An Introduction to Random Geometry系列课程
报告专家：Yi Tian（the University of Cambridge）
课程时间：
2024年12月29日（星期日）上午9:00-12:00
2024年12月30日（星期一）上午9:00-12:00
2024年12月31日（星期二）上午9:00-12:00
2024年12月31日（星期二）下午2:30-5:30
课程地点：四川大学西南中心516
课程摘要：
This mini-course explores foundational topics in random geometry, focusing on Schramm-Löwner Evolution (SLE), Gaussian Free Field (GFF), and Liouville Quantum Gravity (LQG).
1. Schramm-Loewner Evolution (SLE): The first lecture introduces SLE, a family of random, non-self-crossing curves in the upper half-plane parameterized by $\kappa \geq 0$. We define SLE using the Löwner differential equation from complex analysis and discuss its rich phase structure: simple curves $(0 \leq \kappa \leq 4)$, self-touching curves $(4<\kappa<8)$, and space-filling curves $(\kappa \geq 8)$. Additionally, we explore how SLE emerges as the scaling limit of interfaces in critical planar statistical models, including the self-avoiding walk, the Ising model, and the percolation model.
2. Gaussian Free Field (GFF): The second lecture focuses on GFF, the natural generalization of Brownian motion to higher dimensions. We introduce the discrete GFF on graphs and its continuum counterpart, a random generalized function (distribution) on open subsets of $\mathbb{R}^d$. Furthermore, we explore the connection between GFF and SLE via the theory of imaginary geometry, where SLE can be interpreted as flow lines of the GFF.
3. Liouville Quantum Gravity (LQG): In the final lecture, we introduce LQG, a random surface model proposed by Polyakov over forty years ago in the framework of string theory and rigorously constructed only recently. LQG surfaces are equipped with random measures and random metrics, hence can be viewed as "random two-dimensional Riemannian manifolds". This lecture also covers how LQG led to the mathematical proof of Knizhnik-Polyakov-Zamolodchikov formula and mathematical construction of Liouville conformal field theory, the only rigorously defined quantum field theory to date.
专家简介：
Yi Tian is a PhD student in Pure Mathematics and Mathematical Statistics at the University of Cambridge, supervised by Jason Miller. Research focuses on random geometry, particularly conformal loop ensembles and Liouville quantum gravity. Previous studies include an MSc at the University of Bonn and a BSc at Peking University. Awards include a Gold Medal at the 33rd Chinese Mathematical Olympiad.
邀请人：盛利
