Georgia Institute of Technology College of Sciences

I will discuss a mixture of results and conjectures related to the Khovanov homology and Knot Floer homology for ribbon knots. We will explore relationships with fusion numbers (a measure of complexity on ribbon disks) and particular families of symmetric unions (ribbon knots given by particular diagrams). This is joint work with Nathan Dunfield, Sherry Gong, Tom Hockenhull, and Marco Marengon.

Locally conformal symplectic (LCS) geometry is a variant of symplectic geometry in which the symplectic form is locally only defined up to positive scale. For example, for the symplectization R x Y of a contact manifold Y, translation in the R direction are symplectomorphisms up to scale, and hence the quotient (R/Z) x Y is naturally an LCS manifold. The importation of symplectic techniques into LCS geometry is somewhat subtle because of this ambiguity of scale. In this talk, we define a notion of product for LCS manifolds, in which the underlying manifold of an LCS product is not simply the smooth product of the underlying manifolds, but which nonetheless appears to fill the same role in LCS geometry as the standard symplectic product does in standard symplectic geometry. As a proof of concept, with input from an LCS result of Chantraine and Murphy, we use the LCS product to prove that C^0 small Hamiltonian isotopies have a lower bound on the number of fixed points given by the rank Morse-Novikov homology. This is a natural generalization of the classical symplectic proof of the analogous result by Laudenbach and Sikorav which uses the graph of a Hamiltonian diffeomorphism in the product manifold. These results are joint work in progress with Baptiste Chantraine.

A toric vector bundle is a vector bundle over a toric variety equipped with a linear action by the torus of the base. Toric vector bundles pf rank r were famously classified by Klyachko (1989) using certain combinatorial data of compatible filtrations in an r-dimensional vector space E. This data can be thought of as a higher rank generalization of an (integer-valued) piecewise linear function. In this talk, we give an interpretation of Klyachko data as a "piecewise linear map" to a tropical linear space. This point of view leads us to introduce the notion of a "matroidal vector bundle", a generalization of toric vector bundles to (possibly non-representable) matroids. As a special case and by-product of this construction, one recovers the tautological classes of matroids introduced by Berget, Eur, Spink and Tseng. This is a work in progress with Chris Manon (Kentucky).

Langhede and Thomassen conjectured in 2020 that there exists a positive constant c such that every planar graph G with 5-correspondence assignment (L,M) has at least 2^{c v(G)} distinct (L,M)-colourings. I will discuss a proof of this conjecture (which relies on the hyperbolicity of a certain family of graphs), a generalization of this result to some other embedded graphs (again, relying on a hyperbolicity theorem), and a few open problems in the area. Everything presented is joint work with Luke Postle.