A pioneering new nonlinear approach to a fundamental question in algebraic geometry One of the crowning achievements of nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. What Determines an Algebraic Variety? develops a nonlinear version of this theory, offering the first nonlinear generalization of the seminal work of Veblen and Young in a century. While the book uses cutting-edge techniques, the statements of its theorems would have been understandable a century ago; despite this, the results are totally unexpected. Putting geometry first in algebraic geometry, the book provides a new perspective on a classical theorem of fundamental importance to a wide range of fields in mathematics. Starting with basic observations, the book shows how to read off various properties of a variety from its geometry. The results get stronger as the dimension increases. The main result then says that a normal projective variety of dimension at least 4 over a field of characteristic 0 is completely determined by its Zariski topological space. There are many open questions in dimensions 2 and 3, and in positive characteristic.
The authors explain the ways in which uncertainty is an important factor in the problems of risk and policy analysis. This book outlines the source and nature of uncertainty, discusses techniques for obtaining and using expert judgment, and reviews a variety of simple and advanced methods for analyzing uncertainty.
This volume offers a collection of articles written by the renowned conductor and scholar Max Rudolf, together with a selection of his correspondence relating to material in the articles. Max Rudolf's conducting career spanned seventy years, from his first performances in l920-2l to his last in 1990. His life was devoted to performing, scholarship, and teaching. He conducted at the Metropolitan Opera from 1943 to 1937 and was Musical Director of the Cincinnati Symphony from 1938 to 1970, after which he combined guest conducting with teaching opera and conducting at the Curtis Institute in Philadelphia. The articles reflect a lifetime of thought on the art of conducting, musical style, and performance practice. Rudolf, known as an interpreter of the classical repertoire, freely shared his vast knowledge of Mozart's and Beethoven's scores with colleagues and students. His conducting book, The Grammar of Conducting, has been the leading college text in the field for many years. As such it has extended his influence on many generations of conductors. Throughout his life, Rudolf corresponded voluminously with other musicians. The letters included in this volume were selected because they shed a warm, personal light on the formal published articles thus providing an opportunity to share the mind and thoughts of an outstanding human bein
A pioneering new nonlinear approach to a fundamental question in algebraic geometry One of the crowning achievements of nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. What Determines an Algebraic Variety? develops a nonlinear version of this theory, offering the first nonlinear generalization of the seminal work of Veblen and Young in a century. While the book uses cutting-edge techniques, the statements of its theorems would have been understandable a century ago; despite this, the results are totally unexpected. Putting geometry first in algebraic geometry, the book provides a new perspective on a classical theorem of fundamental importance to a wide range of fields in mathematics. Starting with basic observations, the book shows how to read off various properties of a variety from its geometry. The results get stronger as the dimension increases. The main result then says that a normal projective variety of dimension at least 4 over a field of characteristic 0 is completely determined by its Zariski topological space. There are many open questions in dimensions 2 and 3, and in positive characteristic.
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