abstracts and slides of 3 talks in TIFR Bombay, Mumbai, India, February 2013 «3^n» Colloquium in Theoretical Physics, 26 February 2013 slides (pdf, 38 Mo, 273p) abstract: We revisit some old problems of directed animals and compact source animals on a square lattice, whose number was found to be 3^n, and recent extensions with multidirected animals. The connection of such problems with Lorentzian quantum gravity will be discussed. A bijective proof (that we call the "Nordic decomposition" of a heap of dimers) for the formula for multidirected animals, which at the same time give some extensions for Lorentzian triangulations, will be presented. Besides physicists, the topic should be of interest to computer scientists. For computer scientists, we use the model of heaps of pieces, which is equivalent to the concept of trace in computer science, used as model for concurrency access to data structures. The bijective problems show some philosophical considerations about algorithmic constructions of bijections which have a computer science flavor. «The Strahler analysis of binary trees in computer science and in other sciences» Computer science seminar, 27 February 2013 slides (pdf, 33 Mo, 220p) abstract: Computer scientists defined the Strahler number of a binary tree in relation with the minimum number of registers needed for the computation of an arithmetical expression. Beautiful asymptotic analysis for the average Strahler number have been done (Flajolet, Vuillemin, Raoult, Kemp), involving a periodic function coming from number theory. Surprisingly, this parameter appears in hydrogeology (Horton, Strahler) for the morphologic study of river networks, and also in molecular biology in the study of RNA secondary structures (Waterman). Applications have been made in computer graphics (synthetic images of trees), the study of some fractal structures in experimental physics, in radiology and in the domain of visualization of informations. Underlying this asymptotic analysis, there are deep combinatorial mathematics, and some new structures have been introduced, in collaboration with D.Knuth, called Kepler towers. These objects belongs to the "heaps of pieces" theory, which gives a geometric interpretation of equivalence classes of words in the so-called "trace monoid" introduced in computer science by Mazurkiewicz as a model for concurrency access to data structures. «Quadratic algebras, combinatorial physics and planar automata» Random interactions seminar, 28 February 2013 slides (pdf, 16 Mo, 239p) abstract: For certain quadratic algebras Q, we introduce the concept of Q-tableaux, which are certain combinatorial objects drawn on the square lattice. These tableaux are equivalent to the notion of a planar automaton. Planar automata is a new concept (not to be confused with cellular automata) which formalizes the idea of recognizing certain "planar figures" drawn on a 2D lattice. Two quadratic algebras well known in physics are good examples of planar automata: the most simple Weyl-Heisenberg algebra defined by the commutation relation UD=DU+Id (creation-annihilation operators in quantum mechanics) and the so-called PASEP algebra defined by the relation DE=ED+E+D, in the physics of dynamical systems far from equilibrium. The associated Q-tableaux are respectively rook placements, permutations and the so-called alternating, tree-like and permutation tableaux. Other examples include non-crossing configurations of paths, tiling, plane partitions and alternating sign matrices.

$abstracts and slides of 3 talks in TIFR Bombay, Mumbai, India, February 2013 «3^n» Colloquium in Theoretical Physics, 26 February 2013 slides (pdf, 38 Mo, 273p) abstract: We revisit some old problems of directed animals and compact source animals on a square lattice, whose number was found to be 3^n, and recent extensions with multidirected animals. The connection of such problems with Lorentzian quantum gravity will be discussed. A bijective proof (that we call the "Nordic decomposition" of a heap of dimers) for the formula for multidirected animals, which at the same time give some extensions for Lorentzian triangulations, will be presented. Besides physicists, the topic should be of interest to computer scientists. For computer scientists, we use the model of heaps of pieces, which is equivalent to the concept of trace in computer science, used as model for concurrency access to data structures. The bijective problems show some philosophical considerations about algorithmic constructions of bijections which have a computer science flavor. «The Strahler analysis of binary trees in computer science and in other sciences» Computer science seminar, 27 February 2013 slides (pdf, 33 Mo, 220p) abstract: Computer scientists defined the Strahler number of a binary tree in relation with the minimum number of registers needed for the computation of an arithmetical expression. Beautiful asymptotic analysis for the average Strahler number have been done (Flajolet, Vuillemin, Raoult, Kemp), involving a periodic function coming from number theory. Surprisingly, this parameter appears in hydrogeology (Horton, Strahler) for the morphologic study of river networks, and also in molecular biology in the study of RNA secondary structures (Waterman). Applications have been made in computer graphics (synthetic images of trees), the study of some fractal structures in experimental physics, in radiology and in the domain of visualization of informations. Underlying this asymptotic analysis, there are deep combinatorial mathematics, and some new structures have been introduced, in collaboration with D.Knuth, called Kepler towers. These objects belongs to the "heaps of pieces" theory, which gives a geometric interpretation of equivalence classes of words in the so-called "trace monoid" introduced in computer science by Mazurkiewicz as a model for concurrency access to data structures. «Quadratic algebras, combinatorial physics and planar automata» Random interactions seminar, 28 February 2013 slides (pdf, 16 Mo, 239p) abstract: For certain quadratic algebras Q, we introduce the concept of Q-tableaux, which are certain combinatorial objects drawn on the square lattice. These tableaux are equivalent to the notion of a planar automaton. Planar automata is a new concept (not to be confused with cellular automata) which formalizes the idea of recognizing certain "planar figures" drawn on a 2D lattice. Two quadratic algebras well known in physics are good examples of planar automata: the most simple Weyl-Heisenberg algebra defined by the commutation relation UD=DU+Id (creation-annihilation operators in quantum mechanics) and the so-called PASEP algebra defined by the relation DE=ED+E+D, in the physics of dynamical systems far from equilibrium. The associated Q-tableaux are respectively rook placements, permutations and the so-called alternating, tree-like and permutation tableaux. Other examples include non-crossing configurations of paths, tiling, plane partitions and alternating sign matrices.$ shapeimage_1_link_0