Graduate/faculty Seminar
Friday, October 23, 2009, 4:30pm, 119 Physics
Kash Balachandran
The Kakeya Conjecture
Abstract:- In 1917, Soichi Kakeya posed the question: What is the smallest
amount of area required to continuously rotate a unit line segment in
the plane by a full rotation? Inpsired by this, what is the smallest
measure of a set in $\mathbb{R}^n$ that contains a unit line segment
in every direction? Such sets are called Kakeya sets, and can be
shown to have arbitrarily small measure w.r.t. n-dimensional Lebesgue
measure [and in fact, measure zero].
The Kakeya conjecture asserts that the Hausdorff and Minkowski
dimension of these sets in $\mathbb{R}^n$ is $n$.
In this talk, I will introduce at a very elementary level the
machinery necessary to understand what the Kakeya conjecture is
asking, and how the Kakeya conjecture has consequences for fields
diverse as multidimensional Fourier summation, wave equations,
Dirichlet series in analytic number theory, and random number
generation. I'll also touch on how tools from various mathematical
disciplines from additive combinatorics and algebraic geometry to
multiscale analysis and heat flow can be used to obtain partial
results to this problem.
The talk will be geared towards a general audience. [video]
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