Talk 1: Hunter, Cauchy Rabbit, and Optimal Kakeya Sets
Yuval Peres, Microsoft Research
A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1928); we find a new connection to game theory and probability. A hunter and a rabbit move on the integer points in [0,n) without seeing each other. 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. The known optimal randomized strategies for hunter and rabbit achieve expected capture time of order n log n. We show that every rabbit strategy yields a Kakeya set; the optimal rabbit strategy is based on a discretized Cauchy random walk, and it yields a Kakeya set K consisting of 4n triangles, that has minimal area among such sets (the area of K is of order 1/log(n)). Passing to the scaling limit yields a simple construction of a random Kakeya set with zero area from two Brownian motions. (Joint work with Y. Babichenko, R. Peretz, P. Sousi and P. Winkler).
Talk 2: Direct Products and Gap Amplification
Irit Dinur, Weizmann Institute of Science
Probabilistically checkable proofs (PCPs) are proofs in which the distance between right and wrong is greatly amplified: a verifier needs read only O(1) bits of the proof to make its decision. A PCP can be constructed by pushing any proof through a gap-amplifying transformation. One of the most basic ways to get gap-amplification is by parallel repetition, aka direct product. This is a very easy-to-define operation, yet its analysis leaves many open questions. We will introduce PCPs and direct products and then survey existing results and describe some open questions.
Talk 3: Probability and additive Combinatorics
Persi Diaconis, Stanford University
The study of “carries” when we add numbers has surprisingly rich connections with probability and parts of algebra (work with Borodin and Fulman). Simple changes, e.g., using “balanced arithmetic,” lead to new problems. In joint work with Haow and Soundararajan, we show how additive combinatorics (Freiman, Szemeredi) saves the day.
Talk 4: Free Monotone Transport
Alice Guionnet, Massachusetts Institute of Technology
During the last twenty years, the theory of mass transportation and optimal transport was shown to have many applications in diverse fields of mathematics. In this talk, we will discuss its first generalization to a non-commutative setting as well as its applications to derive isomorphisms of some C* algebras. We will introduce some basics of the theory of optimal transport as well as of free probability, the natural set up for such a generalization. This is joint work with D. Shlyakhtenko.
Talk 5: TBD
Gerard Ben Arous, New York University
Abstract TBD