Falconer is known for his work on the mathematics of
fractals and in particular sets and measures arising from
iterated function systems, especially
self-similar and self-affine sets. Closely related is his research on
Hausdorff and other
fractal dimensions. He formulated ''
Falconer's conjecture'' on the dimension of distance sets and conceived the notion of a
digital sundial. In combinatorial geometry he established a lower bound of 5 for the
chromatic number of the plane in the Lebesgue measurable case. ==Education and career==