I will briefly recall the famous theory of multiplicative chaos, including Mandelbrot cascade theory, Kahane's Gaussian multiplicative chaos. Then I will discuss our recent works on the precise formulae for the Fourier dimensions of the Mandelbrot's canonical cascade measures, microcanonical cascade measures and Gaussian multiplicative chaos on the unit intervals. In particular, our works resolve the Mandelbrot-Kahane problem raised by Mandelbrot in 1976 and re-interated by Kahane in 1993. Our work on Gaussian Multiplicative Chaos resolves a conjecture of Garban-Vargas. This talk is based on three works with Xinxin Chen, Yong Han, Zipeng Wang (from Chongqing University) on Mandelbrot's canonical/microcanonical cascades measures and with Zhaofeng Lin, Mingjie Tan on Gaussian multiplicative chaos.
