Modern geometry : a celebration of the work of Simon Donaldson / Vicente Muñoz, Ivan Smith, Richard P. Thomas, editors.

Format:

Book

Contributor:

Muñoz, V. (Vicente), 1971- editor.

Smith, Ivan, 1973- editor.

Thomas, Richard P., 1972- editor.

Series:

Proceedings of symposia in pure mathematics ; Volume 99.

Proceedings of symposia in pure mathematics ; Volume 99

Language:

English

Subjects (All):

Donaldson, S. K.

Manifolds (Mathematics).

Four-manifolds (Topology).

Geometry.

Topology.

Physical Description:

1 online resource (426 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2018]

Summary:

This book contains a collection of survey articles of exciting new developments in geometry, written in tribute to Simon Donaldson to celebrate his 60th birthday. Reflecting the wide range of Donaldson's interests and influence, the papers range from algebraic geometry and topology through symplectic geometry and geometric analysis to mathematical physics. Their expository nature means the book acts as an invitation to the various topics described, while also giving a sense of the links between these different areas and the unity of modern geometry.

Contents:

Cover

Title page

Contents

Preface

Graded linearisations

1. GIT with classical linearisations

2. Extended, graded and torus-graded linearisations

3. Applications

References

Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations

1. Introduction

2. The Extended Moduli Spaces of Flat Connections

3. Equivariant Lagrangian Floer Homology

4. Lagrangian Floer Theory in a Smooth Divisor Complement

5. The Atiyah-Floer Conjecture

6. Atiyah-Floer Conjecture and Moduli Space of Solutions to the Mixed Equation

7. Yang-Mills Gauge Theory and 3-Manifolds with Boundary

8. Lagrangians and _{∞}-categories

9. Cut-down Extended Moduli Spaces for Other Lie Groups

10. Admissible Bundles and Instanton Floer Homology

Weinstein manifolds revisited

1. Weinstein manifolds, domains, cobordisms

2. Weinstein hypersurfaces and Weinstein pairs

3. Operations on Weinstein pairs

4. Looseness and Flexibility

5. Lagrangian submanifolds of Weinstein domains

6. Symplectic topology of Weinstein manifolds

7. Topology of Weinstein fillings

8. Nadler's program of arborealization

Remarks on Nahm's equations

2. Co-Higgs bundles and Nahm's equations

3. Moduli spaces and the Nahm flow

4. Ribbons

Conjectures on counting associative 3-folds in ₂-manifolds

2. Geometry of ₂-manifolds

2.1. ₂-manifolds

2.2. Calabi-Yau 3-folds and ₂-manifolds

2.3. Calibrated submanifolds

2.4. ₂-instantons

2.5. Tamed almost- ₂-manifolds

2.6. Moduli spaces of associative 3-folds

2.7. Associative 3-folds with boundary in coassociatives

3. How to orient moduli spaces of associatives

3.1. Flags and flag structures

3.2. Canonical flags of associatives, and orientations.

4. An index 1 singularity of associative 3-folds

4.1. A family of SL 3-folds in \C³

4.2. Desingularizing immersed associative 3-folds

5. Another index 1 associative singularity

5.1. Three families of SL 3-folds in \C³

5.2. Associative 3-folds with singularities modelled on ₀

5.3. Algebraic topology of desingularizations using ^{ }_{ }

6. \U(1)-invariant associative 3-folds in \R⁷

6.1. Associative 3-folds and -holomorphic curves

6.2. Associative 3-folds with boundary in coassociatives

7. A superpotential counting associative 3-folds

7.1. Set up of situation and notation

7.2. Six kinds of wall-crossing behaviour

7.3. Definition of the superpotential

7.4. How Φ_{ } depends on choices, and on

7.5. Our main conjecture

7.6. ₂ quantum cohomology

7.7. Generalizations

8. Remarks on counting ₂-instantons

8.1. The Donaldson-Segal programme

8.2. Canonical orientations for moduli of ₂-instantons

8.3. '-flags, and canonical '-flags

8.4. Problems with counting ₂-instantons

8.5. A suggestion for how to modify Donaldson-Segal

Toward an algebraic Donaldson-Floer theory

1. introduction

2. Good degeneration of moduli of stable sheaves

3. Moduli of relative stable sheaves

4. The homology groups of the stack of vector bundles

5. A proposed algebraic Donaldson-Floer theory

Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional =4 gauge theories

1. Coulomb and Higgs branches -complex symplectic varieties and deformation quantization

2. Physical background

3. A mathematical definition

4. Not necessarily cotangent type

4(i). Holomorphic Chern-Simons functional

4(ii). Derived symplectic geometry

4(iii). Cutting

5. Examples

5(i).

5(ii).

6. Structures

6(i).

6(ii).

6(iii).

6(iv).

6(v).

6(vi).

7. Quiver gauge theories

8. Quantized Coulomb branches

An overview of knot Floer homology

1. Motivation for the construction

2. Statement of the symplectic constructions

3. First properties

4. Topological applications

5. Heegaard diagrams

6. Bordered preliminaries

7. Bordered algebras and knot invariants

8. Bordered knot algebras and pseudo-holomorphic curves

9. Further remarks

Descendents for stable pairs on 3-folds

0. Introduction

1. Rationality

2. Gromov-Witten/Pairs correspondence

3. Virasoro constraints

4. Virtual class in algebraic cobordism

Acknowledgments

The Dirichlet problem for the complex homogeneous Monge-Ampère equation

2. Preliminaries

3. The HMAE on domains in ℂⁿ

4. The HMAE for compact Kähler manifolds

5. The Hele-Shaw Flow

6. The Duality Theorem

7. Harmonic discs

8. The Strong Hele-Shaw Flow

9. Examples

Acknowledgements

Kähler-Einstein metrics

2. The moment map picture

3. K-stability

4. Geometric and algebraic limits

5. Applications

Donaldson theory in non-Kählerian geometry

1. Introduction.

2. An instanton moduli space on definite 4-manifolds with ₁=1

3. The circles of reductions

4. Curves on class VII surfaces

Two lectures on gauge theory and Khovanov homology

1. Lecture One

2. Lecture Two

Appendix A. Some Physics Background

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-4811-4

OCLC:

1054059182