Georgia Institute of Technology College of Sciences

Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

A result by Ozsvath and Szabo states that the knot Floer complex of an L-space knot is a staircase. In this talk, we will discuss a similar result for two-component L-space links: the link Floer complex of such links can be thought of as an array of staircases. We will describe an algorithm to extract this array directly from the H-function of the link. As an application, we will discuss how to use this and the link surgery formula to compute the knot Floer complex and the tau-invariant of a certain class of satellite knots. This is joint work with Ian Zemke and Hugo Zhou.

Vanishing cycles of Lefschetz fibrations give examples of Lagrangian spheres in the fiber. A natural question, first raised by Donaldson, is whether all Lagrangian spheres arise this way. We focus on this problem for positive rational surfaces, which were shown to admit a geometric structure called almost toric fibrations. I will talk about a work-in-progress showing all Lagrangian spheres here are visible in an almost toric fibration and thus are vanishing cycles of a nodal degeneration.

An Anosov representation of a hyperbolic group $\Gamma$ is a representation which quasi-isometrically embeds $\Gamma$ into a semisimple Lie group - say, SL(d, R) - in a way which imitates and generalizes the behavior of a convex cocompact group acting on a hyperbolic metric space. It is unknown whether every linear hyperbolic group admits an Anosov representation. In this talk, I will discuss joint work with Sami Douba, Balthazar Flechelles, and Feng Zhu, which shows that every hyperbolic group that acts geometrically on a CAT(0) cube complex admits a 1-Anosov representation into SL(d, R) for some d. Mainly, the proof exploits the relationship between the combinatorial/CAT(0) geometry of right-angled Coxeter groups and the projective geometry of a convex domain in real projective space on which a Coxeter group acts by reflections.