On braided fusion categories II

Date:  15:20 - 16:55, Thu, Fri, 2025-10-09 ~ 2025-12-26
Venue: BIMSA, Room A3-3-301 (webpage of the course)
Zoom ID: 242 742 6089, Password: BIMSA

Prerequisite: Familiarity with the concept of fusion categories is assumed; however, key definitions and fundamental results will be reviewed. For further reading, please refer to [ENO05] and [EGNO15], as well as the videos and notes from the previous courses [I], [II], [III], [IV] and [B1] listed in the references.

Introduction: This course is a continuation of "On braided fusion categories" in [B1] and provides an in-depth exploration of the material presented in [DGNO10]. We begin with a comprehensive, self-contained overview of the key results on braided fusion categories, without assuming they are pre-modular or non-degenerate. The main focus of this course is to introduce the concept of the core of a braided fusion category, which allows us to identify the components of a braided fusion category that are not derived from finite groups.

Even if you did not attend last semester, you are welcome to join this course, as it will start with a review of the material from the previous semester.

References:
[DGNO10] Drinfeld, Vladimir; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. On braided fusion categories. I. Selecta Math. (N.S.) 16 (2010), no. 1, 1--119.
[ENO05] Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.

Link to the previous course on braided fusion categories:
[B1] http://bimsa.net/activity/BraFusCat/

Links to the previous courses on fusion categories:
[I] http://bimsa.net/activity/Fusion
[II] http://bimsa.net/activity/FusionII
[III] http://bimsa.net/activity/FusionIII
[IV] http://bimsa.net/activity/fuscatIV

On braided fusion categories

Date:  15:20 - 16:55, Thu, Fri, 2025-03-13 ~ 2025-06-12
Venue: BIMSA, Room A3-3-301 (webpage of the course)
Zoom ID: 242 742 6089, Password: BIMSA

Prerequisite: Familiarity with the concept of fusion categories is assumed; however, key definitions and fundamental results will be reviewed. For further reading, please refer to [ENO05] and [EGNO15], as well as the videos and notes from the previous courses [I], [II], [III], and [IV] listed in the references.

Introduction: This course closely examines the material in [DGNO10]. We begin by providing a thorough and self-contained overview of the known results on braided fusion categories, without assuming they are pre-modular or non-degenerate. Following this, the primary objective of this course is to introduce the concept of the core of a braided fusion category. This concept enables us to distinguish the components of a braided fusion category that are not derived from finite groups.

References:
[DGNO10] Drinfeld, Vladimir; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. On braided fusion categories. I. Selecta Math. (N.S.) 16 (2010), no. 1, 1--119.
[ENO05] Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.

Links to the relevant previous courses:
[I] http://bimsa.net/activity/Fusion
[II] http://bimsa.net/activity/FusionII
[III] http://bimsa.net/activity/FusionIII
[IV] http://bimsa.net/activity/fuscatIV

On braided fusion categories

Date:  15:20 - 16:55, Thu, Fri, 2025-03-13 ~ 2025-06-12
Venue: BIMSA, Room A3-3-301 (webpage of the course)
Zoom ID: 242 742 6089, Password: BIMSA

Prerequisite: Familiarity with the concept of fusion categories is assumed; however, key definitions and fundamental results will be reviewed. For further reading, please refer to [ENO05] and [EGNO15], as well as the videos and notes from the previous courses [I], [II], [III], and [IV] listed in the references.

Introduction: This course closely examines the material in [DGNO10]. We begin by providing a thorough and self-contained overview of the known results on braided fusion categories, without assuming they are pre-modular or non-degenerate. Following this, the primary objective of this course is to introduce the concept of the core of a braided fusion category. This concept enables us to distinguish the components of a braided fusion category that are not derived from finite groups.

References:
[DGNO10] Drinfeld, Vladimir; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. On braided fusion categories. I. Selecta Math. (N.S.) 16 (2010), no. 1, 1--119.
[ENO05] Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581--642.
[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.

Links to the relevant previous courses:
[I] http://bimsa.net/activity/Fusion
[II] http://bimsa.net/activity/FusionII
[III] http://bimsa.net/activity/FusionIII
[IV] http://bimsa.net/activity/fuscatIV

Burnside type results for fusion categories II

Date:  15:20 - 16:55, Thu, Fri, 2024-10-10 ~ 2024-12-27
Venue：BIMSA, Room A3-3-301 (webpage of the course)
Zoom ID：518 868 7656，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. This course serves as a continuation of the previous semester's course. Therefore, the initial sessions will be dedicated to a brief review of the material covered last semester.

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 categories, arXiv:2302.07604,
- 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.

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 categories, arXiv:2302.07604,
- 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. 

• 2009-2010: Full ATER (Teaching assistant, 192 h) 

• 2008-2009: 1/2 ATER (Teaching assistant, 96 h) 

• 2007-2008: Tutor of a blind student (120h) 

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