The Finite Field Kakeya Problem

The Finite Field Kakeya Problem

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Xander Faber, Columbia University
Fine Hall 314

If V is a vector space over the field with q elements, a (finite field) Kakeya set is a subset of V containing a line in every direction. The main problem is to determine how large such a set is forced to be. Recently, Zeev Dvir gave a simple proof of the correct order of magnitude of Kakeya sets by introducing a nice technique from algebraic geometry. This new idea has provided a great deal of excitement. In this talk, I'll survey what is known about the problem (relatively little given that it is 10 years old), and then discuss some of the low-hanging combinatorial fruit in the case where V has dimension 2.