Bernard Maskit – författare

The description for this book, Riemann Surfaces Related Topics (AM-97): Proceedings of the 1978 Stony Brook Conference. (AM-97), will be forthcoming.

Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers is a compendium of papers provided by Bers, friends, students, colleagues, and professors. These papers deal with Teichmuller spaces, Kleinian groups, theta functions, algebraic geometry. Other papers discuss quasiconformal mappings, function theory, differential equations, and differential topology. One paper discusses the results of the rigidity theorem of Mostow and its generalization by Marden in relation to geometric properties of Kleinian groups of the first kind. These results, obtained by planar methods, are presented in terms of the hyperbolic 3-space language, which is a natural pedestal in approaching the action of the Kleinian groups. Another paper reviews Riemann''s vanishing theorem which solves the Jacobi inversion problem, by relating the vanishing properties of the theta function (particularly at half periods) to properties of certain linear series on the Riemann surface. One paper examines the problem of obtaining relations among the periods of the differentials of first kind on a compact Riemann surface. An application of a computer program involves supersonic transport. The program is based on the hodograph transformation and a method of complex characteristics to calculate profiles that are shock-less at a specified angle of attack, or at a specified subsonic free-stream Mach number. The collection can prove useful for engineers, statisticians, students, and professors in advance mathematics or courses related to aeronautics.

The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors' finiteness theorem, and Bers' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geome­ try, and there is now an active school of research using these methods.

The modern theory of Kleinian groups starts with the work of Lars Ahlfors and Lipman Bers; specifically with Ahlfors'' finiteness theorem, and Bers'' observation that their joint work on the Beltrami equation has deep implications for the theory of Kleinian groups and their deformations. From the point of view of uniformizations of Riemann surfaces, Bers'' observation has the consequence that the question of understanding the different uniformizations of a finite Riemann surface poses a purely topological problem; it is independent of the conformal structure on the surface. The last two chapters here give a topological description of the set of all (geometrically finite) uniformizations of finite Riemann surfaces. We carefully skirt Ahlfors'' finiteness theorem. For groups which uniformize a finite Riemann surface; that is, groups with an invariant component, one can either start with the assumption that the group is finitely generated, and then use the finiteness theorem to conclude that the group represents only finitely many finite Riemann surfaces, or, as we do here, one can start with the assumption that, in the invariant component, the group represents a finite Riemann surface, and then, using essentially topological techniques, reach the same conclusion. More recently, Bill Thurston wrought a revolution in the field by showing that one could analyze Kleinian groups using 3-dimensional hyperbolic geome­ try, and there is now an active school of research using these methods.