Practical information
The workshop takes place in
CIRM and
online from February 8th to February 12th 2021. On CIRM's
webpage of the event, you can (pre)register and book your room and meals. Please make sure to contact the organizers vefore booking anything.
Presentation of the workshop
The aim of this workshop is to continue the study of
lowdimensional topology invariants built in parallel from
quantum groups or homologies of configuration spaces, and to
observe their relations. Recent results indeed relate the Verma modules
of
U_{q}(𝖘𝖑
_{2})
very precisely to configuration space homologies on
punctured disks.
On the one hand, recipes for constructing quantum invariants
are known but their topological content is often lost,
giving rise to numerous conjectures (volume conjecture,
conjecture AMU etc.). On the other hand, the
methods for constructing invariants from configuration
spaces are less developed while the homological information
remains apparent. The rich algebraic structures coming from
both homological tools (Poincaré duality, cup and cap
product, etc.) and representation theory (tensor product,
decomposition into weight spaces, etc.) should shed light on
each other. Also, the generalization of this homological
approach to invariants coming from other quantum groups is
still little known. Finally, it is reasonable to think that
this interpretation in terms of configuration spaces will
bring new point of views on quantum knot polynomials
and their
categorifications.
Participants
 Léo BÉNARD (University of Göttingen)
 Anna BELIAKOVA (University of Zurich)
 Giulio BELLETTI (University of Heidelberg)
 Fathi BEN ARIBI (Université Catholique de Louvain)
 Tara BRENDLE (University of Glasgow)
 Christian BLANCHET (IMJ–PRG, Paris)
 Boris COLOMBARI (AixMarseille Université)
 Francesco COSTANTINO (Université de Toulouse Paul Sabatier)
 Marco DE RENZI (University of Zurich)
 Renaud DETCHERRY (Université de Bourgogne)
 Quentin FAES (Institut Mathématiques de Bourgogne)
 Vincent FLORENS (Université de Pau et des Pays de l'Adour)
 Emmanuel GRAFF (Université Caen Normandie)
 John GUASCHI (Université Caen Normandie)
 Benjamin HAÏOUN (Université de Toulouse Paul Sabatier)
 Zhicheng HAN (University of Göttingen)
 Louis IOOS (Tel Aviv University)
 Alexandra KJUCHUKOVA (Notre Dame University)
 Yohan MANDINHUBLÉ (Institut Fourier)
 Jules MARTEL (Max Planck Institute for Mathematics, Bonn)
 Delphine MOUSSARD (AixMarseille Université)
 Cristina PALMERANGHEL (University of Oxford)
 Martin PALMERANGHEL (Mathematical Institute of the Romanian Academy)
 LouisHadrien ROBERT (Université du Luxembourg)
 Adrien RODAU (Université de Pau et des Pays de l’Adour)
 Ramanujan SANTHAROUBANE (Université Paris Sud)
 Thomas SCHICK (University of Göttingen)
 Arthur SOULIÉ (University of Glasgow)
 AnneLaure THIEL (Université de Caen)
 Fiona TORZEWSKA (University of Leeds)
 Vladimir VERCHININE (Université de Montpellier)
 Emmanuel WAGNER (IMJ–PRG Paris)
 Paul WEDRICH (MPIM and Universität Bonn)
 Sonny WILLETTS (Université de Toulouse Paul Sabatier)
Schedule
A large part of the workshop is dedicated to informal discussions.
All hours are given in
CETtime (UTC+1).


10:30 – 12:00, Sonny Willets: Unifying quantum 𝔰𝔩_{2} knot invariant. [Wi20]. Slides.

14:00 – 15:00, Benjamin Haïoun:
Quantum 𝔰𝔩_{N} invariant. Slides.

15:30 – 16:30, Emmanuel Graff:
A homological definition of the HOMFLY polynomial. [Bi07]

17:30 – 19:00, Stephen Bigelow: Constructing quantum groups from homology of configuration spaces.
 I will give a
construction of the Drinfeld–Jimbo quantum
group corresponding to any simple Lie
algebra. It is similar to Martel's
construction
of U_{q}(𝔰𝔩_{2}),
but using configurations of points that are
labelled by simple roots.


09:00 – 10:30, Renaud Detcherry: Volume conjecture for colored Jones invariant. [HuLe, MuMu]

11:00 – 12:00, Informal discussions.

14:30 – 16:00, Fathi Ben Aribi: Relation betwenn L^{2}torsion and hyperbolic volume for cusped manifolds. [LüSc]. Slides.

16:30 – 17:30, Informal discussions.

Thursday

09:45 – 11:15, Cristina PalmerAnghel: Coloured Jones and Alexander polynomials through Lagrangian intersections in configuration spaces. [Ang]. Slides, Video.

11:30 – 12:30, Informal discussions.

14:00 – 15:00, Marco De Renzi: Nonsemisimple quantum invariants of 3manifolds from the Kauffman bracket. [DRMu]. This talk is cohosted by [KOS]. Video.

16:30 – 18:00, Emmanuel Wagner: Grid diagrams and Jones polynomial. [DrWa]. Video.
Friday

09:00 – 10:30, Christian Blanchet: Models for the Jones polynomial and categorification. Video.

11:00 – 12:00, Informal discussions.
References

[AbSm] M. Abouzaid, I. Smith,
Khovanov homology from Floer cohomology,
Journal of the American Mathematical Society, 32 (2019).

[Ang] C. PalmerAnghel,
Coloured Jones and Alexander polynomials through Lagrangian intersections in configuration spaces,
arXiv:2002.09390.
 [AoKi] K. Aomoto, M. Kita,
Theory of hypergeometric Function,
Springer Monographs in Mathematics (2011). Available at CIRM library
 [Big04] S. Bigelow,
Homological representations of the Iwahori–Hecke algebra,
Geometry and Topology Monographs, 7 (2004).
 [Big07] S. Bigelow,
A homological definition of the HOMFLY polynomial,
Algebraic and Geometric Topology, 7 (2007).
 [BVW] V. G. Bardakov, V. V.Vershinin, J. Wu,
On homotopy braids, arXiv:2008.07806.
 [Co14] V. Colin,
Réalisations géométriques de l'homologie de Khovanov par des homologies de Floer,
Séminaire Bourbaki 66ème année, 1079 (2014).
 [DrWa] J.M. Droz, E. Wagner,
Grid diagrams and Khovanov homology,
Algebraic and Geometric topology, 9 (2009).
 [DRMu] Nonsemisimple 3manifold invariants derived from the Kauffman bracket, Quantum topology (2021).
 [HuLe] V. Huynh, T. T. Q. Lê,
On the colored Jones polynomial and the Kashaev invariant,
Fundamentalnaya i Prikladnaya Matematika, 11:5 (2005).

[JaKe] C. Jackson, T. Kerler,
The Lawrence–Krammer–Bigelow representations of the braid groups via quantum sl_{2},
Advances in Mathematics, 228 (2011).

[Ko12] T. Kohno,
Quantum and homological representations of braid groups,
Configuration Spaces  Geometry, Combinatorics and Topology, Edizioni della Normale (2012).

[LüSc] W. Lück, T. Schick,
L^{2}torsion of hyperbolic manifolds of finite volume,
Geometric & Functional Analysis GAFA, 9 (1999).

[Ma06] C. Manolescu,
Nilpotent slices, Hilbert schemes, and the Jones polynomial,
Duke Mathemactical Journal ,132 (2006).
 [Ma20] J. Martel,
A homological model for U_{q}(𝔰𝔩_{2}) Vermamodules and their braid representations,
arXiv:2002.08785.
 [MuMu] H. Murakami, J. Murakami,
The colored Jones polynomials and the simplicial volume of a knot,
Acta Mathematica, 186 (2001).
 [We16] P. Wedrich,
Categorified 𝔰𝔩_{N} invariants of colored rational tangles,
Algebraic and Geometric Topology 161 (2016).
 [Wi20] S. Willets,
A unification of the ADO and colored Jones polynomials of a knot,
arXiv:2003.09854.
Attend online
Except for one talk (see below), we will use the BigBlueButton server from the CIRM.
The only requirement to attend the talk online is an uptodate webbrowser (preferably chrome, chromium or firefox). If you forgot or did not receive the password, send an email to the organizers.
Before you enter the room, you'll be asked if you want to use camera and microphone. We strongly suggest you to answer yes. Note that once you are in the room, you'll be muted and you can switch you camera on or off. CIRM's documentation about BBB for participants and speakers. Link to the worskhop room: https://bbb1.cirmmath.fr/b/orgzj4ygh.
During informal discussion the above mentionned BBB room will be open. An extra room will be open, its address is: https://bbb1.cirmmath.fr/b/orguvl04kyec. Feel free to use it. Its password has been sent to partipants by email.
The talk from Marco De Renzi on February 11th will be hosted on the Zoom room of [KOS]. Details for connections are available here.
Note that for ecological reasons, we do
not plan to ship bouillabaisse to online
participants.
Organizers