Presentation of my research domains
Enumerative combinatorics is the art of counting “finite structures” or “combinatorial objects” by giving explicit formulae, or explicit expressions for the corresponding “generating function”, or at least some equations involving these quantities.
Algebraic combinatorics is the art of relating algebraic structures and combinatorial structures. One can associate algebraic structures to combinatorial objects, as for example the Jones polynomial associated to a knot, or the chromatic and the Tutte polynomial associated to a graph. Conversely, combinatorial objects appear in the study of certain algebraic structures, as for example the classical Young tableaux in the representation theory of symmetric groups. Another example are the so-called Loday-Ronco or Connes-Kreimer Hopf algebras (related to renormalisation in physics) based on combinatorial manipulations of trees.
In bijective combinatorics, the purpose is to “explain” the various formulae or equations appearing in combinatorics with the explicit construction of bijections and correspondences between various classes of combinatorial objects. By working with “weighted objects” and “weight preserving bijections”, a whole universe of identities are accessible via combinatorics, involving identities coming from number theory, algebra and classical analysis (special functions, orthogonal polynomials, ...). Before constructing bijections, one has first to give “combinatorial interpretation”, i.e. to invent a class of weighted combinatorial objects such that the sum of the weights of all these objects will give the polynomial, series or function coming from other banches of pure or applied mathematics.
Related topics are the random generation of combinatorial objects (i.e. to construct efficient algorithms giving a random object of a given “size” with uniform probability), also analytic combinatorics (that is the asymptotic study for the number of objects of a given type). There are several relations between these combinatorics and other part of pure and applied mathematics (algebra, number theory, analysis, probability theory, control theory, ...) and some other fields such as theoretical physics (statistical mechanics, fractals, quantum gravity, ...), computer science (automata theory, analysis of algorithms, computer graphics, ...) and molecular biology.
