Yuezhao Li

I am a PhD candidate in mathematics at Leiden University, studying noncommutative geometry (NCG) and its applications in index theory and mathematical physics. My supervisor is Dr. Bram Mesland.

From 2013 to 2018, I studied physics and mathematics at Peking University, China. From 2018 to 2021, I studied mathematics at Georg-August-Universität Göttingen, Germany, specialising in noncommutative geometry and its application in topological insulators. I wrote my master thesis under the supervision of Prof. Dr. Ralf Meyer.

Currently, my research focuses on two topics. One is on finite-dimensionl computations of index pairings, using techniques from bivariant K-theories. This provides a theoretical framework to talk about index theory of spectral truncations in the sense of Alain Connes and Walter van Suijlekom. In another project I attempt to understand the robustness of topological invariants that occur in topological insulators, using techniques from K-theory, groupoid C*-algebras and coarse geometry.

My CV can be founded here. Last updated: March 2, 2025.

Here are some other things about myself, including running, languages, coding, etc.

My calendar in 2025

In 2025 spring semester, Bram Mesland, Francesca Arici and Marcel de Jeu are running a reading seminar on Hilbert C*-modules on Friday afternoons. Please contact them for the mailing list.

I will participate in the conference Applications of noncommutative geometry to gauge theories, field theories, and quantum space-time at CIRM, 7–11 April, 2025.

Notes and Handouts

I have taken notes for several lectures and seminars. Some of them are below. They may be incomplete and contain mistakes. Please use at your own risk.

• Unbounded KK-theory. The notes originate from Bram Mesland’s lecture course on unbounded KK-theory given at Hausdorff school “Noncommutative Geometry and Operator Algebras” in Bonn, in May, 2023. Unbounded KK-theory treats unbounded Kasparov modules as geometric objects and study their representing noncommutative geometries in the sense of Alain Connes. I took these notes as a first draft during the course and they were made complete by Bram Mesland.
• Groupoid C*-algebras. My notes of Leiden NCG reading seminar on Groupoid C*-algebras, 2022 fall semester. The notes contain fundaments of groupoids and their C*-algebras, as well as some special topics based on the participants’ own interest.
• KK-theory. My notes of Leiden NCG reading seminar on K(K)-theory, 2022 spring semester. We covered the basic knowledge of KK-theory in the first half, including the approaches of Kasparov and Cuntz. In the second half we turned to more research-related fields based on speakers’ own interest.
• Coarse geometry and groupoids. My notes of Nick Wright’s talk on the coarse Baum-Connes conjecture at Autumn School on Large Scale Geometry, with an emphasis on the coarse groupoid construction of Skandalis, Tu and Yu. The interest of mine lies in the fact that the groupoid models of Roe C*-algebras provide more explicit comparison maps between different models of aperiodic topological insulators, and hence have potential physical applications.
• Localisation algebras and K-homology. A handout provided for a talk in the Local NCG seminar at Radboud University Nijmegen, 2023 spring semester. It is mainly based on Guoliang Yu’s article “Localization algebras and the coarse Baum-Connes conjecture”, in which a new picture of K-homology is given to simplify the higher index map.
• Introduction to index theory. A handout provided for a talk at Leiden NCG reading seminar on Deformation quantisation and index theory, 2021 fall semester. It provides a basic introduction to the Atiyah-Singer index theorem, and highlights the role of K-theory of C*-algebras.
• Completely bounded maps. A handout provided for a talk at Leiden Functional Analysis seminar on Completely bounded maps and operator algebras, 2021 fall semester. It is based on Chapter 8 of Vern Paulsen’s book “Completely bounded maps and operator algebras”.

NCG-Leiden

I am a member of the NCG research group in Leiden. I help maintain the website NCG-Leiden which announces its activities and seminars. If you want to be added to the mailing list, please contact Francesca Arici or Bram Mesland.

In previous semesters, we have run reading seminars on noncommutative geometry of foliations (2023 fall), groupoid C*-algebras (2022 fall) and KK-theory (2022 spring).