Georgia Institute of Technology College of Sciences

Legendrian knots are an important kind of knot in contact topology. One of their invariants,  the Thurston-Bennequin number, has an upper bound for any given knot type, called max-tb. Using convex surface theory, we will compute the max-tb of positive torus knots and show that two max-tb positive torus knots are Legendrian isotopic. If time permits, we will show that any non max-tb positive torus knot is obtained from the unique max-tb positive torus knot by a sequence of stabilizations. 

In this talk I will present some recent results in collaboration with Jessica Trespalacios where we consider Einstein-Belinski-Zakharov spacetimes and prove local and global existence, long time behavior of possibly large solutions and some applications to gravisolitons of Kasner type.

The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston. After introducing the relevant definitions, I'll explain a bit about this connection and how I used the geometric structure on this moduli space to exactly identify the kernel of the Burau representation after evaluating its formal parameter at complex roots of unity. There will be many pictures!

In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to “drift". This observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.