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

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

 09h30-10h00 Welcome of the participants 10h00-10h50 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. 10h50-11h10 Tea / Coffee break 11h10-12h00 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). 12h00-12h50 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. 12h50-14h30 Lunch / Free time 14h30-15h20 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: Lengths functions L i , i = 1, 2 related to numbers of inversions and connected components on the permutation (Coxeter) groups (W, S). Positive definite functions of the Poisson type P i (x) = exp(−L i (x)), x ∈ W . Generalized CCR relations related to the Weyl groups of type A, B, D and new type II factorial von Neumann algebras. Riesz product on (W, S) and operator spaces of row and columns related to arbitrary Coxeter groups (W,S). 15h20-15h40 Tea / Coffee break 15h40-16h30 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) 16h30-17h20 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

 09h30-10h00 Welcome Tea / Coffee 10h00-10h50 Gleb Koshevoy Geometric Kashiwara star duality and DT transformations 10h50-11h10 Tea / Coffee break 11h10-12h00 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. 12h00-12h50 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. 12h50-14h00 Lunch / Free time 14h00-14h50 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. 14h50-15h40 Dimitry Grigoryev Tropical combinatorial Nullstellensatz and fewnomials ( pdf ) We give tropical analogues of the combinatorial Nullstellensatz due to N. Alon and of Risler-Ronga conjecture; 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; a universal testing set for sparse polynomials (for classical polynomials con- structed by Grigoriev-Karpinski and Ben-Or-Tiwari); Shub-Smale τ-conjecture. These results were obtained jointly with V. Podolskii. 15h40-16h00 Tea / Coffee break 16h00-16h50 Natalia Iyudu Sklyanin algebras via Groebner bases and finiteness conditions for potential algebras ( pdf ) 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. 16h50-17h40 Pierre Cartier A Combinatorial Presentation of Various Galois Theories

Début

List of participants

• Hélène Airault (Univ. de Picardie, Lamfa, CNRS 7352)
• Daniel Barsky (LAGA, Paris XIII)
• Nicolas Behr (IRIF Université Paris Diderot)
• Joseph Bengeloun (LIPN, Paris XIII)
• Alin Bostan (INRIA Saclay)
• Marek Bozejko (University of Wroclaw)
• Pierre Cartier (IHES)
• Gérard H. E. Duchamp (IHP & Paris 13)
• Justine Falque (Orsay Paris XI)
• Nihar Gargava (EPFL, Lausanne)
• Dimitri Grigoryev (CNRS-Lille 1)
• Dmitry Gurevich (Université de Valenciennes)
• Natalia Iyudu (University of Edinburgh & IHES)
• Richard Kerner (LPTMC)
• Gleb Koshevoy (Poncelet Lab., Moscow)
• Christian Lavault (LIPN, Paris XIII)
• Jerzy Lukierski (LPTMC Paris VI)
• Jean-Marie Maillard (LPTMC Paris VI)
• Ho\ang Ngoc Minh (Lille 2 & LIPN)
• Karol Penson (LPTMC & Paris 6)
• Alban Quadrat (Inria Lille - Nord Europe)
• Vincent Rivasseau (Orsay Paris XI)
• Nicolas Thiéry (Orsay Paris XI)
• Christophe Tollu (LIPN, Paris XIII)
• Pierre Vanhove (CEA/Saclay)