Georgia Institute of Technology College of Sciences

A 3-manifold is called an L-space if its Heegaard Floer homology is "simple." No characterization of all such "simple" 3-manifolds is known. Manifolds obtained as the double-branched cover of alternating knots in the 3-sphere give examples of L-spaces. In this talk, I'll discuss the search for L-spaces among higher index branched cyclic covers of knots. In particular, I'll give new examples of knots whose branched cyclic covers are L-spaces for every index n. I will also discuss an application to "visibility" of certain periodic symmetries of a knot.

Ever since Eliashberg distinguished overtwisted from tight contact structures in dimension 3, there has been an ongoing project to determine which closed, oriented 3–manifolds support a tight contact structure, and on those that do, whether we can classify them. This thesis studies tight contact structures on an infinite family of hyperbolic L-spaces, which come from surgeries on the Whitehead link. We also present partial results on symplectic fillability on those manifolds.

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.

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.

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.

I'm interested in the smooth mapping class group of S^4, i.e. pi_0(Diff^+(S^4)); we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of S^4 is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5-spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory.

Following the approach of Nabutovsky and Rotman for the curve-shortening flow on geodesic nets, we'll show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. On the pinched curvature setting, we prove a bound on the first eigenvalue of the Laplacian and use it to prove a new isoperimetric inequality for pinched 2-spheres sufficiently close to being round. This allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting.

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 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.

This pre-talk will be an introduction to infinite-type surfaces and big mapping class groups. I will have a prepared talk, but it will be extremely informal, and I am more than happy to take scenic diversions if the audience so desires!