+ Venue: Auditorium, 6 Floor, Institute of Mathematics (NTU Campus)

+ Title: Visual probability and connections to analysis

Lecture 1: Gravitational allocation to uniform points on the sphere

Abstract: Given n uniform points on the surface of a two-dimensional sphere, how can we partition the sphere fairly among them? "Fairly" means that each region has the same area. It turns out that if the given points apply a two-dimensional gravity force to the rest of the sphere, then the basins of attraction for the resulting gradient flow yield such a partition—with exactly equal areas, no matter how the points are distributed. (See the cover of the AMS Notices at http://www.ams.org/publications/journals/notices/201705/rnoti-cvr1.pdf or the PNAS article http://www.pnas.org/content/early/2018/09/06/1720804115 ). Our main result is that this partition minimizes, up to a bounded factor, the average distance between points in the same cell. I will also present an application to almost optimal matching of n uniform blue points to n uniform red points on the sphere, connecting to a classical result of Ajtai, Komlos and Tusnady (Combinatorica 1984). I will emphasize open problems on the diameters of the basins and the behavior of greedy matching schemes. Joint work with Nina Holden and Alex Zhai.

Lecture 2: Laplacian growth, sandpiles and scaling limits

Abstract: How can repeating simple local operations lead to an intricate large scale structure? This phenomenon arises in several growth models originating in Physics: Internal diffusion limited aggregation (IDLA) and the Abelian sandpile. The first of these is closely related to free boundary problems for the Laplacian and an algebraic operation introduced by Diaconis and Fulton known as ``smash sum’’. These connections allow a precise description of large scale geometry, using a least action principle. Theabelian sandpile, discovered independently by Statistical Physicists and Combinatorialists is harder to analyze, yet has recently yielded many of its secrets in works of Pegden, Smart and Levine. I will also discuss the rotor-router model, where (with random initial conditions) the range is conjectured to grow like t^{2/3} at time t; recently, with L. Florescu, we showed this holds as a lower bound. (Talk based on joint works with Lionel Levine).

Lecture 3: Search games and Kakeya sets

Abstract: I'll describe a search game with a surprising geometric connection. A hunter and a rabbit move on an n-vertex graph without seeing each other until they meet. At each step, the hunter moves to a neighboring vertex or stays in place, while the rabbit is free to jump to any node. Thus they are engaged in a zero sum game, where the payoff is the capture time. We show that an optimal rabbit strategy for the cycle yields a Kakeya set: a plane set of zero area that contains a unit segment in every direction. Kakeya sets have been studied intensively in harmonic analysis since their discovery by Besicovitch (1919); their connection to search games is new and yields insights in both directions. (Based on joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler, and on earlier work by Adler et al (2003).)