Title:  Algorithms for Invariants of Trisected Branched Covers

Abstract:  We give diagrammatic algorithms for computing the group trisection, homology groups, and intersection form of a 4-manifold, presented as a branched cover of a bridge-trisected surface in S^4.  The algorithm takes as input a tri-plane diagram, labelled with permutations according to the Wirtinger relations.  We apply our algorithm to several examples, including dihedral and cyclic covers of spun knots, cyclic covers of Suciu's ribbon knots, and an infinite family of irregular covers of the Stevedore disk double.  As an application, we give the first fully automated algorithm for computing Kjuchukova's homotopy-ribbon obstruction for a p-colorable knot for arbitrary odd p, given an extension of that coloring over a ribbon surface in the 4-ball.  This is joint work with Alishahi, Matic, Pinzón-Caicedo, and Ruppik, and was initiated at the 2020 Winter Trisector's Workshop.

Title: "Up to isotopy" in trisection and symplectic topology

Abstract: In this talk (mostly philosophical) I will double down on one of my pet themes, the importance of thinking about structures in low-dimensional topology, especially trisections and symplectic and contact structures, up to isotopy rather than just up to diffeomorphism. Going even further, I will try to drive home the point that what we really should be striving for is an understanding of the homotopy type of the space of isomorphisms of whatever structures we are interested in, and I will give evidence that, using this philosophy, trisections and symplectic structures working together have the potential to uncover new information about the underlying smooth topology.

Title: Symplectic trisections and connected sum decompositions

Abstract: This talk will have two parts.  The first half will describe how to construct symplectic structures on trisected 4-manifolds. This construction is inspired by projective complex geometry and completely characterizes symplectic 4-manifolds among all smooth 4-manifolds.  The second half will address a curious phenomenon: symplectic 4-manifolds appear to not admit any interesting connected sum decompositions.  One potential explanation is that every embedded 3-sphere can be made contact-type.  I will outline some strategies to prove this from a trisections perspective, describe some of the obstructions, and give evidence that these obstructions may be overcome.

Title: Lagrangian Realizations of Ribbon Cobordisms

Abstract: Similarly to how every smooth knot has a Legendrian representative (in fact, infinitely many different representatives), in this talk we will discuss why every ribbon cobordism has a Legendrian representative. Meaning, if $C$ is a ribbon cobordism in $[0,1]\times S^3$ from the link $K_0$ to $K_1$, then there are Legendrian realizations $\Lambda_0$ and $\Lambda_1$ of $K_0$ and $K_1$, respectively, such that $C$ may be isotoped to a decomposable Lagrangian cobordism from $\Lambda_0$ to $\Lambda_1$. We will also give examples of some interesting constructions of such decomposable Lagrangian cobordisms. This is joint work with John Etnyre

Title: Symplectic handlebody decompositions of CP^2 and rational homology ball embeddings

Abstract: Using mutations of almost toric fibrations, Vianna and Evans-Smith constructed embeddings of lens spaces into CP^2. Using work of Gay from early 2000s, and inspiration from recent work of Etnyre-Min-Mukherjee on non-loose knots in overtwisted contact structures, we reinterpret the almost toric pictures as symplectic handlebody descriptions of CP^2. These are a geometrically enriched version of horizontal descriptions discovered by Lisca-Parma, which they have used to find several more smooth, non-symplectic embeddings of rational homology balls. Time permitting, we will also see how to turn these into trisections of CP^2 and discuss relevant questions.  This is joint work with Etnyre, Min, and Piccirillo.

Title:  Explicit constructions of antisurgery Weinstein manifolds

Abstract: A  Legendrian knot in the boundary of a Weinstein domain of dimension at least 6 which bounds a Lagrangian disk can be considered the boundary of the co-core of a handle. A Weinstein antisurgery amounts to carving out this handle from the Weinstein domain. For any collection of integers P, Lazarev and Sylvan defined a class of Weinstein manifolds which they called P-flexible, formed from handle attachment along P-loose Legendrians. These have properties that are generalizations of the properties of flexible Weinstein manifolds constructed from handle attachments along loose Legendrians.  The P-flexible manifolds were constructed via and antisurgery, thus their Weinstein diagrams (in particular the front projections of the P-loose Legendrians) were not known. In this talk, I'll explain how to construct handle decompositions of these manifolds and show that this strategy also works more generally to give explicit presentations of Weinstein domains that come from antisurgery. This talk is based on work in progress with Ipsita Datta, Oleg Lazarev, and Chindu Mohanakumar.