Topology

Abstract: We will discuss problems of finding quadrilaterals inscribed on planar curves, such as the inscribed square problem, the inscribed rectangle problem, and the periodic square peg problem. We will examine the effectiveness and limitations of various techniques, from intersection theory…

Abstract: Khovanov homology is a combinatorially-defined invariant which has proved to contain a wealth of geometric information. In 2006 Seidel and Smith introduced a candidate analog of the theory in Lagrangian Floer analog cohomology, which has been shown by Abouzaid and Smith to be…

Abstract:  I compute the knot Floer complex for the regular fiber of \Sigma(2,3,7), and I show that its Seifert genus and genus in a self homology cobordism agree. The key step in this result was providing upgrades to the surgery formula for knot lattice homotopy. Ozsv\'ath, Stipsicz, and…

Abstract: Satellite operators are a well-trodden subject in Heegaard Floer theory. There are a number of algorithms to compute the effect of satellite operations on knot Floer homology. Most of these go via the bordered theory of Lipshtiz, Ozsvath and Thurston. There are some very helpful…

Abstract: In general, the classification of finitely generated subgroups of a given group is intractable. Restricting to two-generator subgroups in a geometric setting is an exception. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and…

Morrison, Walker, and Wedrich’s skein lasagna modules are 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining…

Abstract: This talk has a simple thesis statement: torus surgeries are a powerful tool to study 4-manifolds, we apply this technology to knot traces. The key insight is that annulus twisting a knot's Dehn surgery can be realized 4-dimensionally as a torus surgery on the knot's trace. We will…

There are now many examples of integer homology spheres which cannot be written as surgery on a knot, but examples which cannot be surgery on some 2-component link have remained out of reach. From one perspective, the difficulty is that the trace of the surgery is an indefinite 4-…

In this talk, we explain that (infinite dimensional) Teichmueller spaces associated to hyperbolic surfaces with absolute boundary carry Hamiltonian actions of the Virasoro algebra. If time permits, we will also state some open problems for surfaces with marked points. Our study is motivated on…

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…