Equipe
CALIN
IHES, 35 Route de Chartres,
91440 BuressurYvette
.
Organised par Gérard H.E. DUCHAMP, Maxim KONTSEVICH, Gleb KOSHEVOY and HOANG NGOC MINH
Combinatorics and Arithmetic for Physics: special days
060708 November 2019
The meeting’s focus is on questions of discrete mathematics and number theory with an emphasis on computability. Problems are drawn mainly from theoretical physics (renormalisation, combinatorial physics, geometry) or related to its models. Computation, based on combinatorial structures (graphs, trees, words, automata, semirings, bases) or classic structures (operators, Hopf algebras, evolution equations, special functions, categories) are good candidates for computerbased implementation and experimentation.
Poster
Titles and Abstracts
 Program 
09h0010h00 
Welcome of the participants 

10h0010h50

F. Patras

Operads of partitions,
interacting bialgebras, and planar QFT.
( pdf ) 
There are (at least) two approaches to relations between full and connected Green’s
functions in planar QFT: a combinatorial one, based on the lattice of noncrossing
partitions, an algebraic one, based on shuffle products, exponentials and loga
rithms. We establish and explore a relationship between two approaches. We
achieve this by exhibiting two operad structures on (noncrossing) partitions, dif
ferent in nature: one is an ordinary, nonsymmetric operad whose composition law
is given by insertion into gaps between elements, the other is a coloured, symmet
ric operad with composition law expressing refinement of blocks. We show that
these operad structures interact so as to make the corresponding incidence bialge
bra of the former a comodule bialgebra for the latter (more on
Titles and Abstracts
). (Joint work with Kurusch EbrahimiFard, Loic Foissy and Joachim Kock.) 
10h5011h30 
Tea / Coffee break 

11h3012h20 
B. DelcroixOger 
On the (non)freeness of operads considered as preLie algebras. ( pdf ) 
Operads can be naturally endowed with a structure of preLie algebra. We present
here a work in progress with E. Burgunder and D. Manchon on the freeness of op
erads endowed with this structure. After introducing the basic notions on operads
ans preLie algebras, we will explain why there exists no operads which would be
a free preLie algebra. We moreover present a bipreLie structure on the linear
span of planar rooted trees, linked with the magmatic operad, which is not free
but generated by only two elements.
Work in progress with Emily Burgunder (IMT) and Dominique Manchon (CNRSLMBP) 
12h2013h10 
R. Kerner 
Z_{3}graded extension of LorentzPoincaré Algebra. (pdf ) 
We propose a modification of standard QCD description of the colour triplet of quarks by introducing a 12component colour generalization of Dirac spinor, withbuiltin Z_{3}grading playing an important algebraic role in quark confinement. In “colour Dirac equations” the SU(3) colour symmetry is entangled with the Z_{3}graded generalization of Lorentz symmetry, containing three 6parameter sec tors related by Z_{3} maps. The generalized Lorentz covariance requires simultaneous presence of 12 colour Dirac multiplets which lead to the description of all internal symmetries of quarks: besides SU(3) x SU(2) x U(1), the flavour symmetries and three quark families. 
13h1014h10 
Lunch 

14h1015h00 
G. Koshevoy 
Schur positivity, cluster monomials and lattice polytopes. (pdf ) 

12h2013h10 
M. Bellon 
Relating Taylor expansions and sums over poles: a source of poyzeta identities. (pdf ) 
The evaluation of Feynman diagrams with arbitrary powers of the propagators gives rise to quite special meromorphic functions of many variables. In a the sim plest case of the one loop diagram, I show how the Taylor expansion and the sum over poles can be related through a collection of poyzeta identities best expressed in terms of generating functions. 
15h5016h20 
Tea / Coffee break 

16h2017h00 
M. Kontsevich 
Duality of hypergeometric functions. 
Short communication. 
Thursday 7 November
09h0009h30 
Welcome Tea / Coffee 

09h3010h20 
K. A. Penson 
LévyStable Distributions: Mathematical Properties and Explicit Representations. ( pdf ) 
The evaluation of Feynman diagrams with arbitrary powers of the propagators gives rise to quite special meromorphic functions of many variables. In a the sim plest case of the one loop diagram, I show how the Taylor expansion and the sum over poles can be related through a collection of poyzeta identities best expressed in terms of generating functions (more on Titles and Abstracts ). (Joint work with K. Gorska and A. Horzela, both at IFJ, PAN, Cracow and G. Dat toli at ENEA, Frascati, Roma.) 
10h2011h00 
Tea / Coffee break 

11h0011h50 
A. Sokal 
Coefficientwise Hankeltotal positivity. ( pdf ) 
A matrix M of real numbers is called totally positive if every minor of M is nonnegative. Gantmakher and Krein showed in 1937 that a Hankel matrix H = (a_{i+j})_{i,j≥0} of real numbers is totally positive if and only if the underlying sequence (a_{n})_{n≥0} is a Stieltjes moment.(more on Titles and Abstracts ). (Joint work with Mathias Pétréolle and BaoXuan Zhu.) 
11h5012h40 
N. Behr 
Experimental combinatorics via rulealgebraic methods. ( pdf ) 
A computational approach to combinatorics will be presented that is based upon techniques from the realm of rewriting theories over finitary adhesive and exten sive categories. In the scenario of a given combinatorial structure being spec ified in terms of a generator and an initial configuration, the approach permits to perform ”experiments” on the combinatorial structure in the sense that equa tions describing pattern count distributions of various kinds may be derived in a principled manner. The core mathematical structure of our methodology is the socalled rule algebra, an associative unital algebra encoding sequential composi tions of rewriting steps. The representation theory of the rule algebras as well as elements of the stochastic mechanics framework for rewritingbased continuous time Markov chains form additional building blocks of our framework, revealing en passant close conceptual relationships between problems in combinatorics and in dynamical systems. After providing the general theory, some illustrative re sults for the application example of planar binary rooted trees generated by the socalled Rémy generator will be presented (joint work with N. Zeilberger), in cluding the discovery of a certain form of bisimulation structure for the evolution equations of pattern count distributions. 
12h4013h40 
Lunch 

13h4014h05 
G. Morgado 
Wave fronts with crossdiffusion. ( pdf ) 
In this talk, we present some results concerning the propagation of a chemical wave front ruled out by partial differential (evolution) equations in the space of concentrations. 
14h0514h55 
A. Kiselev 
Poisson bracket deformations using (un)oriented graphs: open problems. ( pdf ) 
We formulate several open problems from the theory of universal  by using the calculus of (un)oriented graphs  infinitesimal symmetries of Poisson brackets on arbitrary finitedimensional affine manifolds (more on Titles and Abstracts ). 
14h5515h10 
Tea / Coffee break 

15h1016h00 
D. Grigoryev 
On a tropical version of the jacobian conjecture. ( pdf ) 
We prove, for a tropical rational map, that if for any point the convex hull of Ja cobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism. This is a joint work with Danylo Radchenko, ETH (Zürich). 
Friday 08 November
09h0009h30 
Welcome Tea / Coffee 

09h3010h20

D. Gurevich

Quantum determinants. ( pdf ) 
I’ll discuss different approaches to defining analogs of determinants of matrices with entries belonging to some noncommutative algebras. In the first turn I am interested in the socalled quantum matrix algebras, Yangianslike algebras and similar ones, related to braidings (solutions to braid relation). I’ll also consider determinants of supermatrices and compare them with Berezinians. 
10h2011h00 
Tea / Coffee break 

11h0011h50 
P. Vanhove 
The CalabiYau geometry of the sunset Feynman graphs. ( pdf ) 
In this talk we will discuss the algebraic and transcendental features of the com putation of multiloop sunset Feynman integrals. Starting from the realization of arbitrary Feynman graph hypersurfaces as (generalized) determinantal varieties, we describe the CalabiYau subvarieties of permutohedral varieties that arise from the multiloop sunset Feynman graphs and some key features of their geometry and moduli. We will explain how the “creative telescoping” algorithm allows to derive the inhomogeneous differential equation when the standard GriffithsDwork al gorithm fails. We will show how the results can be understood by understanding the geometry and the moduli of the CalabiYau varieties of the sunset graph. In particular the how specialization of physical parameters leads to rank jump. We will explain the realization of CalabiYau pencils as LandauGinzburg models mirror to weak Fano varieties. 
11h5012h40 
G. Duchamp 
A localization principle for the Basic Triangle Theorem. ( pdf ) 
The Basic Triangle Theorem, in its classical form, provides a necessary and suf ficient condition for coefficients (local coordinates in the grouplike case) of so lutions of a NCDE (Non Commutative Differential Equation) to be linearly free with respect to a subfield. As this primitive form requires a differential subfield it is, in practice, not easy to handle. In the realm of analytic functions, for instance, one has to cope with germs. In the one of infinitely differentiable functions, it is even worse : a blunt localization kills almost all interesting phenomena. In this talk we give a ready to use form of this theorem with applications the polylogarithms and diverse sub rings of interest. This is a joint work with Hoang Ngoc Minh (LIPN, Paris) and Nihar Gargava (EPFL, Lausanne). 
12h4013h40 
Lunch 

13h4014h30 
V. Hoang Ngoc Minh 
Towards a noncommutative PicardVessiot theory
(with simple applications). ( pdf ) 
We are constructing the first steps of a noncommutative PicardVessiot theory and illustrate this theory with the study of independences of a family of eulerian Gamma functions. 
14h3015h20 
T. Fernique 
Compact packings of spheres. ( pdf ) 
It is well known that the best way to pack oranges in a (very large) box is to place them on a facecentered cubic lattice (also known as checkerboard), although this has been formally demonstrated only in 1998 (with difficulty). This talk focuses on what happens when the dimension or number of different spheres change. In particular, socalled compact packings (the term will be defined properly) seem good candidates to maximize density. In this tallk, we propose a survey of the known mathematical results and a discussion of possible applications in chem istry, including selfassembly of supercrystals ... 
15h2016h00 
Tea / Coffee break and farewell words. 
 List of participants 
First Name
Last Name
Affiliation
Nicolas
Behr
Université de Paris, IRIF
Marc
Bellon
LPTHE Sorbonne UniversitéCNRS
Joseph
Ben Geloun
LIPN, Univ. Paris XIII
Caroline
Brembilla
university Paris 11
Matteo
D'Achille
LIPN, Université Paris Nord, CIRB, Collège de France, LiPARAD, UVSQ
Bérénice
DelcroixOger
Université de Paris, IRIF
Gérard
Duchamp
IHP and LIPN, Univ. Paris XIII
Thomas
Fernique
CNRS & Univ. Paris 13
Stéphane
Gaubert
INRIA and CMAP Ecole Polytechnique
Dima
Grigoriev
CNRS
Alla
Grigorieva
SPGU
Dmitry
Gurevich
Valenciennes University
Vincel
Hoang Ngoc Minh
Université de Lille
Richard
Kerner
SorbonneUniversité
Maxim
Kontsevich
IHES
PierreVincent
Koseleff
SU  IMJPRG  INRIA
Gleb
Koshevoy
ISCP, Moscow
Christian
Lavault
LIPN, Un. Paris 13
Annie
Lemarchand
CNRS Sorbonne University
Claire
Levaillant
IHES, Paris
Pierre
Lochak
cnrs
Gabriel
Morgado
Sorbonne Université
Frédéric
Patras
CNRS Université Côte d'Azur
Karol
Penson
Sorbonne Université, LPTMC, 4 pl. Jussieu, 75005 Paris
Vincent
Rivasseau
LPT, Univ. ParisSud, Orsay
Goodenough
Silvia
université paris13
Alan
Sokal
University College London
Andrea
Sportiello
LIPN, Universtité Paris Nord
JeanYves
Thibon
Université ParisEst MarnelaVallée
Christophe
Tollu
Université de Paris 13
Pierre
Vanhove
IPhT & HSE