Georgia Institute of Technology College of Sciences

I will talk about the application of algebra and algebraic geometry to matroid theory. Baker and Bowler developed the notions of weak and strong matroids over tracts. Later, Baker and Lorscheid developed the notion of foundation of a matroid, which characterize the representability of the matroid. I will introduce a variety of topics under this theme. First, I will talk about a condition which is sufficient to guarantee that the notions of strong and weak matroids coincide. Next, I will describe a software program that computes all representations of matroids over a field, based on the theory of foundations. Finally, I will define a notion of rank for matrices over tracts in order to get uniform proofs of various results about ranks of matrices over fields.

The study of contact and symplectic manifolds has relied heavily on understanding Legendrian and Lagrangian submanifolds in them -- both for constructing the manifolds using these submanifolds, and for computing invariants of the ambient space in terms of invariants of the submanifolds. This thesis explores the construction of Legendrian submanifolds in high dimensional contact manifolds (greater than 3) in two directions. In one, using open book decompositions, we generalise a doubling construction defined by Ekholm and show that the Legendrians obtained are trivial. In the second, which is joint work in progress with Hughes, we use the doubling and twist spun constructions to build a large family of Legendrians, compute their sheaf-theoretic invariants to distinguish them using techniques of Casals-Zaslow, and study their exact Lagrangian fillability properties.

Zoom link:

https://gatech.zoom.us/j/93109756512?pwd=Skljb0tVdjZVNEUvSm9tNnFHZFREUT09