Abstract:
In 1917 a Japanese mathematician named Kakeya posed “the needle problem,” asking: what is the area of the smallest figure in the plane in which a unit line segment (a “needle”) can be rotated 180 degrees? It was conjectured that the smallest such “Kakeya set” was a deltoid with area $\pi/8$. However, in 1928 a Russian mathematician named Besicovitch published a very surprising result: Kakeya sets can be arbitrarily small. More than just a historic and geometric curiosity, Kakeya sets have come to play an important role in analysis today. In this talk, we will construct a Kakeya set with arbitrarily small area and then discuss some problems that are unexpectedly related to these sets.