Title:  Ball swapping in cubes

Abstract:  Let X be a compact smooth codimension-0 submanifold of R^n. I'll show that X is topologically isotopic to a manifold X* which tiles the cube [0, 1]^n and, consequently, also itself. The key idea of the proof is a technique we call ball swapping, which involves trading some n-balls across their orbits under a group action on the cube. This is joint work with Ryan Blair, Patricia Cahn and Hannah Schwartz.

Title:  Boundary Dehn twists and definite fillings

Abstract:  Given any smooth 4-manifold bounding a Seifert manifold, the Seifert action on its boundary can be used to define their boundary Dehn twists. If the given 4-manifold is simply-connected, this Dehn twist is always topologically isotopic to the identity, but usually not smoothly isotopic, making it a very nice potential example of exotic diffeomorphisms. In this talk, we prove that for any Brieskorn homology sphere bounding a positive-definite 4-manifold, their boundary Dehn twists are always infinite-order exotic. This is a joint work with JungHwan Park and Masaki Taniguchi.

Title:  Equivariant Trisections

Abstract:  Since the early days of the trisector's workshop, a question has lingered about how to interface the burgeoning theory of trisections with the old and interesting world of finite group actions on 4-manifolds. The essential questions: if a trisection diagram has some group action, does that correspond uniquely to an action on the trisected 4-manifold? Conversely, if we have a group action on a 4-manifold, can we always "represent" that action by some trisection with a corresponding action? In recent joint work with Jeffrey Meier, we develop a theory of equivariant trisections and answer both questions in the affirmative. This talk aims to explain the toolkit we've developed, show some interesting examples, and posit questions for future research.

Title:  Diffeomorphisms of trisected 4-manifolds

Abstract:  With the recent growing interest in understanding mapping class group and diffeomorphisms of 4-manifolds, it is natural to ask whether trisections can offer any new insights. In this talk, I will present one potential approach: studying pairs (f, T), where a diffeomorphism f interacts nicely with a trisection T. This perspective raises some interesting questions about mapping class groups and the structure of diffeomorphism groups. I’ll discuss some of the challenges that arise, along with motivating examples, open questions, and possible directions for future work.

Title:  An L—type Invariant Using Shadow Diagrams

Abstract:  In 2018, Kirby and Thompson defined the L invariant for closed 4—manifolds by relating trisections to loops in the curve complex.  Since then, Blair, Campisi, Taylor, and Tomova defined an L invariant for bridge trisections of knotted surfaces in S^4 using the pants complex.  Additionally, Aranda, Pongtanapaisan, and Zhang defined the L* invariant for knotted surfaces using the dual curve complex.  In this talk, we discuss a new L—type invariant for bridge multisections of knotted surfaces in S^4, called L^S, which is defined using a CW—complex of shadow diagrams.    This project is joint work with Román Aranda and Nobutaka Asano.

Title:  Irreducible 4-manifolds with order two fundamental group

Abstract:  Let R be a closed, smooth, oriented 4–manifold with order two fundamental group. An irreducible copy of R is a smooth manifold that’s homeomorphic to R and doesn’t decompose into non-trivial connected sums. We’ll show that if R has odd intersection form and non-negative first Chern number, then in all but seven cases, it has an irreducible copy. There are a few different ways to build an irreducible copy. For the bulk of this talk, we’ll focus on my favorite method; we’ll construct manifolds with low first Chern numbers by fiber summing Lefschetz fibrations equivariantly. This is joint work with Mihail Arabadji.