Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, May 12, 2009, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Jiří Matoušek (Charles Univ., Prague)

Kakeya's conjecture over finite fields according to Zeev Dvir

The Kakeya needle problem asks for the smallest area of a planar set in which one can rotate a unit-length needle. One of the surprising results in mathematics is Besicovitch's construction (sketched in the first part of the talk), showing that an arbitrarily small area suffices. A necessary condition for rotating the needle inside a set is that the set contains a unit segment of every direction - such sets are called Kakeya sets. According to Besicovitch, there exist Kakeya sets of measure zero. Yet an important conjecture in analysis asserts that a Kakeya set can't be too small, in the sense of Hausdorff dimension. An analog of that conjecture for finite fields was recently proved by Zeev Dvir, and the amazingly simple and beautiful proof will be reproduced in the second part of the talk.

May serve as an introduction to the forthcoming talk by Micha Sharir in June.

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