The Conformal Dimension and Minimality of Stochastic Objects

YMSC Probability Seminar

The Conformal Dimension and Minimality of Stochastic Objects

Wen-Bo Li 李文博

Peking University

I am a Postdoc Scholar at Beijing International Center for Mathematical Research, Peking University. My Postdoc Mentors are Zhiqiang Li, Xinyi Li and Wenyuan Yang. I was a visiting scholar at the Fields Institute for Research in Mathematical Science from January 2024 to June 2024. I obtained my Ph.D. in Mathematics in 2022 at the University of Toronto under the supervision of Professor Ilia Binder.

My research is focused on Analysis and Geometry on Metric Measure Spaces, Quasiconformal Geometry and Random Geometry. I am also interested in Complex Analysis, Complex Dynamics, Probability, Geometric Measure Theory, Geometric Function Theory, Geometric Group Theory, Metric Geometry, etc.

# Organizers

吴昊，杨帆，姜建平，顾陈琳

# Time

Thur., 3:40 - 4:40 pm, Nov. 21, 2024

# Venue

C548, Shuangqing Complex Building A

#Abstract

The conformal dimension of a metric space is the infimum of the Hausdorff dimension among all its quasisymmetric images. We develop tools related to the Fuglede modulus to study the conformal dimension of stochastic spaces. We first construct the Bedford-McMullen type sets, and show that Bedford-McMullen sets with uniform fibers are minimal for conformal dimension. We further develop this line of inquiry by proving that a "natural" stochastic object, the graph of the one dimensional Brownian motion, is almost surely minimal. If time permits, I will also explore further developments related to Schramm-Loewner evolution (SLE), conformal loop ensembles (CLE), and related questions motivated by an exploration of the renowned Sullivan dictionary. This is a joint work with Ilia Binder(UToronto) and Hrant Hakobyan(KSU), accepted by Duke Math. J.