Equipe
CALIN
GDR CNRS
Renormalisation
IHES, Bures sur Yvette, 2425 October 2018
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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
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 implantation and experimentation.
Wednesday 24 October
09h3010h00 
Welcome of the participants 

10h0010h50

Richard Kerner

Quarks as a combinatorial puzzle: a new approach to quantum chromodynamics ( pdf ) 
Quarks cannot propagate outside the nucleons or mesons, but inside they seem to behave at high energies
as almost free particles. An alternative approach to color dynamics is proposed.
It is based on the observation that the $Z_2 \times Z_3 = Z_6$ cyclic group generated by the sixth root
of unity can be put into onetoone correspondence with three colors and three anticolors.
If we identify 0 as ``colorless", or `white", there are two ternary and three binary combinations of roots
yielding zero. Combining spin with color, and including particleantiparticle symmetry, we arrive at
12component objects, satisfying generalized Dirac equation whose solutions cannot propagate alone
due to the complex wave vectors, but can form propagating combinations via ternary or binary products.
Relativistic invariance realized via nonstandard complex realizations of the Lorentz group is also discussed.

10h5011h10 
Tea / Coffee break 

11h1012h00 
Dimitri Gurevich 
Symmetric polynomials in quantum algebras ( pdf )  I'll
introduce some new quantum algebras which are called Generalized
Yangians. Their definition is based on the notion of compatible
$R$matrices. In these algebras quantum analogs of some
symmetric polynomials (elementary ones, power sums) are
welldefined. These quantum symmetric polynomials generate
commutative subalgebras called Bethe. Also, I plan to exhibit
some quantum analogs of the classical identities
(CayleyHamilton, Newton).

12h0012h50 
Alin Bostan 
Transcendence in the enumeration of lattice walks ( pdf ) 
Classifying lattice walks confined to the quarter plane is an important prob lem in enumerative combinatorics. A key role in the classification is played by an associated group of birational transformations, whose finiteness turns out to be equiv alent to the differential finiteness of the corresponding generating function. We will give an overview of recent results on structural properties and explicit formulas for those generating functions, with an emphasis on their transcendence. 
12h5014h30 
Lunch / Free time 

14h3015h20 
Marek Bozejko 
Positive definite functions on permutation (Coxeter) groups with applications to generalized CCRrelations and operator spaces ( pdf )  In my talk we will consider the following topics:

15h2015h40 
Tea / Coffee break 

15h4016h30 
K.A. Penson 
Integer ratios of factorials as Hausdorff moments versus algebraicity ( pdf ) 
Consider two series of positive integers: a = a_{1} , a_{2} , . . . , a_{K} and b = b_{1} , b_{2} , . . . , b_{K} , b_{K+1} , with Σ a_{i}=Σ b_{i}; K = 1, 2, .... We form the following ratios of factorials u_{n}(a, b) = (a_{1}.n)!(a_{2} .n)! . . . (a_{K}.n)! / (b_{1}.n)!(b_{2}.n)! . . . (b_{K+1}.n)! (1) for n = 0, 1 . . . It turns out that, for many choices of a and b, the ratios u_{n}(a, b) in (1) are themselves integers. In these cases we conceive u_{n}(a, b) as nth moments of the weight functions W(a, b, x) in the Hausdorff moment problem u_{n}(a, b) = ∫_{0}^{R(a,b)} x^{n}.W (a, b, x)dx, where R(a, b) is the upper edge of the support [0, R(a, b)]. We solve exactly and explicitly the above Hausdorff moment problem via the inverse Mellin transform method thus providing the analytic forms of R(a, b) as well as of W (a, b, x) in terms of Meijer Gfunctions and generalized hypergeometric functions. We prove formally the positivity of the weights W (a, b, x) which are all Ushaped and singular at both edges of the support; as such they are generalizations of the arcsin distributions. We discuss a potential link between the proven algebraicity of the ordinary generating functions of u_{n}(a, b) and a possible algebraicity of corresponding weights W(a, b, x). (Joint work with G. H. E. Duchamp, N. Behr and G. Koshevoy) 
16h3017h20 
Nicolas Behr 
Operational Methods in the Study of SobolevJacobi Polynomials ( pdf ) 
Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multivariate version of the socalled umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of SobolevJacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normalordering techniques introduced by Gurappa and Panigrahi, twovariable Hermite polynomials and our integralbased series transforms. This is joint work with G. Dattoli (ENEA Frascati), G.H.E. Duchamp (Paris 13), Silvia Licciardi (ENEA Frascati) and K.A. Penson (Paris 6). 
Thursday 25 October
09h3010h00 
Welcome Tea / Coffee 

10h0010h50

Gleb Koshevoy

Geometric Kashiwara star duality and DT transformations 

10h5011h10 
Tea / Coffee break 

11h1012h00 
Gérard Duchamp 
Combinatorics of characters, Schützenberger's calculus and continuation of Li ( pdf ) 
We start from a new territory, that of (noncommutative) formal power series, to encode polylogarithms and harmonic sums. In this talk, we introduce the object(s), notations and calculus devoted to this very particular Sweedler's dual. In passing, we pay a small tribute to Marcel Paul Schützenberger. 
12h0012h50 
Hoang Ngoc Minh 
A family of Eulerian functions involved in regularization of divergent polyzetas ( pdf ) 
Eulerian functions are most significant for analytic numbertheory and are largely involved in Probably and in Physical sciences (Gamma and Beta densities). In this work, we give an extension of these functions and their relationship with the several parameter zeta function. In particular, starting with the Weierstrass factorization (and the NewtonGirard identity) for Gamma function, we are interested in the ratio of $\zeta(2k)/\pi^{2k}$ and we will obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas. This will be done via the combinatorics of noncommutative rational power series. 
12h5014h00 
Lunch / Free time 

14h0014h50 
Pierre Vanhove 
Mirror symmetry and Feynman integrals ( pdf )  We will describe the connection between Feynman integrals and period integrals of motivic cohomology. We will detail the toric approach to the evaluation of the Feynman integral. We formulate the conjecture that the all loop sunset Feynman integral in two spacetime dimensions is given by the genus zero GromowWitten local prepotential. We prove this conjecture for the case of the twoloop graph. By considering the limiting mixed Hodge structure of the Batyrev dual Amodel, we arrive at an expression for the twoloop sunset Feynman integral in terms of the local GromovWitten prepotential of the del Pezzo surface of degree 6. 
14h5015h40 
Dimitry Grigoryev 
Tropical combinatorial Nullstellensatz and fewnomials 
We give tropical analogues of

15h4016h00 
Tea / Coffee break 

16h0016h50 
Natalja Iyudu 
Sklyanin algebras via Groebner bases and finiteness conditions for potential algebras 
I will discuss how some questions on Sklyanin algebras can be solved using com binatorial techniques, namely, the theory of Groebner bases (rewriting techniques in the ideals of associative algebras). We calculate the Poincaré series, prove Koszulity, PBW, CalabiYau, etc., depending on the parameters of the Sklyanin algebras. There was a gap in the ArtinSchelter classification of algebras of global dimension 3, where Koszulity and the Poincaré series for Sklyanin algebras were proved only generically. It was filled in the Grothendieck Festschrift paper of Artin, Tate and Van den Bergh, using the geometry of elliptic curves. Our point is that we recover these results by purely algebraic, combinatorial means. We use similar methods for other potential algebras as well, including homology of moduli of pointed curves given by Keel relations, and contraction algebras arising in noncommutative resolution of singularities. 
16h5017h40 
Pierre Cartier 
A Combinatorial Presentation of Various Galois Theories 

List of participants