Since the development of the first randomized polynomial-time algorithm for volume computation by Dyer, Frieze, and Kannan in 1989, convex-body sampling has been a central problem at the intersection of algorithms, geometry, and probability. A major milestone came in 1997, when Kannan, Lovász, and Simonovits analyzed the Ball walk and formulated the influential KLS conjecture. This was extended to log-concave distributions by Lovász and Vempala in 2006, and further accelerated by Cousins and Vempala in 2015 through warm-start generation techniques.

In this talk, I will present a gentle introduction to these milestones and how they have been streamlined and improved in the past few years. The talk is based on joint works appearing in SODA'25, STOC'25, and FOCS'25, with Santosh Vempala and Matthew Zhang.

​Yunbum Kook is a Ph.D. candidate in Computer Science at Georgia Tech, advised by Santosh Vempala, and will start a postdoc in Computer Science and Engineering at the University of Michigan. He received his B.S. in Mathematical Sciences from KAIST. His research focuses on the mathematical and algorithmic foundations of high-dimensional sampling, with connections to theoretical computer science, applied mathematics, and machine learning theory.​