Fathi Ben Aribi     Version en français

Research Interests:

• Classical Knot Theory (genus, Alexander polynomials, triangulations of knot complements),
• L²-Invariants (Fuglede-Kadison determinant, L²-torsions, L²-Alexander torsions, L²-Burau maps),
• Hyperbolic Volume (for 3-manifolds with boundary),
• Quantum Invariants and Volume Conjectures (coloured Jones polynomials, Teichmüller TQFT, Turaev-Viro invariants, quantum hyperbolic invariants),
• Braid groups (Burau representations, Lawrence representations, L²-Burau maps),
• Group Theory (combinatorics of Cayley graphs, Lehmer-Lück problem).
• Decompositions of manifolds (triangulations, JSJ decomposition, handle decompositions, Heegaard splittings, trisections of 4-manifolds, higher dimensional analogues).

Preprints:

13. The Andersen-Kashaev volume conjecture for FAMED geometric triangulations with K.H. Wong), 45 pages, arXiv 2410.10776 (october 2024).
12. Multisections of higher-dimensional manifolds (with S. Courte, M. Golla and D. Moussard), 14 pages, arXiv 2303.08779 (march 2023).
11. The Chen-Yang volume conjecture for knots in handlebodies (with J. Gosselet), 34 pages, arXiv 2110.04225 (october 2021).
10. Link invariants from L²-Burau maps of braids, 24 pages, arXiv 2101.01678v4 (july 2021).

Publications:

9. Fuglede-Kadison determinants over free groups and Lehmer's constants, Confluentes Mathematici, 214, 1 (2022) 3--22. Here is the ArXiv preprint (february 2022).
8. Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots (with F. Guéritaud and E. Piguet-Nakazawa), Quantum Topology, 14 (2023), no. 2, pp. 285-406. Here is the ArXiv preprint (february 2020).
7. The leading coefficient of the L²-Alexander torsion (with S. Friedl and G. Herrmann), Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 1993-2035. Here is the ArXiv preprint (june 2018).
6. The Teichmüller TQFT Volume Conjecture for twist knots (with E. Piguet-Nakazawa), C. R. Math. Acad. Sci. Paris 357, 2019, no. 3, 299-305. Here is the preprint (october 2018).
5. L²-Burau maps and L²-Alexander torsions (with A. Conway), Osaka J. Math. 55 (2018), 529-545. Here is the ArXiv preprint (august 2016).
4. The L²-Alexander invariant is stronger than the genus and the simplicial volume, Journal of Knot Theory and Ramifications Vol.28, No.05, 1950030, 2019, 16 pages. Here is the ArXiv preprint (june 2016).
3. Gluing formulas for the L²-Alexander torsions, Communications in Contemporary Mathematics 1850013, 2018, 31 pages, World Scientific Publishing Company, DOI:10.1142/S021919971850013X. Here is the ArXiv preprint (march 2016).
2. The L²-Alexander invariant detects the unknot, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XV (march 2016), 683-708. Here is the ArXiv preprint (november 2013).
1. The L²-Alexander invariant detects the unknot, C. R. Math. Acad. Sci. Paris 351, 2013, no. 5-6, pages 215-219 (march 2013).

My PhD thesis:

My thesis is called "A study of properties and computation techniques of the L²-Alexander invariant in knot theory" and was done under the supervision of Jérôme Dubois. My defense was the 10th July 2015 at the Université Paris 7.

Recent invitations to international conferences:

• Quantum invariants and low-dimensional topology, San Jose, August 2023.
• Let's talk about outreach!, Les Diablerets, October 2022.
• AMS-SMF-EMS International Meeting, Grenoble, July 2022.
• Summer school on Alexander polynomials, Nantes, May-June 2022, Mini-course.
• Conference "Topology & Geometry of Low-dimensional Manifolds", Japan, online, January 2022.

Slides from recent talks:

• The Andersen-Kashaev volume conjecture for twist knots (and more!), Westlake Mathematics Seminar "Recent Progress on the Volume Conjecture", China (Zoom), October 2024.
• Some advice on preparing and giving talks, Université Lyon 1, France, June 2024.
• The Teichmüller TQFT volume conjecture for twist knots, AIM Workshop, Pasadena, August 2023.
• Déterminants de Fuglede-Kadison sur les groupes libres et constantes de Lehmer, Séminaire équipe ADA, Calais, online, October 2022.
• Fuglede-Kadison determinants over free groups and Lehmer's constants, Conference AMS-SMF-EMS International Meeting, special session "Low Dimensional Topology", Grenoble, July 2022.
• The Teichmüller TQFT volume conjecture for twist knots, Conference "Topology & Geometry of Low-dimensional Manifolds", Japon, online, January 2022.
• Link invariants from L²-Burau maps of braids, "Moduli and Friends" Seminar, Bucharest, November 2021.
• L²-torsion and hyperbolic volume, Conference CIRM Workshop "Homological and Quantum Invariants", Marseille, February 2021.

Thesis supervision:

In 2020-2021, at the UCLouvain, I co-supervised (with Pedro Vaz) the Master's thesis of James Gosselet, entitled " The Chen-Yang volume conjecture for knots in handlebodies".

Lecture Notes:

• Advanced Topics in Mathematics: La théorie des nœuds via le polynôme d’Alexander (in French), Master Course, UCLouvain, Spring 2021 (98 pages).

Advice for preparing Math talks:

In 2018 I did a survey of various types of advice for preparing and/or giving talks in mathematical research, and I summarised them in one big list of 50 tips. I hope this can be useful to you or someone you know!

Fictions for scientific outreach:

3. Mystère au Manoir Moutarde (in French), gamebook, 2022, ongoing project.
2. Mystère au Manoir Moutarde (in French), theater play, 2018 (see below).
1. Le noeud du mystère (in French), theater play, 2018 (see below).

Scientific Outreach (in French):

2020-2021: Researchers' night 2020, UCLouvain, with Science Infuse:
    Video promoting my research domain.
    Radio Interview at RTBF La Première.
    Café des sciences virtuel.
2019-2021: Referring researcher for the association MATh.en.JEANS, for 4 groups of high schoolers.
2018-2019: Scientific theater play "Mystère au Manoir Moutarde", within the governmental project DIP-Unige "Et si j'étais scientifique ?":
writing, directing, main acting role, for a total of 17 shows in front of 3500 audience members, including half of the schoolchildren of Geneva of ages 10-11.
    Article in French (CERN).
    Article in French (Scienscope).
    Article in English (SwissMAP).
    Video interview (SwissMAP).
    Radio show "Avis d'experts" (RTS).
2018: Scientific theater play "Le noeud du mystère", for the "Science Me" competition:
writing and directing, with a cast by Guillaume Bertoli, Adrien Laurent and Joana Lopes, and the help of Shaula Fiorelli.
    Video of the show.
2016-2019: Animator at the Mathscope, outreach workshops for all ages.
2016-2018: Referring researcher for the association MATh.en.JEANS, for 11 groups of high schoolers.