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28888 results in Differential and integral equations, dynamical systems and control theory

Page 62 of 1445

  • 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.

  • The localisation theorem for the K-theory of stable ∞-categories

  • Part of
  • Fabian Hebestreit, Andrea Lachmann, Wolfgang Steimle
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 6 / December 2024
    Published online by Cambridge University Press:
    20 July 2023, pp. 1749-1785
    Print publication:
    December 2024
    • Article
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  • We provide a fairly self-contained account of the localisation and cofinality theorems for the algebraic $\operatorname K$-theory of stable $\infty$-categories. It is based on a general formula for the evaluation of an additive functor on a Verdier quotient closely following work of Waldhausen. We also include a new proof of the additivity theorem of $\operatorname K$-theory, strongly inspired by Ranicki's algebraic Thom construction, a short proof of the universality theorem of Blumberg, Gepner and Tabuada, and a second proof of the cofinality theorem which is based on the universal property of $\operatorname K$-theory.

  • 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.

  • 3 - Global action-angle coordinates and torus actions

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 31-44

  • Summary

    We study the class of toric manifolds: integrable Hamiltonian systems with standard period lattices away from certain controlled singularities. We show that the boundary of the image of such an integrable system is piecewise linear and discuss the Atiyah–Guillemin–Sternberg convexity theorem and Delzant existence and uniqueness theorems. We illustrate all of this with many examples.

  • 2 - Lagrangian fibrations

  • Jonny Evans, University of Lancaster
  • Book:
    Lectures on Lagrangian Torus Fibrations
    Published online:
    06 July 2023
    Print publication:
    20 July 2023, pp 17-30

  • Summary

    We study Lagrangian torus fibrations: maps on symplectic manifolds whose fibres are Lagrangian. We show they are locally the same as integrable Hamiltonian systems and then use the Arnold–Liouville theorem to construct an integral affine structure on the image of a Lagrangian torus fibration. We give an interpretation of this in terms of flux integrals and introduce the key concepts of the developing map of an integral affine structure and the affine monodromy. We also discuss the extent to which the integral affine structure on the image determines the symplectic manifold.

Page 62 of 1445