Georgia Institute of Technology College of Sciences

The hyperplane conjecture, raised by Bourgain in 1986, is a major unsolved problem in high-dimensional geometry. It states that every convex set of volume 1 in the Euclidean space has a section that is lower bounded away from 0 uniformly over the dimension. We will present a probabilistic approach to the conjecture. 

We consider the Benjamin Ono equation, modeling one-dimensional long interval waves in a stratified fluid, with a slowly-varying potential perturbation. Starting with near soliton initial data, we prove that the solution remains close to a soliton wave form, with parameters of position and scale evolving according to effective ODEs depending on the potential. The result is valid on a time-scale that is dynamically relevant, and highlights the effect of the perturbation. It is proved using a Lyapunov functional built from energy and mass, Taylor expansions, spectral estimates, and estimates for the Hilbert transform.

Prime-power-fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots: In this talk, we give examples of non-slice knots that have the above property. This is joint work with Aceto, Meier, A. Miller, M. Miller, and Stipsicz.

The cosmetic surgery conjecture states that no two different Dehn surgeries on a given knot produce the same oriented 3-manifold (such a pair of surgeries is called purely cosmetic). For knots in S^3, I will describe how knot Floer homology provides a strong obstruction to the existence of purely cosmetic surgeries. For many knots, including all alternating knots with genus not equal to two as well as all but 337 of the first 1.7 million knots, this is enough to confirm the conjecture. For the remaining knots, all but finitely many surgery slopes are obstructed, so checking the conjecture for a given knot reduces to distinguishing finitely many pairs of manifolds. Using a computer search, the conjecture has been verified for all prime knots with up to 16 crossings, as well as for arbitrary connected sums of such knots. These results significantly improve on earlier work of Ni and Wu, who also used Heegaard Floer homology to obstruct purely cosmetic surgeries. The improvement comes from using the full graded Heegaard Floer invariant, which is facilitated by a recent recasting of knot Floer homology as a collection of immersed curves in the punctured torus.