Colloquium

In 1984, K. Mahler published a paper entitled "Some suggestions for further research", in which he writes the following moving statement: "At the age of 80 I cannot expect to do much more mathematics. I may however, state a number of questions where perhaps further research might lead to interesting results." One of these questions was regarding intrinsic approximation on the Cantor set. In Mahler's words, "how close can irrational elements of Cantor's set be approximated by rational numbers in Cantor's set"? In this talk, I will describe recent advances towards answering Mahler's question on fractals in general and on the Cantor set in particular. The talk is intended for a general audience and no prior knowledge on fractal geometry nor Diophantine approximation will be assumed.