Lectures on Lagrangian torus fibrations

TL;DR

This book provides an in-depth exploration of Lagrangian torus fibrations in symplectic geometry, emphasizing the integral affine structure of the base and its applications to various geometric phenomena.

Contribution

It offers a comprehensive, graduate-level exposition on Lagrangian torus fibrations, including new insights on their applications and open problems in symplectic geometry.

Findings

Base of Lagrangian torus fibration inherits an integral affine structure

Applications to symplectic reduction, toric manifolds, and tropical Lagrangians

Discussion of open problems in the field

Abstract

This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can use to "read off" a lot of interesting geometry of the total space. Topics covered include: action-angle coordinates, symplectic reduction, toric manifolds, visible and tropical Lagrangians, almost toric systems, Milnor fibres of cyclic quotient singularities, mutation of polygons, non-toric blow-up, an almost toric view on Lisca's classification of fillings of lens spaces, resolutions of cusp singularities, Markov triples and Vianna tori. The book ends with a short list of open problems. Throughout there is an emphasis on examples and there are some exercises with solutions.