LIPN UMR CNRS-7030

Equipe CALIN

IHES, 35 Route de Chartres,
91440 Bures-sur-Yvette

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Organised par Gérard H.E. DUCHAMP, Maxim KONTSEVICH, Gleb KOSHEVOY and HOANG NGOC MINH


Combinatorics and Arithmetic for Physics: special days
06-07-08 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 computer-based implementation and experimentation.





Poster
Titles and Abstracts


- Program -

Wednesday 06 November

09h00-10h00


Welcome of the participants
10h00-10h50
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, non-symmetric 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 Ebrahimi-Fard, Loic Foissy and Joachim Kock.)
10h50-11h30


Tea / Coffee break
11h30-12h20
B. Delcroix-Oger
On the (non)-freeness of operads considered as pre-Lie algebras.
( pdf )
Operads can be naturally endowed with a structure of pre-Lie 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 pre-Lie algebras, we will explain why there exists no operads which would be a free pre-Lie algebra. We moreover present a bi-pre-Lie 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 (CNRS-LMBP)
12h20-13h10
R. Kerner
Z3-graded extension of Lorentz-Poincaré Algebra.
(pdf )

We propose a modification of standard QCD description of the colour triplet of quarks by introducing a 12-component colour generalization of Dirac spinor, withbuilt-in Z3-grading playing an important algebraic role in quark confinement. In “colour Dirac equations” the SU(3) colour symmetry is entangled with the Z3-graded generalization of Lorentz symmetry, containing three 6-parameter sec- tors related by Z3 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.
13h10-14h10


Lunch
14h10-15h00
G. Koshevoy
Schur positivity, cluster monomials and lattice polytopes.
(pdf )
12h20-13h10
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.
15h50-16h20


Tea / Coffee break
16h20-17h00
M. Kontsevich
Duality of hypergeometric functions.
Short communication.

Thursday 7 November

09h00-09h30


Welcome Tea / Coffee
09h30-10h20
K. A. Penson
Lévy-Stable 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.)

10h20-11h00


Tea / Coffee break
11h00-11h50
A. Sokal
Coefficientwise Hankel-total 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 = (ai+j)i,j≥0 of real numbers is totally positive if and only if the underlying sequence (an)n≥0 is a Stieltjes moment.(more on Titles and Abstracts ).
(Joint work with Mathias Pétréolle and Bao-Xuan Zhu.)
11h50-12h40
N. Behr
Experimental combinatorics via rule-algebraic 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 so-called 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 rewriting-based 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 so-called 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.
12h40-13h40


Lunch
13h40-14h05
G. Morgado
Wave fronts with cross-diffusion.
( 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.
14h05-14h55
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 finite-dimensional affine manifolds (more on Titles and Abstracts ).
14h55-15h10


Tea / Coffee break
15h10-16h00
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

09h00-09h30


Welcome Tea / Coffee
09h30-10h20
D. Gurevich
Quantum determinants.
( pdf )

I’ll discuss different approaches to defining analogs of determinants of matrices with entries belonging to some non-commutative algebras. In the first turn I am interested in the so-called quantum matrix algebras, Yangians-like algebras and similar ones, related to braidings (solutions to braid relation). I’ll also consider determinants of super-matrices and compare them with Berezinians.
10h20-11h00


Tea / Coffee break
11h00-11h50
P. Vanhove
The Calabi-Yau 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 Calabi-Yau sub-varieties 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 Griffiths-Dwork al- gorithm fails.
We will show how the results can be understood by understanding the geometry and the moduli of the Calabi-Yau varieties of the sunset graph. In particular the how specialization of physical parameters leads to rank jump.
We will explain the realization of Calabi-Yau pencils as Landau-Ginzburg models mirror to weak Fano varieties.
11h50-12h40
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 group-like 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).
12h40-13h40


Lunch
13h40-14h30
V. Hoang Ngoc Minh
Towards a noncommutative Picard-Vessiot theory (with simple applications).
( pdf )
We are constructing the first steps of a noncommutative Picard-Vessiot theory and illustrate this theory with the study of independences of a family of eulerian Gamma functions.
14h30-15h20
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 face-centered 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, so-called 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 self-assembly of supercrystals ...
15h20-16h00


Tea / Coffee break and farewell words.



- List of participants -

First Name Last Name Affiliation
Nicolas Behr Université de Paris, IRIF
Marc Bellon LPTHE- Sorbonne Université-CNRS
Joseph Ben Geloun LIPN, Univ. Paris XIII
Caroline Brembilla university Paris 11
Matteo D'Achille LIPN, Université Paris Nord, CIRB, Collège de France, Li-PARAD, UVSQ
Bérénice Delcroix-Oger 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 Sorbonne-Université
Maxim Kontsevich IHES
Pierre-Vincent Koseleff SU - IMJ-PRG - 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. Paris-Sud, Orsay
Goodenough Silvia université paris13
Alan Sokal University College London
Andrea Sportiello LIPN, Universtité Paris Nord
Jean-Yves Thibon Université Paris-Est Marne-la-Vallée
Christophe Tollu Université de Paris 13
Pierre Vanhove IPhT & HSE

Début