Thanks to the magic (and obvious) relation between the 2n + 1-dimensional Heisenberg group and its vector group [R.

Understanding the nature of these kind of partial differential operators and their invariance on the Heisenberg group requires the admission of solutions.

In this paper we describe a method to derive a Weierstrass-type representation formula for simply connected immersed minimal surfaces in

Heisenberg group [H.

Turhan: Completeness of Lorentz Metric on 3-Dimensional

Heisenberg Group, Int.

Hans-Jurgen Eisler, who heads the DFG

Heisenberg group at the Light Technology Institute.

Ten chapters discuss the skew field of quaternions; elements of the geometry of S3, Hopf bundles, and spin representations; internal variables of singularity free vector fields in a Euclidean space; isomorphism classes, Chern classes, and homotopy classes of singularity free vector fields in 3-space; Heisenberg algebras,

Heisenberg groups, Minkowski metrics, Jordan algebras, and special linear groups; the Heisenbreg group and natural C*-algebras of a vector field in 3-space; the Schrodinger representation and the metaplectic representation; the

Heisenberg group as a basic geometric background of signal analysis and geometric optics; quantization of quadratic polynomials; and field theoretic Weyl quantization of a vector field in 3-space.

This

Heisenberg group has many important applications on Sub-Riemannian geometry and has very important role in physics.

Specific topics include Hormander operators and non-holonomic geometry, Weyl transforms and the inverse of the sub-Laplacian on the

Heisenberg group, corner operators and applications to elliptic complexes, semilinear pseudo-differential equations and traveling waves, trace ideals for Fourier integral operators with non-smooth symbols, the S-transform and why to use it, inversion formulas for two-dimensional Stockwell transforms, Shannon-type sampling theorems on the

Heisenberg group, and a unified point of view on time-frequency representatives and pseudo-differential operators.

n] the

Heisenberg group of dimension 2n+1, so that we get the existence theorem for these operatrs.

Geometric analysis on the

Heisenberg group and its generalizations.

moment results for the

Heisenberg group interpreted using the Weyl calculus, anisotropic blow up and compacting, planar complex vector fields and infinitesimal bending of surfaces with non-negative curvature.

Topics of the 37 papers include biographies of Professor Zalcman, a multiplicator problem and characteristics of growth of entire functions, quasinormal families with periodic points, univalent functions starlike and with respect to a boundary point, the Wiman- Valiron theory, holomorphic extendability and the argument principle, entire functions with no unbounded Fatou components, the boundary properties of convex functions, modules of vector measures on the

Heisenberg Group, characteristic problems for the spherical mean transform.