# Topology

Organizers: Ciprian Manolescu, Cole Hugelmeyer, and Luya Wang

## Past Events

Work of Hanselman-Rasmussen-Watson has shown that the bordered Floer invariants for a 3-manifold with torus boundary can be represented as a collection of (decorated) immersed curves in the punctured torus; moreover, the Heegaard Floer homology of the closed manifold obtained by gluing two such…

For each compact, orientable surface whose connected components have non-empty boundary, we define a dg category that categorifies the Temperley--Lieb skein of the surface. In the case when the surface is a disk, our categories are quasi-equivalent to the so-called "Bar-Natan category": the…

Given a four-manifold with non-vanishing Seiberg-Witten invariants, the adjunction inequality provides a lower bound on the genus of any smoothly embedded surface representing a fixed homology class. Stabilization (that is, taking a connected sum with a product of two 2-spheres) always kills the…

Given a Liouville manifold M, we can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group, important for relating (for instance) compactly supported symplectomorphisms of M and…

Following Kronheimer and Mrowka’s approach, we show that Khovanov homology detects the unknot and the projective unknot in RP^3. I’ll explain the idea of the proof. Time permitting, I’ll discuss potential further detection results.

Twist-roll spun knots are a well-studied family of knotted 2-spheres in the 4-sphere. The inputs for determining a twist roll spun knots are an integer number of twists, a number of rolls, and a choice of classical knot in the 3-sphere. I will explain why adding twists, while changing the…

Knot invariants are typically used to give a negative answer to the question of when two embeddings are ambiently isotopic, and rarely to give a positive answer. An exception is the celebrated result of Freedman and Quinn that if the complement of a 2-sphere embedded in the 4-sphere has…

Let S be a finite set of invertible 2-by-2 matrices with algebraic entries. Is there an algorithm to determine a presentation for <S>? We shall pose this question, and suggest a possible avenue to answering it. Note that every finite-volume hyperbolic 3-manifold arises as above. Beyond…

Given n points and a smooth Jordan curve in the complex plane, what is the minimum degree of a non-constant polynomial which maps all of the points to the curve? It is easy to bound the degree above by n-1, while if the points are collinear and the curve is an ellipse, then the degree is…