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.
Knots in contact manifolds are interesting objects to study. In this talk, I will focus on knots in overtwisted manifolds. There are two types of knots in an overtwisted manifold, one with overtwisted complement (known as loose) and one with tight complement (known as non-loose). Not very surprisingly, non-loose knots behave very mysteriously. They are interesting in their own right as we still do not understand them well.
This week, we'll continue discussing the rational blowdown and use it to construct small exotic 4-manifolds. We will see how we can view the rational blowdown as a "monodromy substitution." Finally, if time allows, we will discuss knot surgery on 4-manifolds.
Contact 3-manifolds arise organically as boundaries of symplectic 4-manifolds, so it’s natural to ask: Given a contact 3-manifold Y, does there exist a symplectic 4-manifold X filling Y in a compatible way? Stein fillability is one such notion of compatibility that can be explored via open books: representations of a 3-manifold by means of a surface with boundary and its self-diffeomorphism, called a monodromy.
Are you tired of having to read a bunch of words during a seminar talk? Well, you’re in luck! This talk will be a (nearly) word-free exploration of a construction called unicorn paths. These paths are incredibly useful and can be used to show that both the curve graph and the arc graph of a surface are hyperbolic.
In high dimensional contact and symplectic topology, finding interesting constructions for Legendrian submanifolds is an active area of research. Further, it is desirable that the constructions lend themselves nicely to computation of invariants. The doubling construction was defined by Ekholm, which uses Lagrangian fillings of a Legendrian knot in standard contact R^{2n-1} to produce a closed Legendrian submanifold in standard contact R^{2n+1}. Later Courte-Ekholm showed that symmetric doubles of embedded fillings are "uninteresting".
This talk includes an interactive prop demonstration. There exist non-trivial loops in SO(3) (the familiar group of real life rotations) which can be visualized with Dirac's belt trick. Although the belt trick offers a vivid picture of a loop in SO(3), a belt is not a proof, so we will prove SO(n) is not simply connected (n>2), and find its universal covering group Spin(n) (n >2). Along the way we'll introduce the Clifford algebra and study its basic properties.
