题目: Random planar geometry with conformal invariance
报告人: Prof. Wei Qian (The University of Hong Kong)
时间: 2025年10月18, 19日, 上午9:30-11:30
地点: 光华楼东主楼1801
报告摘要: In the past decades, random curves, loops and surfaces with conformal symmetry has been the subject of intense study, and has led in particular to three Fields medals. For example, shortly after the introduction of SLE (Schramm-Loewner evolutions) by Schramm in 1999, Werner obtained the Fields medal in 2006 for the study of SLE.
SLE and its loop variant CLE (conformal loop ensemble) universally arise in the scaling limits of a large class of statistical physics models. For instance, the Bernoulli percolation at criticality was proved by Smirnov to converge to a conformally invariant limit, which is described by SLE(6) and CLE(6). In a recent joint work with Lis and Duminil-Copin, we prove that the double random currents model converges to SLE(4) and CLE(4).
In the first part of this course, we start with a gentle introduction of percolation, SLE and CLE. We show some of their basic properties, such as conformal restriction. This property is further related to the Virasoro algebra representation of SLE. We then show an application of SLE in computing the arm exponents for percolation. We also mention a recent joint work with Nolin, Sun, Zhuang reported in Quanta Magazine, where we have computed the backbone exponent for percolation.
In the second part of this course, we introduce the Brownian loop soup, and highlight its close relations with CLE and the Gaussian free field. We survey a series of our recent works on the geometric structure of the loop-soup clusters, and point out some open questions.
This course is for students who are interested in getting a glimpse of the field of random geometry. It starts by a gentle introduction of the most basic objects, and ends up surveying some recent developments.
We also advertise one (or possibly several) open PhD position(s) at University of Hong Kong in this theme of research.
报告人简介: Wei Qian is an Associate Professor in the Department of Mathematics of the University of Hong Kong. She is currently on leave from the position of Chargée de Recherche in CNRS in France.
She obtained her PhD at ETH Zürich in 2017, under the supervision of Wendelin Werner. Subsequently, she was a Junior Research Fellow at Churchill college in the University of Cambridge, a Chargée de Recherche of CNRS in Université Paris-Saclay, and an Assistant Professor in the City University of Hong Kong.
