Geometry in Advanced Pure Mathematics

Description

This book leads readers from a basic foundation to an advanced level understanding of geometry in advanced pure mathematics. Chapter by chapter, readers will be led from a foundation level understanding to advanced level understanding. This is the perfect text for graduate or PhD mathematical-science students looking for support in algebraic geometry, geometric group theory, modular group, holomorphic dynamics and hyperbolic geometry, syzygies and minimal resolutions, and minimal surfaces. Geometry in Advanced Pure Mathematics is the fourth volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Edited by the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas. Contents Preface Chapter 1. Algebraic Geometry 1. Introduction 1.1. Solving polynomial equations 1.2. Historical development 2. Varieties and Schemes 2.1. Affine varieties 2.2. Morphisms 2.3. Sheaves 2.4. Schemes 2.5. Projective varieties 2.6. First properties of schemes 3. Local Properties 3.1. Non-singular schemes 3.2. Multiplicities 3.3. Divisors 3.4. Riemann{Roch theorem 4. Weil Conjectures 4.1. The zeta function 4.2. Rationality for curves 4.3. Cohomological interpretation of Weil conjectures 5. Further Reading 6. Exercises 6.1. Solutions and hints to selected exercises References Chapter 2. Introduction to the Modular Group and Modular Forms 1. Introduction to the Main Characters 1.1. Prologue 1.2. The modular group and M¨obius transformations in the plane 1.3. Hyperbolic geometry in the upper half-plane 1.4. The upper half-plane as parameter space: Lattices in the plane 1.5. Geometry of real M¨obius transformations 1.6. Exercises for Section 1 2. Discontinuity of the Group Action: The Modular Surface 2.1. Fundamental sets for discrete groups 2.2. A fundamental domain for SL(2, Z) 2.3. The modular surface 2.4. Cone points and cusps 2.5. Automorphic forms: The definition and examples 2.6. Exercises for Section 2 3. Modular Forms: Eisenstein Series and Moduli of Lattices 3.1. Cusps and horodiscs: Cusp neighbourhoods 3.2. Eisenstein series 3.3. Elliptic curves in Weierstrass form 3.4. The discriminant form 3.5. The space of modular forms and the j-invariant 3.6. Exercises for Section 3 4. Horocycles, Ford Circles and Farey Fractions 4.1. Circle geometry and horocycles of Γ(1) 4.2. Ford circles 4.3. Farey sequences 4.4. Farey sequences and Ford circles 4.5. Diophantine approximation of irrational numbers 4.6. The Farey graph 4.7. Exercises for Section 4 5. Topics in Automorphic Forms 5.1. Historical remarks 5.2. Finite type and finite area surfaces 5.3. The space of lattices: A trefoil knot complement 5.4. Conformal field theory and Moonshine 5.5. Sundry topics 5.6. Final exercises References Chapter 3. Geometric Group Theory 1. Free Groups and Presentations 2. Two Basic Constructions 3. Graphs and CW-complexes 4. The Bass–Serre Theory 5. Some Classes of Groups 6. Further Reading Solutions to Exercises References Chapter 4. Holomorphic Dynamics and Hyperbolic Geometry 1. Introduction 1.1. Overview 1.2. The family of maps qc : z → z2 + c 1.3. Examples of Kleinian groups 2. Dynamics of Rational Maps 2.1. The Riemann sphere 2.2. Basic essentials from complex analysis 2.3. Rational maps and critical points 2.4. Conformal automorphisms of ˆC , C and D 2.5. The Poincar´e metric on the upper half-plane 2.6. Conjugacies, fixed points and multipliers 3. The Fatou and Julia Sets of a Rational Map 3.1. Equicontinuity and the Fatou and Julia sets 3.2. Equicontinuity and normality 3.3. Counting critical points, and the exceptional set 3.4. Properties of Julia sets 3.5. An algorithm for plotting J(f) 3.6. Julia sets and repelling periodic points 3.7. The Julia set of qc : z → z2 + c when qnc (0) tends to ∞ 4. Fatou Components and Linearisation Theorems 4.1. Linearisation theorems 4.2. The classification of types of Fatou component 5. Hyperbolic 3-space and Kleinian Groups 5.1. Types of isometries of hyperbolic 3-space 5.2. The ordinary set of a Kleinian group 5.3. The action of a Kleinian group on H3+ 5.4. Limit sets of Kleinian groups 5.5. Elementary Kleinian groups 5.6. Properties of ordinary and limit sets 5.7. Analogies between rational maps and Kleinian groups: Sullivan’s dictionary 6. The Family of Quadratic Polynomials qc : z → z2 + c 6.1. The Mandelbrot set and the MLC conjecture 6.2. The cardioid M0 6.3. The intersection of M with the real axis 6.4. Internal and external rays 7. Quasiconformal Mappings: The Measurable Riemann Mapping Theorem and its Applications 8. Further Reading 9. Exercises 10. Outline Solutions to Exercises (3), (5), (7) and (9) References Chapter 5. Minimal Surfaces and the Bernstein Theorem 1. Introduction 1.1. Minimal surfaces 1.2. First variation of area 1.3. Convex hull and other properties 1.4. Minimal graphs 2. The Bernstein Theorem 2.1. Proof of the Bernstein theorem 3. Quadratic Area Growth and Parabolicity 4. Stable Minimal Surfaces 4.1. Second variation formula 4.2. Quadratic area growth for stable minimal surfaces 5. Exercises References Chapter 6. Syzygies and Minimal Resolutions 1. Introduction 2. Some Categorical Preliminaries 3. Splitting and Projectives 4. Some Standard Diagrams 5. A Comparison Theorem for Resolutions 6. Finiteness Conditions and Stability 7. A Strong Comparison Theorem for Syzygies 8. Uniqueness of Minimal Resolutions 9. The Structure of the Stable Syzygies Ωn(M) 10. Realizing Elements of Ωn(M) as Syzygies 11. Minimal Epimorphisms 12. An Existence Criterion References