The impact of the Kakeya problem on combinatorial geometry and vice versa

Nets Katz
California Institute of Technology

The Kakeya problem had been previously introduced by Neil Lyall. Dvir proved the finite field version using a simple polynomial argument and this set off a slew of applications of the polynomial to analogous problems to the Kakeya problem in incidence geometry. Considerable work is going on aimed at the reverse direction of obtaining results on the real Kakeya conjecture. We discuss some of the ideas behind these as well as the challenges.

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