2024-2025
Session 2024–2025
Autumn Session
Jean Delhaye (Université Paris-Saclay) - November 18, 2024
Title: Cutoff for the Brownian Motion on the Unitary Quantum Group.
Abstract: We introduce an analog of the Brownian motion on free unitary quantum groups \(UN+\). We will discuss the construction of this Brownian motion, computing its cutoff, where convergence to equilibrium undergoes a sharp transition. We will also examine the cutoff profile, analyzing the fine-scale behavior of the total variation distance around the cutoff. Unlike classical or orthogonal quantum groups, the study of \(UN+\) has additional challenges, such as non-absolute continuity, distinct properties of its central algebra and inabilities to clearly identify a Brownian motion.
Malte Leimbach (Radboud University) - December 16, 2024
Title: Convergence of Peter–Weyl Truncations of Compact Quantum Groups.
Abstract: A fundamental principle of noncommutative geometry is to encode geometric information by spectral data, formalised in the notion of spectral triples. In physical practice there are, however, always obstructions on the availability of such data, and one might be led to considering truncated versions of spectral triples instead. In this talk we will take a closer look at this formalism and explore it within the framework of compact quantum metric spaces. In particular we will consider compact quantum groups as compact quantum metric spaces when equipped with an invariant lip-norm. We will discuss complete Gromov–Hausdorff convergence of truncations arising from the Peter–Weyl decomposition of a compact quantum group.
Winter Session
Julio Cáceres (Vanderbilt University) - February 3, 2025
Title: New hyperfinite subfactors with infinite depth.
Abstract: We will present new examples of irreducible, hyperfinite subfactors with trivial standard invariant and interesting Jones indices. These are obtained by constructing new finite dimensional commuting squares. We will use two graph planar algebra embedding theorems and the classification of small index subfactors to show that our commuting square subfactors cannot have finite depth. We also present one-parameter families of commuting squares that, by a classification result of Kawahigashi, will also yield irreducible infinite depth subfactors. This is joint work with Dietmar Bisch.
Roberto Hernández Palomares (University of Waterloo) - March 17, 2025
Title: Quantum graphs, subfactors and tensor categories.
Abstract: We will introduce equivariant graphs with respect to a quantum symmetry along with examples such as classical graphs, Cayley graphs of finite groupoids, and their quantum analogues. These graphs can be presented concretely by modeling a quantum vertex set by an inclusion of operator algebras and the quantum edge set by an equivariant endomorphism, idempotent with respect to convolution/Schur product. Equipped with this viewpoint and tools from subfactor theory, we will see how to obtain all these idempotents using higher relative commutants and the quantum Fourier transform. Finally, we will state a quantum version of Frucht’s Theorem, showing that every quasitriangular finite quantum groupoid arises as certain automorphisms of some categorified graph.
Spring Session
Sang-Gyun Youn (Seoul National University) - March 31, 2025 at 10:00 am (CEST)
Title: A Khintchine inequality for central Fourier series on non-Kac compact quantum groups.
Abstract: The study of Khintchine inequalities has a long history in abstract harmonic analysis. While there is almost no possibility of non-trivial Khintchine inequality for central Fourier series on compact connected semisimple Lie groups, it has turned out that a strong contrast holds within the framework of compact quantum groups. Specifically, a Khintchine inequality with operator coefficients is proved for arbitrary central Fourier series in a large class of non-Kac compact quantum groups. The main examples include the Drinfeld-Jimbo \(q\)-deformations, the free orthogonal quantum groups, and the quantum automorphism groups.
Heon Lee (Harbin Institute of Technology) - April 7, 2025
Title: First-order differential calculi and Laplacians on \(q\)-deformations of compact semisimple Lie groups.
Abstract: In this talk, we suggest a simple definition of Laplacian on a compact quantum group (CQG) associated with a first-order differential calculus (FODC) on it. Applied to the classical differential calculus on a compact Lie group, this definition yields classical Laplacians, as it should. Moreover, on the CQG \(K_q\) arising from the \(q\)-deformation of a compact semisimple Lie group \(K\), we can find many interesting linear operators that satisfy this definition, which converge to a classical Laplacian on \(K\) as \(q\) tends to \(1\). In the light of this, we call them \(q\)-Laplacians on \(K_q\) and investigate some of their operator theoretic properties. In particlar, we show that the heat semigroups generated by these are not completely positive, suggesting that perhaps on the CQG \(K_q\), stochastic processes that are most relevant to the geometry of it are not quantum Markov processes. This work is based on the preprint arXiv:2410.00720.
Hua Wang (Harbin Institute of Technology) - April 14, 2025
Title: A Theory of Locally Convex Hopf Algebras – I. Basic Theory and Examples.
Abstract: This is the first of two talks on a recent theory of locally convex Hopf algebras. After a brief introduction to some relevant facts on locally convex spaces as well as their topological tensor products, we will describe the main theory with an emphasis on duality. We will see that besides the usual strong dual, the theory encompasses naturally a new type of dual called the polar dual. After presenting the main theoretical results, we will illustrate the theory with various examples. In particular, we will see how to resolve the duality problem for classical Hopf algebras, how to describe a Lie group as well as its dual using smooth functions, and how to incorporate compact and discrete quantum groups into this framework.
Hua Wang (Harbin Institute of Technology) - April 21, 2025
Title: A Theory of Locally Convex Hopf Algbebras – II. More Duality Results and Examples.
Abstract: This is the second of two talks on a recent theory of locally convex Hopf algebras. We will start by presenting a generalized version of the Gelfand duality, and later apply it in various situations to obtain the underlying topological group from the corresponding locally convex Hopf algebras. Surprisingly, we can go much beyond the locally compact case in this classical situation, and make the theory work for all topological groups with compactly generated topology. Then we shift to some categorical considerations, allowing us to obtain new topological quantum groups as well as their dualities that seem not in the locally compact framework of Kustermans-Vaes. If time permits, we will conclude by mentionning how some deep structural results related to Hilbert’s fifth problem can be applied in this theory.
Farrokh Razavinia (Non Resident Researcher at IPM) - May 19, 2025
Title: \(C^*\)-graph algebras and beyond.
Abstract: Graph \(C^*\)-algebras have shown their importance in mathematics and other disciplines. For instance, recall the theory of quantum groups and quantum graphs, they can provide us with required structures in proving or disproving some interrelated problems. For example, in our recent papers, we showed their importance in looking at some very well-known wonder questions in mathematics from a different direction. In this talk, we will present some elementary definitions and results concerning graph \(C^*\)-algebras, and then we will try to study some constructive examples, and after that we will take a look at the concept of \(C^*\)-colored graph algebras, and finally we will see how these structures will help us to move into some very abstract mathematical objects!