WHAT IS THE GENUS?

1. The ?´e?o??e´?o? According to Aristotle
2. Descartes and the New World of Curves
3. Newton and the Classification of Curves
4. When Integrals Hide Curves
5. Jakob Bernoulli and the Construction of Curves
6. Fagnano and the Lemniscate
7. Euler and the Addition of Lemniscatic Integrals
8. Legendre and Elliptic Functions
9. Abel and the New Transcendental Functions
10. A Proof by Abel
11. Abel’s Motivations
12. Cauchy and the Integration Paths
13. Puiseux and the Permutations of Roots
14. Riemann and the Cutting of Surfaces
15. Riemann and the Birational Invariance of Genus
16. The Riemann–Roch Theorem
17. A Reinterpretation of Abel’s Works
18. Jordan and the Topological Classification
19. Clifford and the Number of Holes
20. Clebsch and the Choice of the Term “Genus”
21. Cayley and the Deficiency
22. Noether and the Adjoint Curves
23. Klein, Weyl, and the Notion of an Abstract Surface
24. The Uniformization of Riemann Surfaces
25. The Genus and the Arithmetic of Curves
26. Several Historical Considerations by Weil
27. And More Recently?
28. The Beginnings of a Theory of Algebraic Surfaces
29. The Problem of the Singular Locus
30. A Profusion of Genera for Surfaces
31. The Classification of Algebraic Surfaces
32. The Geometric Genus and the Newton Polyhedron
33. Singularities Which Do Not Affect the Genus
34. Hodge’s Topological Interpretation of Genera
35. Comparison of Structures
36. Hilbert’s Characteristic Function of a Module
37. Severi and His Genera in Arbitrary Dimension
38. Poincaré and Analysis Situs
39. The Homology and Cohomology Theories
40. Elie Cartan and Differential Forms
41. de Rham and His Cohomology
42. Hodge and the Harmonic Forms
43. Weil’s Conjectures
44. Serre and the Riemann–Roch Problem
45. New Ingredients
46. Whitney and the Cohomology of Fibre Bundles
47. Genus Versus Euler–Poincaré Characteristic
48. Harnack and Real Algebraic Curves
49. The Riemann–Roch–Hirzebruch Theorem
50. The Riemann–Roch–Grothendieck Theorem

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.

Features
• 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