Conference: Trisections Workshop: Connections with Symplectic Topology

University Of California-Davis, Davis CA

Investigators

Abstract

This award provides support for the conference “Trisections Workshop: Connections with Symplectic Topology” that will take place at the University of California, Davis, during June 26-30, 2023. The conference will focus on developing connections between trisection theory, an emerging area of research in four-manifold topology, and the study of the topology and geometry of symplectic four-manifolds. It will catalyze research developments by bringing together mathematicians of all career stages – including established experts, early-career researchers, and students – and will have a number of features that are designed to actively engage the participants. First, there will be pre-workshop, virtual, introductory mini-courses given by experts in the field and designed to preview and motivate the central topics of the workshop. Second, there will be both plenary talks and shorter, lightning-style talks aimed at highlighting recent developments by a wide range of researchers during the morning sessions. Third, the workshop will feature afternoon working-group sessions in which participants will collaborate on open problems relating to the theory of trisections and symplectic topology and geometry of four-manifolds. Symplectic geometry originally arose from Hamiltonian mechanics but has since developed into a rich field of its own with applications to several areas of mathematics, such as complex geometry, algebraic geometry, exotic smooth structures on four-manifolds, singularity theory, dynamics, and mirror symmetry. Despite decades of development and many seminal advances, seemingly basic problems – such as the symplectic isotopy problem – remain wide open. Trisection theory provides new combinatorial methods to study symplectic manifolds and offers a new perspective to shed light on difficult open problems. This provides a starting point for many new directions and applications and a rich opportunity for making advances through the synergy of these two fields. In the ten years since its introduction, trisection theory has developed to touch nearly every facet of four-manifold topology – from comparing smooth structures and introducing new invariants to representing knotted surfaces and describing surgery operations. The workshop will build on recent successes of the theory highlighting the interplay between the symplectic and complex geometry of four-manifolds and the combinatorial features of trisections. These include: a characterization of symplectic surfaces in the complex projective plane as those admitting transverse shadow diagrams; a proof of the existence of trisections on symplectic four-manifolds that are compatible with the ambient symplectic structure; and an established correspondence between certain complex curves in the complex projective plane and hexagonal lattices on the torus that yields a combinatorial approach to the symplectic isotopy problem. The conference website is at: https://sites.google.com/view/tw2023/ This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.