In this colloquium, I will discuss the fractal structure of various special points on two- and three-dimensional random walk and Brownian motion. We will then focus on two examples: cut points and loop-erased random walk and report some recent progresses on the study of these models.

Xinyi Li, Assistant Professor at Beijing International Center for Mathematical Research，Peking University. His research interests are Probability theory, statistical physics, random geometry. Xinyi Li got his Doctor Degree from Department of Mathematics, ETH Zurich in 2016. After that, he had been L. E. Dickson instructor at Department of Mathematics, the University of Chicago. In 2019 he is assistant professor at BICMR, Peking University.