Lectures on Lagrangian Torus Fibrations: (Series Number 105 London Mathematical Society Student Texts)

Symington's almost toric fibrations have played a central role in symplectic geometry over the past decade, from Vianna's discovery of exotic Lagrangian tori to recent work on Fibonacci staircases. Four-dimensional spaces are of relevance in Hamiltonian dynamics, algebraic geometry, and mathematical string theory, and these fibrations encode the geometry of a symplectic 4-manifold in a simple 2-dimensional diagram. This text is a guide to interpreting these diagrams, aimed at graduate students and researchers in geometry and topology. First the theory is developed, and then studied in many examples, including fillings of lens spaces, resolutions of cusp singularities, non-toric blow-ups, and Vianna tori. In addition to the many examples, students will appreciate the exercises with full solutions throughout the text. The appendices explore select topics in more depth, including tropical Lagrangians and Markov triples, with a final appendix listing open problems. Prerequisites include familiarity with algebraic topology and differential geometry.

Table of Contents:
1. The Arnold–Liouville theorem; 2. Lagrangian fibrations; 3. Global action-angle coordinates and torus actions; 4. Symplectic reduction; 5. Visible Lagrangian submanifolds; 6. Focus-focus singularities; 7. Examples of focus-focus systems; 8. Almost toric manifolds; 9. Surgery; 10. Elliptic and cusp singularities; A. Symplectic linear algebra; B. Lie derivatives; C. Complex projective spaces; D. Cotangent bundles; E. Moser's argument; F. Toric varieties revisited; G. Visible contact hypersurfaces and Reeb flows; H. Tropical Lagrangian submanifolds; I. Markov triples; J. Open problems; References; Index.

About the Author :
Jonny Evans is Reader in Pure Mathematics at Lancaster University. He is a geometer, and in 2021 he was awarded an LMS Whitehead Prize for his work on symplectic geometry and singularities.

Review :
'Lagrangian torus fibrations are an interesting source of examples in symplectic geometry, since their symplectic features are encoded by the geometry of certain half-dimensional base diagrams. Enriched by many pictures and exercises with solutions, this book provides an accessible and well-written introduction to this topic, which is of interest to a broad audience through its connections with integrable systems and algebraic geometry. This work will be appreciated by students and experts alike, since it fills a crucial gap in the literature by giving an excellent discussion of almost toric fibrations, which have attracted a lot of attention in recent years.' Felix Schlenk, Université de Neuchatel 'This is a lucid and engaging introduction to the fascinating world of (almost) toric geometry, in which one can understand the properties of Lagrangian and symplectic submanifolds in four dimensions simply by drawing suitable two-dimensional diagrams. The book has many illustrations and intricate examples.' Dusa McDuff, Barnard College, Columbia University