Some questions in Integral Geometry were considered on Euclidean spaces and Heisenberg groups. Functions whose spherical averages are zero over conical manifolds of spheres were studied in detail. Using tools from PDE we prove that the family of spheres in Rⁿ. which intersects a given subset A of Rⁿ uniquely determines a continuous function provided that the set A is not contained in any (n-2) dimensional affine plane. In particular given n points in general positions, ie: affinely independent, the spheres passing through these points uniquely determine any continuous function by its averages, while no (n-2) points do the job.

On the Heisenberg group we prove a local version of the two radius theorem for the class of continuous functions. More precisely, if f is a continuous function on the Heisenberg group and the spherical means
f &lowast _&mur1 = f &lowast_&mur2 = 0, in a cylindrical region of the type B_R × R then f is the zero function provided r₁+r₂ < R and no radial eigenfunction of the Heisenberg sublaplacian vanishes simultaneously on the spheres of radius r₁ and r₂ in Cⁿ.

Publications:

1. Multipliers for the twisted Laplacian Colloq. Math., 97 (2003), no.2, 189-205.

Preprints:

1. An optimal theorem for the spherical maximal operator on the Heisenberg group (jointly with S. Thangavelu), to appear in Contemporary Mathematics series.

2. A local two radius theorem for the twisted spherical means on Cⁿ (jointly with M. L. Agranovsky), to appear in Contemporary Mathematics series.

3. Injectivity of the spherical mean operator on conical manifolds of spheres (jointly with M. L. Agranovsky).

Conference/Workshops Attended:

Spectral Theory and Geometry, in memory of Robert Brooks. Technion, Israel, Jan 2004.

Visits to other Institutes:

1. Academic visit to Bar-Ilan university, Israel, from Nov 2003 to April 2004.