Georgia Institute of Technology College of Sciences

I’ll describe various aspects of my joint work with Oliver Lorscheid on the foundation of a matroid.

Symmetric unions are an interesting class of knots. Although they have not been studied much for their own sake, they frequently appear in the literature. One such instance regards the question of whether there is a nontrivial knot with trivial Jones polynomial. In my talk, I will describe a class of symmetric unions, constructed by Tanaka, such that if any are amphichiral, they would have trivial Jones polynomial. Then I will show how such a knot not only answers the above question but also gives rise to a counterexample to the Cosmetic Surgery Conjecture. However, I will prove that such a knot is in fact trivial and hence cannot be used to answer any of these questions. Finally, I will discuss how the arguments that go into this proof can be generalized to study amphichiral symmetric unions.

Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work of O. Gomide, M. Guardia, T. Seara, and C. Zeng. 

Since Jones introduced his knot polynomial using representation theory, there has been a wide variety of invariants defined this way, e.g., HOMFLY-PT and Reshetikhin-Turaev. Recently, through the work of Bonahon-Wong and Constantino-Le, some of these invariants are reinterpreted as quantum matrices. In this talk, we will review the history of these representation theoretical knot invariants. Then we will discuss one particular connection to the quantum special linear group.