Dvir's solution to the Kakeya conjecture in finite fields

A Kakeya set in the Euclidean space is a set which contains a unit-length line segment in any direction. A. Besicovitch in the early 20th century constructed an example of Kakeya sets which has zero measure. The Kakeya conjecture, on the other hand, asserts that a Kakeya set must have full dimension as the ambient space. In particular, it cannot be ``too small''. In the past century, mathematicians have found deep connections between the Kakeya sets and problems in analysis, partial differential equations, combinatorics, algebraic geometry, and more, such as the restriction conjecture in Fourier analysis. Lying at the center of modern mathematics, the Kakeya conjecture remains open. T. Wolff, a pioneer in the study of the Kakeya conjecture, proposed a finite-field version of the conjecture in the late 1990s. Only partial results were achieved towards this conjecture by T. Wolff, J. Bourgain, T. Tao, etc. In 2008, Z. Dvir settled Wolff's Kakeya conjecture in the finite fields. His concise proof relies on a polynomial method that is completely different from the arguments used before. In this thesis, we review the results and arguments before Dvir: Bourgain's bush argument and Wolff's hairbrush argument. Then we present the polynomial method and Dvir's solution to the Kakeya conjecture in finite fields.