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27 results

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  • 8 - Almost toric manifolds

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 100-121

  • Summary

    We set up Symington’s theory of almost toric fibrations, including a discussion of the three basic operations (nodal trades, nodal slides, and change of branch cut/mutation). We prove Symington’s fundamental theorems that these operations have no effect on the symplectomorphism type of the ambient manifold. We give a range of examples including the Markov–tree of mutations of the standard almost toric fibration on the complex projective plane.

  • 5 - Visible Lagrangian submanifolds

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 64-72

  • Summary

    We study visible Lagrangian submanifolds, that is, Lagrangian submanifolds whose projection under a Lagrangian fibration has positive codimension. We show that projections of visible Lagrangians are affine subspaces of rational slope. We give some examples and explain how a visible Lagrangian can intersect the toric boundary.

  • 7 - Examples of focus-focus systems

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 86-99

  • Summary

    We construct examples of focus–focus systems which occur naturally on Milnor fibres of surface singularities. This allows us to read off aspects of the geometry and topology of these Milnor fibres. We also give a detailed analysis of Auroux’s integrable system on complex 2–space, which is a local model for understanding focus–focus fibrations.

  • 6 - Focus-focus singularities

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 73-85

  • Summary

    We study integrable Hamiltonian systems with “focus–focus” singularities. We give an exposition of Vu Ngoc’s results which characterise the integral affine structure in a punctured neighbourhood of a focus–focus critical value. We give picures to illustrate the affine monodromy around a loop of regular values that encircle a focus–focus value.

  • 9 - Surgery

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 122-137

  • Summary

    We introduce non–toric blow–up and rational blow–up/down, with many examples. We also discuss how to use almost toric diagrams to visualise the symplectic fillings of lens spaces that were classified by Lisca.

  • 1 - The Arnold-Liouville theorem

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 1-16

  • Summary

    We first introduce integrable Hamiltonian systems on symplectic manifolds. We show that if a Hamiltonian system on a two–dimensional phase space has all of its orbits closed then we can modify the Hamiltonian by a diffeomorphism to ensure all the orbits have the same period. The rest of the chapter explains how to generalise this to Hamiltonian systems with more degrees of freedom, culminating in the Arnold–Liouville theorem, which underpins everything else in the book.

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