Student Probability and Mathematical Physics Seminar
Cut points of Brownian motion
Speaker: Zhenfeng Tu, CIMS
Location: Warren Weaver Hall 201
Date: Tuesday, March 24, 2026, 12:30 p.m.
Synopsis:
Brownian motion is one of the most fundamental objects in probability, yet its geometric structure is remarkably rich. In this talk, we focus on cut points—points along the path where the past and future do not intersect. These special points reveal hidden fractal structure inside Brownian motion. We will introduce cut points and discuss their basic properties, including the surprising fact that they form a random fractal set of Hausdorff dimension (3/4). We then explain the key idea behind this exponent: it is governed by the probability that two independent Brownian paths avoid each other. If time permits, we will also discuss how one can go beyond dimension and define a natural measure on the set of cut points using Minkowski content. Along the way, we highlight connections to intersection exponents, conformal invariance, and modern developments such as Schramm–Loewner Evolution.