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Livestream:

InÌý1919, Besicovitch constructed a compact setÌýinÌýtheÌýplane with Lebesgue measure 0 that contains a unit line segment pointingÌýinÌýevery direction. Such objectsÌýareÌýnow called measure 0 Besicovitch sets (aka Kakeya sets). By replacing a measure zero Besicovitch set by its \delta-thickening, one obtains a collection of 1 x \delta rectangles pointingÌýinÌýdifferent directions,ÌýtheÌýsum ofÌýwhoseÌýareas is 1, butÌýwhoseÌýunion has very small volume.ÌýTheÌýexistence of such collections of rectangles is calledÌýtheÌýBesicovitch compression phenomenon.
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TheÌýKakeya set conjecture is a quantitative statement controllingÌýtheÌýstrength ofÌýtheÌýBesicovitch compression phenomenon.ÌýInÌýthis talk, I will discuss connections betweenÌýtheÌýBesicovitch compression phenomenon,ÌýtheÌýKakeya set conjecture, and questionsÌýinÌýharmonic analysis and PDE. This talk is intendedÌýforÌýa general mathematical audience.