Yoonje Jeong

Explanation An expository talk at the KIAS APP Seminar on July 25, 2025. Title Free Probability in Operator Algebras Abstract Free probability has proved to be an unexpectedly powerful tool for longstanding questions in operator algebras. In this expository talk, based on the speaker’s personal selection, we will survey three examples that illustrate the theme: Metric characterisations of freeness:  Beginning from Kesten’s result for group von Neumann algebras, we will see how Cadilhac and Collins generalized this result in the setting of free probability, giving a new route to embeddings of free group factors. Haagerup–Thorbjørnsen’s Strong Convergence Result:  We will see how their work for strong convergence of suitable unitary random-matrices was able to solve a problem in operator algebras. Ultrapowers, free entropy and asymptotic freeness:  We discuss how free entropy and asymptotic freeness results can probe the elementary isomorphism problems for II_1 factors.

Explanation A research presentation at the 2025 CJK Conference on Functional Analysis on July 10, 2025. Title Probabilistic approaches to von Neumann algebras generated by discrete (quantum) groups Abstract An W*-probability space is defined as a von Neumann algebra equipped with a faithful normal tracial state, with group von Neumann algebras serving as typical examples. In this talk, we discuss how probabilistic methods in group von Neumann algebras—such as the study of moments of (non-commutative) random variables—can shed light on the structure of underlying discrete groups. As a generalization, we introduce analogous results in the setting of discrete quantum groups.

Explanation A series of introductory seminars on free probability theory in June ~ August 2025. Title Seminar on Free Probability Abstract This series of seminar will follow the content of following two textbooks. [1] James Mingo and Roland Speicher. Free Probability and Random Matrices. Springer, 2017. [2] Alexandru Nica and Roland Speicher. Lectures on the Combinatorics of Free Probability. Cambridge University Press, 2006.

Explanation A research talk at the SNU Operator Seminar on June 20 and June 24, 2025. Title Traicial ultraproducts of von Neumann algebras and free probability Abstract Two von Neumann algebras are said to be elementary equivalent if they have a pair of isomorphic ultrapowers. In this series of talks, we review how the authors of [2] proved that the II_1 factor, which was constructed in [1] and proven to be without the Gamma property, is not elementary equivalent to any free product of diffuse tracial von Neumann algebras. On the first day, we review basic terminologies of free probability theory and how they are related to (tracial) von Neumann algebras. I will also introduce the problem and main results of [2]. On the second day, we continue to follow the main steps of the proof for the results, particularly focusing on how the language of free probability was used. [1] I. Chifan, A. Ioana, S. Kunnawalkam Elayavalli, An exotic II_1 factor without property Gamma. To appear in Geom. Fu...

Explanation A research talk at the young session of NPRML 2025 at the Institute of Statistical Mathematics (ISM) on March 27, 2025.  Title An analytic characterization of freeness for finitely generated discrete quantum groups Abstract The group algebras, which are non-commutative probability spaces associated with finitely generated discrete quantum groups, have been successful in studying how free the underlying groups are. One can characterize `freer' groups by investigating the moments of the self-adjoint main characters, which are specific elements of the group algebras. In 1959, Kesten proved that the norms of these elements determine the `freest' group. The result was generalized in the framework of free probability in 2022 by Collins and Cadilhac. In this talk, we will see how the concepts of `freer' and `freest' can be captured similarly in the wider category of finitely generated discrete quantum groups.

Explanation A series of peer talks in Fall 2024 at Seoul National University Title Free probability: an operator algebraic probability theory Abstract This talk will begin by reviewing the basic principle of operator algebra theory. As an important application, we begin our study of the non-commutative probability spaces, e.g., an algebra of random matrices. Freeness in non-commutative probability theory is an analog of independence in a classical probability theory (i.e., theory with a probability measure). We will learn some noticeable applications (random matrices & central limit theory) and Voiculescu's motivation for freeness. If time permits, we will see the connection between free probability theory and operator algebras, particularly quantum groups. Title An introduction to compact matrix quantum groups Abstract In this talk, we will breifly review operator algbera and free probability theory. Then, we introduce compact matrix groups and their generalization, compact ma...

Explanation A series of research talks in April 2024 at the SNU Operator Seminar Title A metric characterization of freeness Abstract The Free Probability theory is a branch of operator algebra theory studying a non-commutative analogy of independence in classical probability theory. Although it mainly deals with analytical objects, its definition and properties are usually algebraic and combinatorial. However, in 2021, L. Cadilhac and B. Collins discovered an analytical necessary and sufficient condition of freeness for Haar unitary elements in a finite von-Neumann algebra. The purpose of these talks is to give details of this characterization.