Abstract:  In the study of (1+1)-dimensional random growth processes, a foundational model is the longest-increasing-subsequence problem: from n independent uniform points in the unit square, find the longest chain whose horizontal and vertical coordinates are both increasing. It is known that the maximal length is asymptotically 2n^(1/2), and the longest chain concentrates near the diagonal line y=x. In this talk, we add a geometric constraint: the chain must be convex. I will discuss the history of this problem, as well as recent joint work with Arnab Sen.

Abstract: The four-corner Cantor set is a planar analogue of the classical Cantor set and arises in several areas of analysis, including the study of Kakeya sets and removable singularities for analytic functions. A central problem is to understand how this set behaves when projected onto lines. This turns out to be a very difficult question, so we study a random variant of the Cantor set, where we are able to obtain sharp estimates. This is joint work with Pablo Shmerkin and Ville Suomala. 

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