Take a grid in Euclidean space and delete every edge, independently, with probability p. For which p is there an infinite cluster? How does it look? How big are the finite clusters and how do they look? Such questions have fascinated probabilists and mathematical physicists for over 50 years, and they continue to be an unending source of challenging problems. Most exciting, perhaps, is the connection between the behavior of various probabilistic structure such as the just-mentioned percolation, random walks, self-interacting walks and interacting particle systems with the large-scale geometry of the object. This question links the fields above with differential geometry and geometric group theory, creating a rich mathematical nexus.
And now to something completely different. Take a sum of harmonics converging to some function f, outside of some bad, but small set K. Must this series be f's Fourier expansion? Uniqueness theory tries to answer this question, depending on the set K, the function f and conditions one might impose on the sum. It exemplifies the triple face of harmonic analysis - size, probability and algebra. Some K-s can be analyzed by their size. Random K-s can be analyzed differently. And for some K-s, delicate algebraic properties of the set determine the final outcome. Uniqueness theory traces its roots to works of Riemann and Cantor, but still provides many fascinating problems.
- [with Alexander Olevskii] Analytic representation of functions and a new quasi-analyticity threshold. Ann. of Math. (2) 164: 3 (2006) 1033-1064.
- [with Asaf Nachmias] The Alexander-Orbach conjecture holds in high dimensions.
- The scaling limit of loop-erased random walk in three dimensions. Acta Math. 199: 1 (2007) 29-152.