Lines, Points, and Dimensions:
A Tour of the Kakeya Problem in Algebra and Analysis
Evan Dummit, PhD
University of Rochester
Friday, March 24, 2017
4:30 - 5:30 PM
The Kakeya "needle" problem, posed by S. Kakeya in 1918, is a classical question in analysis that asks how small a subset of the Euclidean plane can be, if it contains a unit line segment in every possible direction. This question was answered by A. Besicovitch in 1919, who gave the rather surprising answer that such sets can have area zero, but necessarily still had dimension 2. It remains open to this day, however, whether the minimal dimension of such a "Kakeya set" in n-dimensional Euclidean space must be n. In the late 1990s, T. Wolff proposed an algebraic version of the Besicovitch-Kakeya problem in the setting of a finite field, and a major advance in this area was made by Z. Dvir in 2008 using a stunning idea known as the "polynomial method". In this broad-audience talk, I will discuss the Kakeya problem in algebra and analysis along with some work of myself and others on bridging the gap between the algebraic and analytic versions of this problem.
ADA: Individuals requiring accommodations please contact Doreen Taylor at 802-656-3166