Address: Bureau S3 240
Laboratoire de Mathématiques Nicolas Oresme
Université de Caen - Normandie
BP 5186, 14032 Caen Cedex, France
E-mail: lebed (at) unicaen.fr
I am a Maître de conférences at the University of Caen Normandy.
You'll find here are my (rarely updated) CVs in English and in French.
Research interests
My interests revolve around the vast notions of structure and braiding, in particular their homological, categorical, and topological aspects. This includes the Yang–Baxter equation, braid groups, knots and their generalized versions (virtual, welded, twisted, handlebody, 2-dimensional), knotted graphs, quantum invariants,coloring (e.g., quandle) invariants, quantum (quasi-)shuffle algebras, braided systems, self- and muti-distributivity, Laver tables, cycle sets, Young tableaux, categorification of algebraic structures, Hopf algebras and Hopf bimodules, Yetter–Drinfel'd modules, Rota–Baxter algebras, operads, etc. This small video introduces some of these notions for non-mathematicians.
I like elegant and elucidating mathematics. Sharing it is a real pleasure. That is why I engage in outreach activities whenever possible. Currently I am a part of Labosaïque. In the past, I co-organized the Mathematics Club of the University of Nantes, and took part in the 5 minutes Lebesgue adventure.
Conferences co-organised
- October 2024, Caen. From Garside to Yang-Baxter.
- March 2024, Caen. Algebra Days in Caen 2024: Algebraic aspects of configuration spaces and moduli spaces.
- Juin–July 2023, Bedlewo. LOOPS'23.
- July 2022, Bedlewo. The algebra of the Yang–Baxter equation.
- March 2022, Caen. Algebra Days in Caen 2022: from Yang–Baxter to Garside.
- September 2021, Caen. Braids and beyond / Tresses et perspectives, Conference in memory of Patrick Dehornoy.
- November 2019, Oberwolfach. Mini-Workshop: Algebraic Tools for Solving the Yang–Baxter Equation.
- October 2019, Caen. Journées Normandes en Topologie.
- October 2017, Dublin. Geometry and combinatorics of associativity.
Teaching
I created a game (in French) to help my students revise basic notions of Liner Algebra. It includes game rules and printable cards. Feel free to use it or modify it as you need!
Papers, preprints, and other works
- Conjugation groups and structure groups of quandles, arXiv:2407.02955.
- (With Markus Szymik) The homology of permutation racks, to appear in Math. Z.
- (With Paolo Bellingeri and Hugo Chemin) Cactus groups, twin groups, and right-angled Artin groups, J. Algebr. Comb. 59, 153-178 (2024).
- (With Santiago Ramírez and Leandro Vendramin) Involutive Yang–Baxter: cabling, decomposability, Dehornoy class, Rev. Mat. Iberoamericana 40 (2024), no. 2, 623–635.
- The word problem for Hecke–Kiselman monoids of type An and A˜n, arXiv:2102.08647.
- (With Simon Covez, Marco Farinati, and Dominique Manchon) Bialgebraic approach to rack cohomology, Algebraic & Geometric Topology 23:4 (2023), 1551–1582.
- (With Leandro Vendramin) Reflection equation as a tool for studying solutions to the Yang–Baxter equation, J. Algebra 607, part B (2022), 360–380.
- (With Arnaud Mortier) Abelian quandles and quandles with abelian structure group, J. Pure Appl. Algebra 225 (2021), no. 1, 106474, 22 pp.
- Plactic Monoids: a Braided Approach, J. Algebra 564 (2020), 325–352.
- (With Leandro Vendramin) On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation, Proc. Edinb. Math. Soc. (2) 62 (2019), no. 3, 683–717.
- (With Scott Carter and Seung Yeop Yang) A Prismatic Classifying Space for Foam Homology, Nonassociative mathematics and its applications, Contemp. Math., 721 (2019).
- Applications of Self-Distributivity to Yang–Baxter Operators and Their Cohomology, J. Knot Theory Ramifications 27 (2018), no. 11, 1843012, 20 pp.
- (With Leandro Vendramin) Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation, Advances Math. 304 (2017), 1219–1261.
- (With Friedrich Wagemann) Representations of crossed modules and other generalized Yetter–Drinfel'd modules, Appl. Categ. Structures 25 (2017), no. 4, 455–488.
- Cohomology of Idempotent Braidings, with Applications to Factorizable Monoids, Int. J. Algebra Comput. 421 (2017), no. 27, 421–454.
- Braided Systems: a Unified Treatment of Algebraic Structures with Several Operations, Homology, Homotopy Appl. 19 (2017), no. 2, 141–174.
- (With Leandro Vendramin) Cohomology and extensions of braces, Pacific J. Math. 284 (2016), no. 1, 191–212.
- Cohomology of Finite Monogenic Self-Distributive Structures, J. Pure Appl. Algebra 220 (2016), no. 2, 711–734.
- (With Seiichi Kamada and Kokoro Tanaka) The Shadow Nature of Positive and Twisted Quandle Cocycle Invariants of Knots, J. Knot Theory Ramifications 24 (2015), no. 10, 1540001, 15 pp.
- Qualgebras and Knotted 3-Valent Graphs, Fund. Math. 230 (2015), no. 2, 167–204.
- Knotted 3-Valent Graphs, Branched Braids, and Multiplication-Conjugation Relations in a Group, Proceedings of Intelligence of Low-Dimensional Topology 2014: 86–100. pdf
- (With Patrick Dehornoy) Two- and Three-Cocycles for Laver Tables, J. Knot Theory Ramifications 23 (2014), no. 4, 1450017, 30 pp.
- R-Matrices, Yetter–Drinfel'd Modules and Braided Systems, Axioms 2013, 2(3), 443–476. pdf
- Categorical Aspects of Virtuality and Self-Distributivity, J. Knot Theory Ramifications 22 (2013), no. 9, 1350045, 32 pp.
- Symmetric Categories as a Means of Virtualization for Braid Groups and Racks, arXiv:1206.3916 (an extended version of the above JKTR publication, containing in articular a chapter on free virtual shelves and quandles).
- Homologies of Algebraic Structures via Braidings and Quantum Shuffles, J. Algebra 391 (2013), 152–92.
- The final version (in English) and a beamer (in French) of my PhD thesis. It was completed in 2012 at Paris 7 University, under the supervision of Marc Rosso.
- Invariants d'enchevêtrements avec des connexions plates dans le complémentaire, Master's dissertation, supervisor Marc Rosso, pdf in French
- (With V.Bernik) Algebraic Points on the Plane, Journal Math. Sciences 146 (2007), 5680–5685.
Upcoming talks
-
2025
- The Yang–Baxter equation and All That, Bedlewo, Poland
Past talks
-
2008
- Conference Algebraic Geometry and Representation Theory, Minsk, Belarus. Introductory talks for students: Knot theory; Hopf algebras 2009
- Working group Quantum Algebra and Topology, IMJ (Institut Mathématique de Jussieu), Paris: 2-categories associated with quantum sl(2). Notes
- Mathematical conference, Minsk, Belarus. Introductory talk for students: Introduction to abelian categorification 2010
- Working group Quantum Algebra and Topology, IMJ: Diagrammatic categorification of quantum sl(n). Notes
- A talk Who is who in the family of Homologies at the informal PhD student seminar, IMJ
- Co-organizing the working group Quantum Groups, IMJ; two talks Quantum knot invariants. Notes 2011
- Co-organizing the working group Canonical Bases, IMJ; two talks Canonical bases in the gl(n) case and their connection to Young diagrams. Notes, notes 2
- Working group Geometric Representation Theory, IMJ: Category O: the action of the center. Notes
- PhD Student Seminar, IMJ: Forget the group multiplication to get a weak universal knot invariant. Notes
- A talk Topological complexity of algorithms at the informal PhD student seminar, IMJ. Notes 2012
- PhD Students' Day, IMJ: Braidings, algebraic structures and homologies: an entangled story
Working
group Semantics, laboratory PPS (Preuves, Programmes,
Systèmes),
IMJ: Homologies
of Algebraic Structures via Braidings. Notes- A talk Billiards: a set of balls rounding down the number pi at the informal PhD student seminar, IMJ. Notes
- Algebra Seminar, IHP (Institut Henri Poincaré), Paris: Braided systems: generalities and applications to Hopf theory. Notes
- Young Mathematicians' Forum, IHP: Homologies of algebraic structures: a unifying "braided" approach. Notes
- Topology Seminar, GWU, Washington DC: Self- and multi-distributivity with a braided flavor. Notes
- Knots in Washington XXXV: Hochschild, Chevalley-Eilenberg and quandle homologies are braided homologies
- Algebra Seminar, University Lyon 1: Braided systems: generalities and applications to Hopf theory 2013
- Algebraic Topology Seminar, University Paris 13: Koszul, Hochschild, Leibniz and quandle homologies are braided homologies
- Colloque tournant du GDR TLAG, University of Cergy-Pontoise: A unifying study of algebraic structures using "braided" tools. Notes
- Algebra, Topology and Geometry Seminar, University of Nice: Koszul, Hochschild, Leibniz and quandle homologies are braided homologies
- Topology Seminar, University Montpellier 2: Categorical aspects of virtuality. Notes
- TQFT
and Categorification Seminar, IMJ: Categorical
aspects of virtuality
- British Mathematical Colloquium, Sheffield, UK: Associative Algebras, Bialgebras and Leibniz Algebras as Braided Objects. Beamer
- Algebra and Topology Seminar, University of Strasbourg: Associative algebras, bialgebras and Leibniz algebras as braided objects
- Knots, Manifolds and Group Actions, Słubice, Poland: Distributive homology is a braided homology. Notes
- Topology Seminar, University of Grenoble: Braids and homology of algebraic structures: a round trip
- KOOK Seminar, Osaka: Braids and homology of algebraic structures: a round trip. Notes 2014
- Joint Mathematics Meetings AMS, Special Session on Algebraic Structures Motivated by Knot Theory, Baltimore, USA: Self-, multi- and G-distributivity with a braided flavor. Beamer
- Knots in Washington XXXVII: Qualgebras and knotted 3-valent graphs. Beamer
- Logic and topology seminar, GWU, Washington DC: Towards topological applications of Laver tables. Notes
- Intelligence of low-dimensional topology, Kyoto: Knotted 3-valent graphs, branched braids, and multiplication-conjugation relations in a group. Beamer
- Topology Seminar, Tsukuba University, Japan: A bridge between knotted graphs and axiomatizations of groups. Beamer
- Friday Seminar, Osaka City University: Towards braid-theoretic applications of Laver tables. Notes and hand-outs
- TAPU-KOOK Seminar, Daejeon, Korea: Categorical aspects of virtuality. Beamer
- Topology Symposium, Tohoku University, Japan: Laver Tables: from Set Theory to Braid Theory. Beamer and (very) extended abstract
- Knots and Low Dimensional Manifolds, Busan, Korea: Laver tables: topological applications of set-theoretic constructions. Beamer
- FMSP Lectures, University of Tokyo: A bridge between knotted graphs and axiomatizations of groups. Beamer
- Topology, geometry and algebra seminar, University of Nantes: Braids and homology of algebraic structures: a round trip. Notes
- Topology and geometry seminar, University of Geneva: Braids and homology of algebraic structures: a round trip
- Symposium on Mathematical Physics, University of Zürich: New aspects of the Yang-Baxter equation. Beamer
- Working group Algebraic topology, University of Nantes: Braid groups and free groups. Notes
- Algebra and geometry seminar, University of Caen: Laver tables: set-theoretic, algebraic and topological aspects. Beamer
- Topology seminar, University of Montpellier: Braids and homology of algebraic structures: a round trip 2015
- Knots in Dallas: Twisted multi-distributivity and Lawrence representations of braid groups. Beamer
- Algebra and topology seminar, University of Strasbourg: Lawrence representations of braid groups via self-distributivity. Notes
- Joint topology seminar of Paris 7 and Paris 13 Universities: Lawrence representations of braid groups via self-distributivity
- Algebraic topology seminar, Université catholique de Louvain, Belgium: Braided diagrams as a unifying tool in homology theory. Notes
- University of Nantes: Cohomology of Finite Monogenic Self-Distributive Structures. Notes
- New trends in
Hopf algebras and tensor categories, Bruxelles: On braidings, Yetter-Drinfel’d modules,
crossed modules, and self-distributivity
- Lens topology and geometry meeting, Lens: Braided aspects of the Yang-Baxter equation. Beamer
- Talks for the working group Lawrence-Krammer-Bigelow representations of braid groups, Nantes. Notes
- Workshop Applied Representation Theory, Amiens: Representations of braid groups: homological and self-distributivity approaches
- Geometry and Algebra, Geometry and Analysis, Utrecht, Netherlands: Unexpected facets of the Yang-Baxter equation. Beamer
- Algebra and geometry seminar, Caen: On the set-theoretic solutions to the Yang-Baxter equation. Notes
- Algebra and Group Theory, Mulhouse: On set-theoretic solutions to the Yang-Baxter equation. Beamer
- Mathematics Club, Nantes: Origamics, or 折り紙. Notes
- Mathematics
seminar, Vannes: Representations of braid groups via
self-distributivity
- University of Nantes: A decomposition theorem for skew cubical homology. Notes 2016
- Workshop on Hopf algebras, Turin, Italy: On set-theoretic solutions to the Yang-Baxter equation. Beamer
- Topology Seminar, Grenoble: Representations of braid groups via self-distributivity
- Seminar of mathematical physics and algebraic topology, Angers: On set-theoretic solutions to the Yang-Baxter equation
- Mathematics Club, Nantes: Billiard as a tool for computing the digits of the number π. Notes
- Mathematics Club, Nantes: Erdős distinct distances problem, or Vote for diversity!
- 3+1
dimensional topological phases of matter, Leeds, UK: Crossed modules and beyond. Beamer
- XXI Coloquio Latinoamericano de Álgebra, Buenos Aires, Argentina: On set-theoretic solutions to the Yang-Baxter equation. Beamer
- Algebra seminar, Buenos Aires, Argentina: Cohomology of idempotent solutions to the Yang-Baxter equation, with applications
- Mathematics, Statistics and Applied Mathematics Seminar, Galway, Ireland: How forgetting group laws leads to a universal knot invariant. Notes
- Mathematics Colloquium, TCD, Dublin, Ireland: Unexpected facets of the Yang-Baxter equation. Beamer
- La réunion annuelle du GDR topologie algébrique et applications, Amiens, France: Équation de Yang-Baxter, tableaux de Young, factorisations de groupes. Beamer
- Semantics seminar, Paris: What do braids know about Young tableaux? Beamer
- Cohomological Methods in Geometry, Freiburg, Germany: How braids can help to compute Hochschild cohomology. Beamer
- Marburg, Germany: What do braids know about Young tableaux? Beamer
- Algebra days 2017 in honour of Patrick Dehornoy, Caen, France: A journey into the green book. Beamer
- MathSoc, TCD, Ireland: Mathematics of Origami. Poster
- CINVESTAV, Mexico:
- How forgetting group laws leads to a universal knot invariant (Colloquium). Beamer
- Yang–Baxter equation, Young tableaux, and group factorizations (Algebra Seminar)
- Self-distributive system and quandle (co)homology theory in algebra and low dimensional topology. Busan, South Korea: Applications of self-distributivity to Yang–Baxter operators and their cohomology. Beamer
- Groups, Rings and the Yang–Baxter Equation, Spa, Belgium: Structure groups of YBE solutions: new properties, and cohomological applications. Beamer
- 4th Mile High Conference on Nonassociative Mathematics, Denver, USA:
- Self-distributive cohomology: Why care, and how to compute. Beamer
- More algebraic connections: SD and the Yang–Baxter equation, Leibniz algebras etc. Beamer
- Categories in Homotopy and Rewriting, CIRM, France: Yang–Baxter equation up to homotopy
Yang–Baxter equation, knots, cohomology: a golden triangle
2017 
- Cohomology of set-theoretic solutions to the Yang–Baxter equation
- The structure groups of racks. Notes
