Integral geometry in affine and projective spaces

A survey is given of results of integral geometry related to the integration of sections of one-dimensional bundles and differential forms over planes in affine and projective spaces.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 16, pp. 53–226, 1980.     

The horospherical duality

We discuss the horospherical duality as a geometrical background of harmonic analysis on semisimple symmetric spaces.     

Harmonic analysis on symmetric Stein manifolds from the point of view of complex analysis

The classical theory of finite dimensional representations of compact and complex semisimple Lie groups is discussed from the perspective of multidimensional complex geometry and analysis. The key tool is the complex horospherical transform which establishes a duality between spaces of holomorphic functions on symmetric Stein manifolds and dual horospherical manifolds. Communicated by: Toshiyuki Kobayashi     

Kink-antikink collisions in the φ⁶ model

We study kink-antikink collisions in the one-dimensional nonintegrable scalar φ? model. Although the single-kink solutions for this model do not possess an internal vibrational mode, our simulations reveal a resonant scattering structure, thereby providing a counterexample to the standard belief that the existence of such a mode is a necessary condition for multibounce resonances in general kink-antikink collisions. We investigate the two-bounce windows in detail, and present evidence that this structure is caused by the existence of bound states in the spectrum of small oscillations about a combined kink-antikink configuration.