Title: Introduction to bridge trisections

Abstract: In 2015, Meier and Zupan showed that trisections can be used to describe smooth surfaces in S^4 in a manner analogous to bridge position of knots in S^3. In 2017, they improved this to apply to surfaces in arbitrary closed, compact, connected, oriented 4-manifolds. In this talk, I will define this bridge position of surfaces in 4-manifolds (with emphasis on S^4), go through some examples, discuss the move that relates distinct bridge trisections of isotopic surfaces, and discuss open questions about bridge trisections.

Title: Introduction to relative trisections II

Abstract: Trisections of 4-manifolds relative to their boundary were introduced by Gay and Kirby in 2012. They are decompositions of 4-manifolds that induce open book decomposition in the bounding 3-manifolds. I think of relative trisections as fillings of open book decompositions, in analogy to symplectic fillings of contact manifolds. The following is the plan for the second talk:

• Explicit constructions of relative trisections,

• Recovering open book decompositions from trisection diagrams,

• Connections to contact/symplectic topology.

Title: Trisections and the geometry of CP2 II

Abstract: The complex projective plane has a simple genus 1 trisection. The aim of these talks will be to unpack all of the ways in which the geometry of CP2 interacts with this trisection. Starting from the basics, we will explore how different types of geometric structures — such as symplectic, contact, holomorphic, Kahler, Hermitian and foliations — fit together naturally using the trisection decomposition. We will then use this explicitly-worked example to motivate questions and ideas about trisections of symplectic 4-manifolds.

In small groups, we'll explore exercises related to the content of today's talks.