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What is the Genus?

  • Book
  • © 2016

Overview

Authors:
  1. Patrick Popescu-Pampu

    1. UFR de Mathématiques, Universitè Lille 1, Villeneuve d’Ascq, France

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  • Presents, through the works of the pioneers, the sophisticated evolution of one of the most important notions of geometry and topology
  • Features many illustrations which aid the understanding of the more complicated constructions
  • Excepting M. Audin's novel La formule de Stokes, there are no similar books
  • Explains many connections between analysis, algebra, number theory, geometry and topology
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2162)

Part of the book sub series: History of Mathematics Subseries (HISTORYMS)

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Table of contents (50 chapters)

  1. Algebraic Curves

    1. Clifford and the Number of Holes

      • Patrick Popescu-Pampu
      Pages 53-57

    2. Clebsch and the Choice of the Term “Genus”

      • Patrick Popescu-Pampu
      Pages 59-61

    3. Cayley and the Deficiency

      • Patrick Popescu-Pampu
      Pages 63-64

    4. Noether and the Adjoint Curves

      • Patrick Popescu-Pampu
      Pages 65-66

    5. Klein, Weyl, and the Notion of an Abstract Surface

      • Patrick Popescu-Pampu
      Pages 67-68

    6. The Uniformization of Riemann Surfaces

      • Patrick Popescu-Pampu
      Pages 69-70

    7. The Genus and the Arithmetic of Curves

      • Patrick Popescu-Pampu
      Pages 71-72

    8. Several Historical Considerations by Weil

      • Patrick Popescu-Pampu
      Pages 73-75

    9. And More Recently?

      • Patrick Popescu-Pampu
      Pages 77-77

  2. Algebraic Surfaces

    1. Front Matter

      Pages 79-79
      Download chapter PDF

    2. The Beginnings of a Theory of Algebraic Surfaces

      • Patrick Popescu-Pampu
      Pages 81-84

    3. The Problem of the Singular Locus

      • Patrick Popescu-Pampu
      Pages 85-89

    4. A Profusion of Genera for Surfaces

      • Patrick Popescu-Pampu
      Pages 91-92

    5. The Classification of Algebraic Surfaces

      • Patrick Popescu-Pampu
      Pages 93-95

    6. The Geometric Genus and the Newton Polyhedron

      • Patrick Popescu-Pampu
      Pages 97-98

    7. Singularities Which Do Not Affect the Genus

      • Patrick Popescu-Pampu
      Pages 99-101

    8. Hodge’s Topological Interpretation of Genera

      • Patrick Popescu-Pampu
      Pages 103-104

    9. Comparison of Structures

      • Patrick Popescu-Pampu
      Pages 105-106

  3. Algebraic Surfaces

    1. Front Matter

      Pages 107-107
      Download chapter PDF

    2. Hilbert’s Characteristic Function of a Module

      • Patrick Popescu-Pampu
      Pages 109-111
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Keywords

About this book

Exploring several of the evolutionary branches of the mathematical notion of genus, this book traces the idea from its prehistory in problems of integration, through algebraic curves and their associated Riemann surfaces, into algebraic surfaces, and finally into higher dimensions. Its importance in analysis, algebraic geometry, number theory and topology is emphasized through many theorems. Almost every chapter is organized around excerpts from a research paper in which a new perspective was brought on the genus or on one of the objects to which this notion applies. The author was motivated by the belief that a subject may best be understood and communicated by studying its broad lines of development, feeling the way one arrives at the definitions of its fundamental notions, and appreciating the amount of effort spent in order to explore its phenomena.

Reviews

“This book provides a fascinating—yet technically quite demanding—tour of the ideas and generalizations that feed into our modern conception of the genus. It should therefore serve as a useful starting point for historical and even philosophical investigations into past mathematical practice of forming and shaping concepts. It may also be useful in teaching advanced mathematical content with a view to historical development … .” (Henrik Kragh Sørensen, ISIS, Vol. 109 (2), June, 2018)


“This book is addressed to any person with a little knowledge of mathematics (such as an undergraduate student) who wants to get an idea of some of the most important concepts that arose in geometry and topology in the last couple of centuries: the author has made a commendable effort to explain all the more advanced concepts almost from scratch (but obviously this was not possible for the last chapter).” (Pietro De Poi, Mathematical Reviews, July, 2017)



“This clear book thoroughly covers the metamorphosis and theoretical significance of the genus by using several historical approaches that incorporate discussions related to analysis, algebraic geometry, number theory, and topology. … The book contains 20 figures and concludes with a list of 200 references and an index, which will help those who wish to delve into deeper studies. Summing Up: Recommended. Upper-division undergraduates and above; faculty and professionals.” (D. P. Turner, Choice, Vol. 54 (9), May, 2017)

Authors and Affiliations

  • UFR de Mathématiques, Universitè Lille 1, Villeneuve d’Ascq, France

    Patrick Popescu-Pampu

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