Contact and symplectic topology are exciting areas of current research, born out of ideas in classical mechanics. In this introductory talk we review the basic notions and set up some general research directions within the field -- in particular relating to classifications of Legendrians, Lagrangians, symplectic fillings, and understanding contact surgery. 

Searching for triple grid diagrams

Abstract: Grid diagrams have emerged as a powerful combinatorial tool for encoding knots and links, finding applications in Grid Homology and relevance in Contact topology. This talk focuses on Triple Grid diagrams, as defined by Blackwell, Gay, and Lambert-Cole, which encode embedded surfaces in CP2. Specifically, we explore the motivation, definition, and role of Triple Grid diagrams in representing Lagrangian surfaces within CP2. Furthermore, we discuss the challenges encountered when searching for examples of Triple Grid diagrams and highlight the significance of cubic Tait-colored graphs in the spine of the bridge trisection, which gives rise to a moduli space of Geometric Triple Grid diagrams. 

Metric contact topology

Abstract: Metric contact topology is the intersection of Riemannian geometry and contact topology. It is a fairly little known nook of (low dimensional) geometry and topology but it actually connects various subfields in ways few people know about. I will aim to give a brief intro into some of the ideas and a couple of results in the first talk and briefly outline one of my own results as well as potential future directions in metric contact topology in the second talk.

https://arxiv.org/abs/2211.06916

Classifying Stein Fillings of Contact Structures on L(p,q)

Abstract: Classifying symplectic fillings of contact manifolds is an active area of research that brings together ideas from different areas in topology. Wendl proved that if the open book supporting a contact manifold is planar, then its Stein fillings are in 1-1 correspondence with positive factorizations of its monodromy up to some equivalences. Thus the problem of classifying Stein fillings of L(p,q) spaces essentially turns into a mapping class group problem. We will look specifically at the mapping class group of the disk with n holes removed and invariants on positive factorizations of its elements to classify Stein fillings of L(nm-1,m), which is integer surgery on the Hopf link.

Classification of tight contact structures

Abstract: Part 1: Contact structures come in two flavors: tight and overtwisted. Classification of overtwisted contact structures is well understood as opposed to tight contact structures. We will in particular look at the classification of tight contact structure on Seifert fibered manifold with 4 exceptional fibers. These are interesting manifolds because they have incompressible tori, which allows for non zero Giroux torsion in the tight contact structures. Part 2: We will look at some techniques like Legendrian surgery and convex surface theory that are used for classification. I will end by stating what more classification results can we hope to get using the same techniques and what is far-fetched.

https://doi.org/10.48550/arXiv.2303.09490