Christopher-Lloyd SIMON

List of refereed works (mostly articles accepted in a peer-review journal).
They are grouped by themes and choronological order. The dates refer to those of my first prepreprint.
The preprint versions (often on arxiv) may have more appealing layouts
The corresponding talks are collected in the following rubrick.

• Topology and combinatorics of curves in surfaces
  • On the topology of a real analytic curve in the neighborhood of a singular point (with Etienne Ghys)
    Journal: Astérisque 415, 2020, p. 1-33. Preprint: link to PDF.
  • Topologie et dénombrement des courbes algébriques réelles
    Journal Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 2, pp. 383-422. Preprint: arXiv.
  • Loops in surfaces, chord diagrams, interlace graphs: operad factorisations and generating grammars
    Preprint: arXiv.
• Character varieties of surface groups
  • Automorphisms of character varieties (with Julien Marché)
    Journal Annales Henri Lebesgue 4 (2021) 591-603 Preprint: arXiv.
  • Valuations on the character variety: Newton polytopes and Residual Poisson Bracket (with Julien Marché)
    Journal: Geometry & Topology, to appear. Preprint: arXiv.
• Arithmetic and topology of the modular group
  • Arithmetic and Topology of modular knots
    PhD Thesis, under the direction of Etienne Ghys and Patrick Popescu-Pampu.
    A compressed manuscript and the version on HAL.
    
  • Conjugacy classes in PSL(2;K)
    This article synthetises and extends the results in chapter 1 of my thesis.
    Journal: Mathematics Research Reports, Volume 4 (2023), pp. 23-45. Preprint: arXiv.
  • Linking numbers of modular knots
    This article synthetises and extends the results in chapter 4 and 5 of my thesis.
    Journal: Geometry & Topology, to appear. Preprint: arXiv.
• Arithmetics and Dynamics of continued fractions
  • Hexponential dynamics and transcendence (with Scott Schmieding)
    Preprint: arXiv.

Some of the slides i used for beamer talks are collected below, and maybe some recordings.
They are grouped by topics and then appear in chronological order.
There are variations on the same theme, adapted for different audiences and durations.

• Miscelaneous talks
  Some slides in French about the Théorème de Brower via le Lemme de Sperner for the Séminaire de la détente, ENS Lyon.
• Topology and combinatorics of plane real analytic curves
  Some slides in French about Combinatorics of analytic chord diagrams, for the Séminaire de Combinatoire, ENS Lyon.
  Some slides in French about the Geometry of real analytic singular curves, for the Séminaire de Géométrie des Espaces Singuliers, Laboratoire Painlevé, Lille.
  More slides in French about the Algebraic geometry of real analytic singular curves, for the Séminaire Géométrie et Topologie IMJ-PRG, Paris.
  Some slides in English about the Topology and enumeration of real algebraic curves, for the Conference in enumerative, real and birational geometry, Le Croisic.
  
• Character varieties of surface groups.
  Some slides in French about Automorphisms of character varieties, for the Séminaire de Géométrie complexe, Nancy.
  Some slides in English about Newton Polygons and Poisson brackets, for the GAP seminar, PSU and the SSAAEC in Bedlewo, Poland.
  
• Arithmetic and topology of modular knots.
  The slides in French about Arithmetic intersection and topological linking of modular knots, for my PhD defense, in Lille.
  Similar slides in French about Intersection arithmétique et enlacement topologique des noeuds modulaires, for the SVF Groupes et Géométrie, UJF.
  Similar slides in English, for the Department of Mathematics Colloquium, PSU and the Geometry/Topology Seminar, Brown University.
  Other slides in English, about Quasicharacters of the modular group from linking numbers of modular knots for a short talk
  in the Conference on Monodromy and its Applications, Princeton in honor of Nicholas Katz.
  

Some of my works in progress are listed below, and the most mature come along with some preprints.
I welcome (m)any discussions about those topics, and you are welcome to contact me to exchange about them.

• Chord diagrams of loops in surfaces
  Part 1: operad factorisations and generating grammars (submitted)
  Part 2: Hopf algebra and polynomial invariants
  Part 3: homotopy groupoids and cluster algebras
• Pin the loop and other topologames
  A new algorithm for computing intersection of loops in a punctured sphere
  The group of m-trivial loops in F(n)
  Intersection functions on free groups and the lower central series
• Continued fractions, and symbolic dynamics
  Dynamics and transcendance of the Hexponential map
  Rational equivalence of rotation numbers for subshifts
• Linking numbers of modular knots and scalar products of modular forms
  Representations of the adelic modular group: equidistribution of structure coefficients and periods of automorphic forms
  Gauss linking forms for lengendrian knots
• Mapping class groups of mapping tori
• Surgery distances between 3-manifolds

Various expository notes from courses, conference proceedings, holiday gatherings, personal reflections, ...
Some are unfinished (like many things including this website), but they may still be useful.
They are organized by type and appear in chronological order.

• Undergraduate research reports and presentations
  • Etude statistique des éléments du groupe symmétrique
    The research report of my TIPE for applying to the ENS Lyon.
    It concerns the distribution of cycle lengths appearing in the factorisation of random permutations as the order of the symmetric group tends to infinity.
    In retrospect, most of the results follow from Polya's counting theorem, the symbolic method for generating functions, and asymptotic analysis.
  • Symmetries et invariants en geometrie algébrique
    A naïve and short open-ended reflection about Hilbert's invariant theory.
  • Modularité des courbes elliptiques
    The report of my 1 month research experience in 2016 at IRMAR with Jean-Pierre Wintenberger.
  • Topology and geometry of curves in hyperbolic surfaces
    The report of my 3 month research experience in 2017 at Stony Brook with Moira Chas, Tony Phillips and Dennis Sullivan.
  • Birational geometry and Jung's theorem after Stéphane Lamy
    This presentation of an article by Stéphane Lamy was written for my class mates of the master program in geometry.
• Personnal notes and accounts of topics
  • Dynamics and classical entropies of Z-actions
    These lectures notes of a short master course in 2018 by Mikael de la Salle were written with, and mostly by, Matthieu Joseph.
  • Survival Guide to First Order Logic
    These unfinished notes on first order logic were written with, and mostly by, Tristan Sterin.
  • Elementary number theory and quadratic arithmetics
    Various notes and thoughts on elementary number theory and arithmetics of quadratic forms, mostly after
    the book Topology of Numbers by Allen Hatcher, the book A Classical Invitation to Algebraic Numbers and Class Fields by Harvey Cohn,
    the article Sur la théorie de la divisibilité by Jean Dieudonné, and the Salam lectures on Modular forms by Don Zagier.
  • Géométrie des algèbres de quaternions réelles
    These personal notes were written for, and in retrospect of, a mathematical week during the summer 2020
    organised with my friends Felipe Bellio da Nobrega, Martin Donati, Colin Jahel, Matthieu Joseph, Mickael Nahon, Quentin Vila.
  • Algebras of loops in surface orbifolds and volumes in moduli spaces
    Some almost finished notes about geodesic currents and character varieties of surface groups.
• Conference Symposiums and Surveys
  • Introduction au calcul différentiel quantique et aux idées de la géométrie non commutative
    I organised the traditional week-end du Goutelas in 2018 with Nalini Anantharaman, and in 2019 with Alain Connes.
    For the latter I coordinated the edition of the notes in a special issue of the JME journal.
    You can access the prepublication and the JME publication.

Informatic notebooks serving as experimental basis for conjecturing or checking results.

• Pin the loop (with Ben Stucky)
  A first jupyter notebook (2024) computing topological intersetion number of a loop in a punctured sphere.
  A second jupyter notebook (2024) computing the pinning poset of a loop in a punctured sphere.
• ChorDiaGraPoloops (with Mayank Yadav)
  A first jupyter notebook (2023) computing various invariants of chord diagrams and their interlace graphs
  to compare various invariants in particular the genus and the spectrum.
  A second jupyter notebook (2023) with a naive implementation of the Cunningham decomposition for graphs.
  
• Arithmetic and topology of modular knots
  A first jupyter notebook (2022) containing various experimentations
  comparing the arithmetic of quadratic extensions and the topology of modular geodesics.
  A second jupyter notebook (2023) computing higher Poisson brackets of modular geodesics.
• Loops in the punctured torus and Markov numbers (with Suzanne Schlish)
  A jupyter notebook (2020) to decompose a multicurve inside the punctured torus into the basis of simple multicurves .

Various outreach documents. They appear in chronological order.

• Topolo-geometrical entries to the blog of the 2017 geometry workshop at Stony Brook.
  While taking part in the 2017 geometry workshop held by Moira Chas, Tony Phillips and Dennis Sullivan,
  I wrote a couple of blog entries about Möbius, Euler and Dirichlet series, iterating Thurston's simple curve, moduli space of hyperbolic metrics, Dehn twists.
• An online article in the expository journal Image des mathématiques about elastic energy and bouncing balls.
• A podcast interview by Etienne Ghys for his podcast series Petites histoires de sciences.