TRISECTIONS AND THEIR GENERALIZATIONS LES TRISECTIONS ET LEURS GÉNÉRALISATIONS

The notion of trisection for a smooth 4-manifold, introduced by David Gay and Robion Kirby, is an analogue of a Heegaard splitting in dimension 3, a key notion in the study of 3-manifolds. The theory of trisections provides a new tool to study smooth 4-manifolds, where we are very much in need of new approaches that could eventually lead to progress on some of the most important open problems in topology, such as the smooth 4-dimensional Poincaré conjecture and the smooth 4-dimensional Schoenflies problem. Since the 2016 publication of Gay and Kirby’s original paper on trisections, fundamental questions have been answered and extensive generalizations have been developed, broadening the reach of the field to include contact and symplectic topology, higher dimensional manifolds, piecewise linear topology, and other geometric structures on manifolds.

This Jean Morlet chair will focus on four specific areas in which there is exciting work to be done on trisections and their generalizations: (1) the relationship between trisections and contact and symplectic topology, (2) using trisections to produce new 4-dimensional invariants, appropriating tools from Floer theory and algebraic geometry, (3) generalizations of trisections to higher dimensions, building on recent work on higher dimensional multisections and (4) understanding trisections and their generalizations from the non-smooth piecewise linear perspective. In addition to these topics, other areas of significant interest are the potential to use trisections to study diffeomorphisms of 4-manifolds, studying Goeritz groups of trisected 4-manifolds, and setting up a careful theory of parameterized families of trisections, parameterized by arbitrarily many dimensions, to address naturality issues in invariant constructions.