Teaching

Burnside type results for fusion categories

Date:  15:20 - 16:55, Thu, Fri, 2024-03-14 ~ 2024-06-14
Venue：BIMSA, Room A3-3-301 (webpage of the course)
Zoom ID：242 742 6089，Password: BIMSA

Prerequisite: Be a bit familiar with the finite group representation theory, and the notions of fusion categories, fusion rings and hypergroup, but the definitions will be recalled.

Abstract：We extend a classical vanishing result of Burnside from the character tables of finite groups to the character tables of commutative fusion rings, or more generally to a certain class of abelian normalizable hypergroups. We also treat the dual vanishing result. We show that any nilpotent fusion categories satisfy both Burnside's property and its dual. Using Drinfeld's map, we obtain that the Grothendieck ring of any weakly-integral modular tensor category satisfies both properties. As applications, we prove new identities that hold in the Grothendieck ring of any weakly-integral fusion category satisfying the dual-Burnside's property, thus providing new categorification criteria. We also prove some new results on the perfect fusion categories.

References：
- S. Burciu, S. Palcoux. Burnside type results for fusion rings, arXiv:2302.07604, [in a future version, "rings" should be replaced by "categories"],
- P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor categories, volume 205. Mathematical Surveys and Monographs, AMS, Providence, RI, 2015,
- P. Etingof, D. Nikshych, and V. Ostrik. On fusion categories. Ann. of Math. (2), 162(2):581–642, 2005,
- P. Etingof, D. Nikshych, and V. Ostrik. Weakly group-theoretical and solvable fusion categories. Adv. Math., 226(1):176–205, 2011,
- S. Gelaki and D. Nikshych. Nilpotent fusion categories. Adv. in Math., 217(3):1053–1071, 2008.

Triangular Prism Equations and Categorification

Date:  15:20 - 16:55, Thu, Fri, 12/10/2023 - 19/01/2024
Venue：BIMSA, Room A3-2-303 (webpage of the course)
Zoom ID：293 812 9202，Password: BIMSA

Prerequisite:  Be a bit familiar with the notion of monoidal category. Otherwise, consider the review part beginning the last semester course (videos and slides are available here or on YouTube) and EGNO book in reference.

Abstract：This course introduces to the pictorial reformulation of the Pentagon Equations (of a monoidal category) called Triangular Prism Equations, and apply it to the classification program of simple integral fusion categories, but also multiplicity-free or near-group fusion categories.

References：
- P. Etingof, D. Nikshych, V. Ostrik, On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
- P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, Tensor categories. Mathematical Surveys and Monographs, 205 (2015), xvi+343 pp. Online Book.
- Z. Liu, S. Palcoux, Y. Ren, Triangular prism equations and categorification, arXiv:2203.06522, 33 pages, under review.
- Z. Liu, S. Palcoux, Y. Ren, Interpolated family of non group-like simple integral fusion rings of Lie type (with Zhengwei Liu and Yunxiang Ren), Internat. J. Math. 34 (2023), no. 6, Paper No. 2350030, 51 pp.
- Z. Liu, S. Palcoux, Y. Ren, Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six, Lett. Math. Phys. 112 (2022), no. 3, Paper No. 54, 37 pp.
- Z. Liu, S. Palcoux, Y. Ren, Polynomial Equations for Irrational Near Group Categories, work in progress.

On Fusion Categories IV

Date:  15:20 - 16:55, Thu, Fri, 16/03/2023 - 09/06/2023
Venue：BIMSA, Room 1137 (webpage of the course)
Zoom ID：537 192 5549，Password: BIMSA

Abstract：This is the sequel of the course "On Fusion Categories III" given last semester. It introduces to the notion fusion category, which can be seen as a representation theory of the (finite) quantum symmetries. It follows the book "Tensor Categories" mentioned in References. The notes and videos of the previous parts are available at Part I, Part II, Part III, or in this YouTube playlist.
If you did not attend the previous semesters, you can still attend this one because the course will start by a review of the previous parts.  

References：
Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205 (2015), xvi+343 pp. Online Book.

On Fusion Categories III

Date:  15:20 - 16:55, Mon,Tue, 9/13/2022 - 12/12/2022
Venue：BIMSA, Room 1120 (webpage of the course)
Zoom ID：518 868 7656，Password: BIMSA

Abstract：This is the sequel of the course "On Fusion Categories II" given last semester. It introduces to the notion fusion category, which can be seen as a representation theory of the (finite) quantum symmetries. The notes and videos of the previous parts are available at Part I, Part II, or in this YouTube playlist.

References：
Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205 (2015), xvi+343 pp. Online Book.

On Fusion Categories II

Date:  15:20 - 16:55, every Monday, Thursday, 3/14/2022 - 6/9/2022
Venue：BIMSA, Room 1120 (webpage of the course)
Zoom ID：388 528 9728，Password: BIMSA

Abstract：This is the sequel of the course "On Fusion Categories" given on first semester. It introduces to the notion fusion category, which can be seen as a representation theory of the (finite) quantum symmetries. The notes and videos of the first part are available here or this YouTube playlist.

References：
Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205 (2015), xvi+343 pp. Online Book.

On Fusion Categories

Date:  Every Monday & Tuesday 2021-09-13~12-03, 15:20-16:55
Venue：BIMSA, Room 1120 (webpage of the course)
Zoom ID：638 227 8222，Password: BIMSA

Abstract：This course will introduce to the notion fusion category, which can be seen as a representation theory of the (finite) quantum symmetries.

References：
Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205 (2015), xvi+343 pp. Online Book.

Quantum Symmetries and Quantum Arithmetic

Material: slides, videos.
Date：Monday & Tuesday, 2021-02-22~05-11, 15:20-16:55
Venue：BIMSA, Room 1120 (webpage of the course)
Abstract：This mini-course will introduce to the Galois-like symmetries of von Neumann algebras: subfactor and planar algebra, analogous of field extension and Galois group, which highly generalizes the notion of (finite) quantum group.
In this framework, we will generalize Ore's theorem (about cyclic groups), extend the prime and natural numbers, generalize the Euler's totient function, and apply to the representation theory of finite groups.  A quantum Riemann hypothesis is stated at the end. 

Previous teaching experience

I taught three years in France, in the Mathematics Department of Luminy (Aix-Marseille University), between 2007 and 2010. 

Subjects taught: Arithmetic, algebra, analysis, discrete mathematics, probability and statistics, differential geometry.