Georgia Institute of Technology College of Sciences

We study the construction of exponential bases and exponential frames on general $L^2$ space with the measures supported on self-affine fractals. This problem dates back to the conjecture of Fuglede. It lies at the interface between analysis, geometry and number theory and it relates to translational tilings. In this talk, we give an introduction to this topic, and report on some of the recent advances. In particular, the possibility of constructing exponential frames on fractal measures without exponential bases will be discussed.