Victor Lie

Our talk investigates the subtle relationship be-tween the size of the level sets of the (bilinear) Kakeya func-tion and the corresponding geometric distribution of the pointswithin these level sets. Under suitable conditions, our goal isto characterize the situation in which the size of these level setsis maximal and thus to provide qualitative and quantitative in-formation about the extremizers associated with the (bilinear)Kakeya function. Our analysis will involve additive combina-torics (e.g. Pl ̈unnecke sum-set estimate) and incidence geometry(e.g. Szemeredi-Trotter) techniques and relates with a class ofproblems including Bourgain’s sum-product theorem and Katz-Tao ring conjecture. This is a joint work with Michael Bateman.