lie algebra from the setting of complex analysis. Then I
will use the Laguerre calculus to solve the $\bar\partial$-Neumann
problem in the non-isotropic Siegel Domain. If time permits, I will
also talk about its application to the Heat kernel, power of sub-Laplacian
and Riesz transform on the Heisenberg group.