One of interesting topic in low-dimensional topology is to study exotic smooth structures on closed 4-manifolds. In this talk, we will see an example to distinguish exotic smooth structure using H-slice knots.
Embedding problems, of an n-manifold into an m-manifold, can be heuristically thought to belong to a spectrum, from rigid, to flexible. Euclidean embeddings define the rigid end of the spectrum, meaning you can only translate or rotate an object into the target. Symplectic embeddings, depending on the object, and target, can show up anywhere on the spectrum, and it is this flexible vs rigid philosophy, and techniques developed to study them, that has lead to a lot of interesting mathematics. In this talk I will make this heuristic clearer, and show some examples and applications of these embedding problems.
Knot Floer homology is an invariant for knots introduced by Ozsváth-Szabó and, independently, Rasmussen. We will give a general introduction to the theory, sketching the definition and highlight its major properties and applications.
We will explain some basic ways in which a sequence of groups/spaces can be said to be stable and give some natural examples of stability.
Bluejeans link: https://bluejeans.com/872588027
The algebraic structure of mapping class groups is deep and beautiful; in this talk, we'll explore some curious conjectures and definite theorems about the structure and quality of different subgroups of the mapping class group.
The group of homeomorphisms on a surface is called the mapping class group. This talk will cover some background and key results to provide a foundation for related talks that will occur later in the semester.
The Teichmüller space is the space of hyperbolic structures on surfaces, and there are different flavors depending on the class of surfaces. In this talk we consider the enhanced Teichmüller space which includes additional data at boundary components. The enhanced version can be parametrized by shear coordinates, and in these coordinates, the Weil-Peterson Poisson structure has a simple form. We will discuss a construction of the quantum Teichmüller space corresponding to this Poisson structure.
Bluejeans: https://bluejeans.com/872588027
I will introduce Serre spectral sequences, then compute some examples. The talk will be in most part following Allen Hatcher's notes on spectral sequences.
A knot in the 3-sphere is slice if it bounds a smooth disc in the 4-ball. A knot is ribbon if it bounds a self-intersecting disc with only singularities that are closed arcs consisting of intersection points of the disc with itself. Every ribbon knot is a slice knot; the converse is a famous unsolved conjecture of Fox. This talk will show some recent interesting progress around the slice-ribbon conjecture.
Grid homology is a purely combinatorial description of knot Floer homology in which the counting of psuedo-holomorphic disks is replaced with a counting of polygons in grid diagrams. This talk will provide an introduction to this theory, and is aimed at an audience with little to no experience with Heegaard Floer homology.
