Schedule and Abstracts

Abstract: A surface embedded in a 4-manifold can be depicted via a banded unlink diagram, which consists of decorated links in S^3 with some bands attached. In 2002, Swenton showed that diagrams of isotopic surfaces in S^4 are related by a short list of moves. In 2019, Hughes, Kim and I extended this to isotopic surfaces in a general 4-manifold. In this talk, I will describe singular banded unlink diagrams of regularly immersed surfaces in 4-manifolds, and again show that there is a complete set of moves on these diagrams. These diagrams can be used to construct homotopies of surfaces and study the intersection sets, which is useful for computing certain surface invariants. This is joint work with Mark Hughes and Seungwon Kim.

Abstract: A link L in the 3-sphere is called chi-slice if it bounds a properly embedded surface F in the 4-ball with Euler characteristic 1. If L is a knot, then this definition coincides with the usual definition of sliceness. One feature of such a link L is that if the determinant of L is nonzero, then the double cover of the 3-sphere branched over L bounds a rational homology ball. In this talk, we will explore the chi-sliceness of quasi-alternating 3-braid links. In particular, we will construct explicit families of chi-slice quasi-alternating 3-braids using band moves and we will obstruct the chi-sliceness of most other quasi-alternating 3-braid links by showing that their double branched covers do not bound rational homology 4-balls. This is a work in progress joint with Vitaly Brejevs.

Abstract: In this talk I'll discuss two notions of an unknotting number for knotted 2-spheres in the 4-sphere, with the goal of comparing them. One is the minimal number of stabilizations needed to produce an unknotted surface, and the other is the minimal number of pairs of finger and Whitney moves needed during a regular homotopy to the unknotted sphere. Most of the times that these numbers can be computed, they are the same; however, I'll present a general inequality between them and give a family of examples where they differ. This is joint work with Michael Klug, Benjamin Ruppik, and Hannah Schwartz.

https://gather.town/i/pveFgsrs

Abstract: Gromov's celebrated Non-Squeezing Theorem states that a ball of radius R cannot embed symplectically into a cylinder of radius 1 unless R ≤ 1. Nowadays, there is an industry of understanding when symplectic embeddings exist between all sorts of domains. We study a slightly different question -- how much do we need to remove from the ball of radius R so that it squeezes into a cylinder of radius 1? In four dimensions, we prove that the (lower) Minkowski dimension of the region removed must be at least 2, and that this result is optimal for R^2 ≤ 2. Time permitting, we discuss a few related results and open problems. This work is joint with Antoine Song, Umut Varolgunes, and Jonathan Zhu.

Abstract: One of the earliest fundamental applications of Lagrangian Floer theory is detecting the non-displaceablity of a Lagrangian submanifold. Many progress and generalisations have been made since then but little is known when the Lagrangian submanifold is disconnected. In this talk, we describe a new idea to address this problem. Subsequently, we explain how to use Fukaya-Oh-Ohta-Ono and Cho-Poddar theory to show that for every S^2 \times S^2 with a non-monotone product symplectic form, there is a continuum of disconnected, non-displaceable Lagrangian submanifolds such that each connected component is displaceable. This is a joint work with Ivan Smith.

Abstract: A trisection of a smooth, compact, 4–manifold is a decomposition into three diffeomorphic pieces, where the complexity of the 4–manifold lies in how these pieces are attached to one another. In the case of a manifold with boundary, a relative trisection induces a structure on the boundary known as an open book decomposition. In this talk, we will provide a correspondence between relative trisections of 4–manifolds with boundary and commutative cubes of groups, known as relative group trisections. The interesting feature of a (relative) group trisection is that it encodes all of the smooth topology of the 4–manifold, including the induced boundary data. This extends group trisections of closed 4–manifolds, due to work of Abrams, Gay, and Kirby, to the relative setting. This work is joint with Jason Joseph and Patrick McFaddin.

https://gather.town/i/pveFgsrs

Abstract: In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk) in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Abstract: Symplectic manifolds arise naturally in abstract formulations of classical mechanics, and symplectic geometry is an interesting mixture of the “soft” and the “rigid.” In this talk, we will focus on symplectic rigidity phenomena. In particular, we will discuss recent developments for the Symplectic mapping class groups of 4-dimensional symplectic manifolds, and how these root in dynamics and topology in dimension 2. Those are based on joint works with T-J Li, Weiwei Wu, and with Olga Buse. Time permitting, we'll outline some open problems for the informal discussion session.

Abstract: Using Heegaard Floer homology, one can associate to a rational homology 3-sphere Y, equipped with a spinc-structure s, a rational number, commonly referred to as the d-invariant of (Y, s). d-invariants have been useful in answering a range of questions in low-dimensional topology. A nice source of rational homology 3-spheres comes from considering double branched covers Sigma_2(K) of knots K. If Sigma_2(K) is an L-space, then the d-invariant of Sigma_2(K), at the unique spin-structure s_0, is well-understood: it's a multiple of the signature of K.

One could ask about branching over multi-component links. In 2015, Lisca-Owens proved that double branched covers of quasi-alternating links satisfy a similar phenomenon (at each spin structure). In this talk, we show that for a certain infinite family of non-quasi-alternating links whose double branched covers are L-spaces, this property also holds. This is work in progress with M. Marengon.

Abstract: Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. In this talk I'll show that no Legendrian knot which is both concordant to and from the unstabilized Legendrian unknot can be the closure of an index 3 braid except the unknot itself.

https://gather.town/i/pveFgsrs