Yi Tian

I am Yi Tian, a PhD student in mathematics at the University of Cambridge (since 2024) advised by Jason Miller. I completed my master's at Universität Bonn in 2024 with Eveliina Peltola, and my undergraduate studies at Peking University in 2022 with Zhiyu Tian. Here is my CV.

Email: CRSid@cam.ac.uk

CRSid: yt425

Papers

Dual of stable carpets

• The conformally invariant metric on CLE\(_4\) I, II, III (with E. Kammerer, K. Kavvadias, and J. Miller). arXiv:2609.04138, arXiv:2609.04139, arXiv:2609.04140

Quasisymmetric geometry of random fractals

• Quasisymmetric rigidity of the Brownian sphere (with J. Miller). arXiv:2606.10973
• The conformal dimension of the Brownian tree is one (with J. Miller). arXiv:2604.25769
• The conformal dimension of the Brownian sphere is two (with J. Miller). arXiv:2603.24473

Intrinsic metric and diffusion/heat flow on simple conformal loop ensemble carpets

• Existence and uniqueness of the Brownian motion on simple conformal loop ensemble carpets (with J. Miller). In preparation.
• Existence and uniqueness of the conformally covariant geodesic metric on simple conformal loop ensemble carpets (with J. Miller). arXiv:2511.16208

Talks

• The conformally invariant metric on CLE\(_4\) (poster). Random Explorations (IMSI Chicago Jul 2026).
• Quasisymmetric geometry of the Brownian sphere. Peking Summer Program in Probability (PKU Jun 2026).
• Analysis on conformally invariant random fractals. Jane Street Graduate Research Fellowship Workshop (online Apr 2026).
• The conformally covariant geodesic metric on simple conformal loop ensemble carpets. Combinatorial and hyperbolic random planar geometry (CIRM Marseille Aug 2025).
• The conformally covariant geodesic metric on simple conformal loop ensemble carpets. Probabilistic methods in quantum field theory (Bonn Jul 2025).

Images

Simulations of the uniform exploration (first two panels), a geodesic, and the geodesic tree for the conformally invariant metric on CLE\(_4\). Based on the work [KKMT26].

Simulations of metric balls for the conformally covariant geodesic metric on CLE\(_\kappa\) carpets, with \(\kappa = 2.72, 2.85, 3, 10/3, 3.99\), respectively. Based on the work [MT25].