In this talk, we will present a spectral flow proof of Llarull's theorem in odd dimensions, together with a noncompact generalization. Let (M, g^TM) be a compact n‑dimensional spin Riemannian manifold whose scalar curvature k^TM satisfies k^TM > n(n − 1). Llarull's theorem states that any smooth area‑decreasing map f : M → Sⁿ(1) of nonzero degree must be an isometry. We provide a direct proof of this theorem in odd dimensions using a spectral flow argument. This new argument allow us to extend this result to the noncompact setting. More precisely, we show that if (M, g^TM) is a connected, oriented, complete, noncompact spin Riemannian manifold of odd dimension n ≥ 3, and f : M → Sⁿ(1) is a smooth map of nonzero degree that is locally constant near infinity, then the condition k^TM ≥ n(n − 1) on supp(df) implies that inf_M k^TM < 0. This extension is established using a similar spectral flow argument involving deformed Dirac operators. The first part of the talk will focus on Llarull's theorem and the main ideas underlying the proof. The second part will address the analytical details of the argument, with particular emphasis on the eta invariant associated with a smooth family of deformed Dirac operators. This talk is based on joint works with Guangxiang Su, Xiangsheng Wang and Weiping Zhang.

Constructing new subfactors beyond representation theory of groups or quantum groups is the most challenging and important problem in subfactor theory. Haagerup constructed the first exotic example in 1994. Sooner Izumi proposed his equations to construct a potentially infinite sequence of near group subfactors as a generalization of the Haagerup subfactor. In this talk, we will explain our recent construction of irrational near group subfactors for all cyclic groups based on our triangle prism equations and the Belousov‑Sarkissian‑Spiridonov identity of Gamma functions. The work is partially assisted by AI.

In this talk, we will present the skein theoretical construction and classification of subfactors and planar algebras. We propose a systematic classification of exchange relation fusion bialgebras. In our classification up to rank 6, we discover a new one‑parameter family of subfactor planar algebras and another two non‑unitary families.

In these two lectures, we will introduce some basic ideas in index theory, focusing on Witten deformation and the Bismut–Lebeau analytic localization method. Starting from elementary examples, we will explain how Witten deformation leads to spectral localization and how the Bismut–Lebeau method can be used to relate Dirac‑type operators on different manifolds. If time permits, we will discuss an application to leafwise positive scalar curvature and the vanishing of the Rosenberg index.

Free probability is a noncommutative probability theory in which free independence plays the role of classical independence. It originated in the study of operator algebras and has close connections with the limiting behavior of large random matrices. In this talk, we will give an introduction to free probability in two parts. The first and main part is devoted to the basic concepts and examples. In the second part, we will briefly discuss some of our earlier results, focusing on joint work with Tobias Mai and Roland Speicher on the realization of the free field via unbounded operators. These results will provide the background for the following talk on noncommutative Kronecker's criterion.

Kronecker's criterion characterizes rational series by finite rank of the associated Hankel matrix. Connes conjectured a noncommutative analogue for free group algebras, replacing the Hankel‑rank condition with a finite‑rank commutator condition. Duchamp and Reutenauer proved the conjecture, Linnell extended the criterion to affiliated operators, and Miyagawa established a corresponding criterion for free semicircular families. For tuples of operators in a finite tracial von Neumann algebra with maximal Δ, we characterize rational affiliated operators by the existence of a finite‑rank solution to a commutator equation. For normal generators whose dual operators satisfy a common domain condition, we characterize rationality by finite‑rank fraction commutators, covering bounded dual systems and selfadjoint tuples admitting conjugate variables. These results thus give a common generalization of the earlier criteria of Duchamp and Reutenauer, Linnell, and Miyagawa. These results were obtained using an autonomous research framework powered by OpenAI's llm ChatGPT‑6 Astra. The framework consists of a lightweight set of rules written in Markdown and can be used across different agent harnesses. We will spent the last 10 minutes to introduce the framework and share our experience using it to investigate the noncommutative rationality problem.