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Riemann Surfaces and Algebraic Curves A First Course in Hurwitz Theory 1st Edition by Renzo Cavalieri, Eric Miles ISBN 110714924X 9781107149243

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Riemann Surfaces and Algebraic Curves A First Course in Hurwitz Theory 1st Edition Renzo Cavalieri Digital Instant Download

Author(s): Renzo Cavalieri, Eric Miles
ISBN(s): 9781107149243, 110714924X
Edition: 1
File Details: PDF, 3.31 MB
Year: 2016
Language: english
SKU: EB-6983090 Category: Tags: ,

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ISBN 10: 110714924X 
ISBN 13: 9781107149243
Author: Renzo Cavalieri, Eric Miles

Hurwitz theory, the study of analytic functions among Riemann surfaces, is a classical field and active research area in algebraic geometry. The subject’s interplay between algebra, geometry, topology and analysis is a beautiful example of the interconnectedness of mathematics. This book introduces students to this increasingly important field, covering key topics such as manifolds, monodromy representations and the Hurwitz potential. Designed for undergraduate study, this classroom-tested text includes over 100 exercises to provide motivation for the reader. Also included are short essays by guest writers on how they use Hurwitz theory in their work, which ranges from string theory to non-Archimedean geometry. Whether used in a course or as a self-contained reference for graduate students, this book will provide an exciting glimpse at mathematics beyond the standard university classes.

Riemann Surfaces and Algebraic Curves A First Course in Hurwitz Theory 1st Table of contents:

1 From Complex Analysis to Riemann Surfaces
1.1 Differentiability
1.2 Integration
1.3 Cauchy’s Integral Formula and Consequences
1.4 Inverse Functions
2 Introduction to Manifolds
2.1 General Definition of a Manifold
2.2 Basic Examples
2.3 Projective Spaces
2.4 Compact Surfaces
2.5 Manifolds as Level Sets
3 Riemann Surfaces
3.1 Examples of Riemann Surfaces
3.2 Compact Riemann Surfaces
4 Maps of Riemann Surfaces
4.1 Holomorphic Maps of Riemann Surfaces
4.2 Local Structure of Maps
4.3 Maps of Compact Riemann Surfaces
4.4 The Riemann–Hurwitz Formula
4.5 Examples of Maps of Compact Riemann Surfaces
5 Loops and Lifts
5.1 Homotopy
5.2 The Fundamental Group
5.3 Covering Spaces
6 Counting Maps
6.1 Hurwitz Numbers
6.2 Riemann’s Existence Theorem
6.3 Hyperelliptic Covers
7 Counting Monodromy Representations
7.1 From Maps to Monodromy
7.2 From Monodromy to Maps
7.3 Monodromy Representations and Hurwitz Numbers
7.4 Examples and Computations
7.5 Degeneration Formulas
8 Representation Theory of
8.1 The Group Ring and the Group Algebra
8.2 Representations
8.3 Characters
8.4 The Class Algebra
9 Hurwitz Numbers and
9.1 Genus 0
9.2 Genus and Commutators
9.3 Burnside Formula
10 The Hurwitz Potential
10.1 Generating Functions
10.2 Connected Hurwitz Numbers
10.3 Cut-and-Join
A Hurwitz Theory in Positive Characteristic
A.1 Introduction
A.2 Algebraic Curves in Positive Characteristic
A.3 Smooth Algebraic Curves
A.4 The Genus and the Riemann–Hurwitz Formula
A.5 The Affine Line Is Not Simply Connected Over k
B Tropical Hurwitz Numbers
B.1 Tropical Geometry: Where Does It Come From?
B.2 Axiomatic Tropical Geometry
B.3 Tropical Covers
B.4 Tropical Covers as Shadows – Movies of Monodromy Representations
B.5 Bending and Breaking
C Hurwitz Spaces
C.1 Moduli Spaces
C.2 Hurwitz Spaces
C.3 Applications of Hurwitz Spaces
D Does Physics Have Anything to Say About Hurwitz Numbers?
D.1 Physical Mathematics
D.2 Hurwitz Numbers and String Theory

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Tags: Renzo Cavalieri, Eric Miles, Riemann Surfaces, Algebraic Curves, Hurwitz Theory