Equipe CALIN
GDR CNRS Renormalisation
IHES, Bures sur Yvette, 24-25 October 2018



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.

- Program -

Wednesday 24 October


Welcome of the participants
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 one-to-one correspondence with three colors and three anti-colors. 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 particle-antiparticle symmetry, we arrive at 12-component 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 non-standard complex realizations of the Lorentz group is also discussed.

Tea / Coffee break
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 well-defined. These quantum symmetric polynomials generate commutative subalgebras called Bethe. Also, I plan to exhibit some quantum analogs of the classical identities (Cayley-Hamilton, Newton).
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.

Lunch / Free time
Marek Bozejko
Positive definite functions on permutation (Coxeter) groups with applications to generalized CCR-relations and operator spaces ( pdf ) In my talk we will consider the following topics:
  1. Lengths functions L i , i = 1, 2 related to numbers of inversions and connected components on the permutation (Coxeter) groups (W, S).
  2. Positive definite functions of the Poisson type P i (x) = exp(−L i (x)), x ∈ W .
  3. Generalized CCR relations related to the Weyl groups of type A, B, D and new type II factorial von Neumann algebras.
  4. Riesz product on (W, S) and operator spaces of row and columns related to arbitrary Coxeter groups (W,S).

Tea / Coffee break
K.A. Penson
Integer ratios of factorials as Hausdorff moments versus algebraicity ( pdf )
Consider two series of positive integers: a = a1 , a2 , . . . , aK and b = b1 , b2 , . . . , bK , bK+1 , with Σ ai=Σ bi; K = 1, 2, ....
We form the following ratios of factorials

un(a, b) = (a1.n)!(a2 .n)! . . . (aK.n)! / (b1.n)!(b2.n)! . . . (bK+1.n)!   (1)
for n = 0, 1 . . .

It turns out that, for many choices of a and b, the ratios un(a, b) in (1) are themselves integers. In these cases we conceive un(a, b) as nth moments of the weight functions W(a, b, x) in the Hausdorff moment problem

un(a, b) = ∫0R(a,b) xn.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 G-functions and generalized hypergeometric functions. We prove formally the positivity of the weights W (a, b, x) which are all U-shaped 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 un(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)
Nicolas Behr
Operational Methods in the Study of Sobolev-Jacobi 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 multi-variate version of the so-called 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 Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based 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


Welcome Tea / Coffee
Gleb Koshevoy
Geometric Kashiwara star duality and DT transformations

Tea / Coffee break
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.
Hoang Ngoc Minh
A family of Eulerian functions involved in regularization of divergent polyzetas ( pdf )
Eulerian functions are most significant for analytic number-theory 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 Newton-Girard 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.

Lunch / Free time
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 Gromow-Witten local prepotential. We prove this conjecture for the case of the two-loop graph. By considering the limiting mixed Hodge structure of the Batyrev dual A-model, we arrive at an expression for the two-loop sunset Feynman integral in terms of the local Gromov-Witten prepotential of the del Pezzo surface of degree 6.
Dimitry Grigoryev
Tropical combinatorial Nullstellensatz and fewnomials
We give tropical analogues of
  1. the combinatorial Nullstellensatz due to N. Alon and of Risler-Ronga conjecture;
  2. Schwartz-Zippel lemma providing an exact bound on the number of points in a finite grid at which a polynomial of a fixed degree can vanish;
  3. a universal testing set for sparse polynomials (for classical polynomials con- structed by Grigoriev-Karpinski and Ben-Or-Tiwari);
  4. Shub-Smale τ-conjecture.
These results were obtained jointly with V. Podolskii.

Tea / Coffee break
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, Calabi-Yau, etc., depending on the parameters of the Sklyanin algebras. There was a gap in the Artin-Schelter 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.
Pierre Cartier
A Combinatorial Presentation of Various Galois Theories


List of participants