Developments in Contact and Symplectic Topology

Date: 20 - 24 June 2016

Location: University of Glasgow

Symplectic manifolds arise as phase spaces in the Hamiltonian formulation of classical dynamics.  In this setting, fixed energy levels naturally give rise to contact structures on odd-dimensional manifolds.  The global topology of symplectic and contact manifolds has been a central area of study since the groundbreaking work of Gromov, Donaldson, Floer and others in the 1980s and 1990s.  Contact topology has steadily increased in importance as a subject in its own right, and has been an essential ingredient in some of the biggest results in low-dimensional topology this century, including Kronheimer-Mrowka’s proof of Property P and the deep links exhibited by Ozsváth-Szabó and others between Floer-theoretic invariants such as Heegaard Floer homology and the topology of 3-manifolds.

Several key themes have emerged in three dimensional contact topology.  These include: Gromov’sh-principle, and the resulting flexible/rigid dichotomy; the importance of open book decompositionsof manifolds; fillability questions concerning symplectic manifolds bounded by a given contact manifold; and the intimate relationship between symplectic and contact topology and gauge-theoretic invariants of smooth manifolds.

A key result of Borman-Eliashberg-Murphy in 2014 gave a dramatic advance in our understanding of contact topology in higher dimensions: using an h-principle argument, they extended Eliashberg’s classification of flexible contact structures in three dimensions to arbitrary dimension.  This result generated a great deal of activity in high-dimensional contact topology, leading to a rapidly growing understanding of which aspects of the theory in three dimensions can be generalised to higher dimensions, and of new phenomena which arise in high dimensions.

This LMS-CMI Research School will give students a comprehensive and accessible introduction to key aspects of contact topology in three dimensions and to the new frontier of high-dimensional contact topology.  This is a unique opportunity for students and early-career researchers to get a hands-on guided tour of an exciting and fast-developing area of research from a world-leading team of experts.

Lecturers
Vincent Colin, Université de Nantes
Emmy Murphy, Massachusetts Institute of Technology
András Stipsicz, Alfréd Rényi Institute of Mathematics

Each of the above will give a minicourse of 5 lectures, aimed at graduate students with a fairly broad background.  These will be supported by teaching assistants who will lead tutorials/problem sessions.  Additional lectures will be given by Patrick Massot, Gordana Matić, and Ivan Smith.

Image taken, with permission, from Ken Baker’s blog Sketches of Topology