Codes, combinatorial and enumerative combinatorics, monomial curves

Coordinator: Santiago Zarzuela Armengou (Universitat de Barcelona)

Members
  • Maria Bras Amorós (Universitat Rovira i Virgili)
  • Teresa Cortadellas Benítez (Universitat de Barcelona y Universitat Pompeu Fabra)
  • Juan Elías García (Universitat de Barcelona)
  • Julio Fernández González (Universitat Politècnica de Catalunya)
  • Roser Homs Pons (estudiante de doctorado, Universitat de Barcelona)
  • Raheleh Jafari (Mosaheb Institute of Mathematics, Kharazmi University, Irán)
  • Francesc Planas Vilanova (Universitat Politècnica de Catalunya)
  • Alonso Sepúlveda Castellanos (Universidade Federal de Uberlândia, Brasil)
  • José Miguel Serradilla Merinero (estudiante de doctorado, Universitat Rovira i Virgili)
  • Klara Stokes (Hamiltonian Institute, Maynooth University, Irlanda)
  • Francesco Strazzanti (INdAM y Università degli Studi di Catania, Italia)
  • Santiago Zarzuela Armengou (Universitat de Barcelona)
Description of the research activity
The main activity of this group is related with several applications of numerical semigroups. First coding theory, particularly applications to error correcting codes, which are used to detect and correct the errors that appear in the communication and storage of data, through channels or defective devices that may distort the information to be sent and stored. Error correcting codes are also used in the distributive storage for the cloud. The aim of the theory is the design and implementation of codes with a good capacity for error correcting, but having a low cost in sending encoded information, and the corresponding correcting algorithms that allow recovering the original information. Numerical semigroups are a very useful tool to parametrize the corrective capacity and the redundancy of the algebraic codes. They are also useful to control the behavior of the decoding algorithms.
Algebraic and enumerative combinatorics are relatively recent disciplines. The first one uses algebraic, topological and geometric techniques for solving combinatorial problems or, the other way around, uses combinatorial techniques to address problems in those areas. On the other side, enumerative combinatorics deals with counting elements in a finite set. Typically, one has an infinite collection of finite sets indexed by some set (usually the naturals) and one studies simultaneously the number of elements in each set as a function of the set of indices. Our group works on the general problem of counting numerical semigroups, or the more specific of counting semigroups that satisfy special properties, in the junction of both disciplines: algebraic and enumerative combinatorics.
Each numerical semigroup provides the parametrization of a monomial curve whose ring of functions is the corresponding numerical semigroup ring. In this way, it is possible to establish a rich correspondence between the arithmetical properties of the monomial curve (those of its ring of functions) and the ones of the associated numerical semigroup. Monomial curves are a paradigmatic case from the point of view of the algebraic geometry and the commutative algebra. This is due to both their complexity and to the availability of an effective computation. In this way, one can check for them results for more general curves or varieties of higher dimension. Our group studies some of the most fundamental properties of monomial curves, as for instance the Hilbert function, the arithmetical properties of the tangent cones (Cohen-Macaulay, Gorenstein, Buchsbaum, complete intersection), free resolutions, or the behavior of these properties under constructions such as the gluing. The techniques come from the direct study of the numerical semigroup by using Apéry bases and from the study of the definition ideal of the numerical semigroup ring: this is a toric ideal for which study one can use several methods coming from algebraic combinatorics.

Monoids and Applications Network | Design inspired in HTML5 UP