Title: Pseudo-isotopy theory and diffeomorphisms of 4-manifolds

Abstract: The goal will be to indicate how pseudo-isotopy theory can be used to attempt to understand diffeomorphisms of S^1\times S^3 modulo passing to finite coverings.  See https://arxiv.org/abs/2212.02004.

Title: Pants Distances of Knotted Surfaces in 4-Manifolds

Abstract: Kirby and Thompson’s “L-invariant” is an invariant of (smooth, connected, closed, oriented) 4-manifolds produced by minimizing the complexity of trisection diagrams for that manifold over all possible trisections. This invariant has since been generalized by various authors to other settings, including 4-manifolds with boundary and smoothly knotted surfaces in the 4-sphere. In this talk, I will describe an extension of the L-invariant to the case of smoothly knotted surfaces in general (smooth, connected, closed, oriented) 4-manifolds. Perhaps unsurprisingly, these types of invariants are in general difficult to calculate, but I will present some results and calculations for small values. This is joint work with Roman Aranda, Devashi Gulati, Homayun Karimi, Geunyoung Kim, Nicholas Meyer, and Puttipong Pongtanapaisan.

Title : A gauge theoretic invariant of embedded surfaces in 4-manifolds and exotic P^2-knots 

Abstract: When two embeddings of surfaces on a 4-dimensional manifold are given, if they are topologically isotopic but not smoothly isotopic, we call them a pair of exotic surfaces. While there is a great deal of study of exotic surfaces in 4-manifolds, studies of closed exotic surfaces in S^4 are limited. In particular, the existence of orientable exotic surfaces in S^4 remains unknown to date. There are some examples of non-orientable exotic surfaces in S^4, including the first example given by Finashin-Kreck-Viro in 1988, but all such cases have genus greater than or equal to 5. The difficulty in detecting exotic surfaces in S^4 is to prove that two embeddings of surfaces are not smoothly isotopic. All examples of exotic non-orientable surfaces in S^4 have been detected by proving the 4-manifolds obtained by the double branched covers are exotic. If we attempt to apply this technique to low-genus non-orientable surfaces in S^4, we have to discover exotic small 4-manifolds, which is known to be difficult. We construct an invariant for embedded surfaces in 4-manifolds using Real Seiberg-Witten theory, that is a variant of Seiberg--WItten theory. As an application, we give an infinite family of exotic embeddings into S^4 for the real projective plane.

Title: Skein lasagna modules and computations with Kirby belts  

Abstract: Skein lasagna modules, defined for a chosen TQFT, are a relatively new kind of invariant of smooth 4-manifolds. In 2024, these purely algebraic invariants have been shown to detect non-trivial exotic phenomena, providing some of the first examples of analysis-free exotic detection. In this talk, we define and describe relevant properties of these invariants for Khovanov and Lee homology. We then explain a new computational technique on the chain level using colimits of tangle complexes called Kirby belts. These computational results were obtained in joint work with Melissa Zhang.

Title: Computing Invariants of Trisected Branched Covers of the 4-Sphere

Abstract: Every closed, connected, oriented 4-manifold is a branched cover of S^4 along an embedded surface. When the branching surface is placed in a bridge position with respect to the standard trisection of the 4-sphere, a branched cover is determined by a (compatible) permutation labelling of a corresponding tri-plane diagram. The resulting 4-manifold then comes with a natural trisection. Given such a presentation, we give a diagrammatic algorithm for computing the group trisection, homology groups, and intersection form of the branched cover. We apply our algorithm to several examples, including dihedral and cyclic covers of spun knots, cyclic covers of Suciu’s ribbon knots with the same knot group, and an infinite family of irregular covers of the Stevedore disk double, as well as an automated algorithm for computing Kjuchukova’s homotopy-ribbon obstruction. 

This is joint work with Patricia Cahn and Ben Ruppik.

Title: Surfaces as loops in complexes of curves.

Abstract: Triplane diagrams were introduced by Meier and Zupan as a new way to encode knotted surfaces in 4-space using 3-dimensional information: triples of trivial tangles satisfying some conditions. In this talk, we explain how to describe knotted surfaces as loops in curve complexes. We derive a uniqueness result for bridge multisections: tuples of trivial tangles determining surface in 4-space. The work discussed is a collaboration with Carolyn Engelhardt.