Showing papers on "Geometry and topology published in 2023"

Lectures on Lagrangian Torus Fibrations

Abstract: 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.

Topologie

Abstract: The lectures in the workshop covered various topics in modern topology, including algebraic and geometric topology, homotopy theory, geometric group theory, and manifold topology, as well as connections to neighboring areas, most prominently symplectic topology/geometry. The following current research topics received more attention during the workshop: manifolds and K-theory, symplectic topology and Floer homology, generalizations of hyperbolic techniques in geometric group theory, and equivariant and motivic homotopy theory. The aim of the various topics was to foster communication and provide chances for participants to see and experience driving questions and important methods in nearby fields within the realm of topology.

Introduction to contact complete integrability

Abstract: Integrable Hamiltonian systems on symplectic manifolds have been well-studied. However, an intrinsic property of these kind of systems is that they can only live on even dimensional manifolds. To introduce a similar notion of integrability on odd dimensional manifolds, one needs another type of geometry: contact geometry. These lecture notes, aimed towards graduate students, serve as an introduction to this form of integrabilty. We will start by introducing the necessary concepts from contact geometry, which is the sister geometry of symplectic geometry. In the third chapter, we discuss the contact Hamiltonian vector field (which is similar to the standard Hamiltonian vector field) and the Jacobi bracket (which has similar dynamical properties as the Poisson bracket in the symplectic world). After this, we are able to introduce contact complete integrability: there exist (at least) two different notions of such integrability in the literature: one by Khesin & Tabachnikov (which has a more geometric nature) and one by Jovanovic & Jovanovic. We will show that both notions coincide. Subsequently, we will give an overview of the semi-local aspects of contact complete integrability, namely the behaviour of the dynamics near regular fibres (an Arnold-Liouville-like theorem by Jovanovic) and singular fibres (a local normal form as described by Miranda). Finally, we will introduce contact toric G-manifolds, which where classified by Lerman in 2003 (similar to the Delzant-classification of toric integrable systems). Throughout these notes, we will focus on the link with integrable Hamiltonian systems in symplectic geometry.