Professor Pedro A. García Sánchez, from the Department of Algebra at the University of Granada, welcomes postdoctoral candidates interested in applying for an Athenea3i Research Fellowship in 2018 at this University. The information about the Fellowship conditions, how to apply, Eligibility Criteria, Selection Process, Evaluation Process, etc. is available in https://athenea3i.ugr.es/. Please note that applicants must comply with the Eligibility Criteria (https://athenea3i.ugr.es/?page_id=23).

Brief description of the institution:

The University of Granada (UGR), founded in 1531, is one of the largest and most important universities in Spain. It serves more than 60000 students per year, including many foreign students, as UGR is the leader host institution in the Erasmus program. UGR, featuring 3650 professors and more than 2000 auxiliary personnel, offers a total of 75 degrees through its 112 departments and 28 centers.

UGR is also a leading institution in research, located in the top 5/10 of Spanish universities by a variety of ranking criteria, such as national R&D projects, fellowships awarded, publications, or international funding. UGR is one of the few Spanish Universities listed in the Shanghai Top 500 ranking (http://www.arwu.org/), and it is also well recognized for its web presence (http://www.4icu.org/top200/).

Internationally, we bet decidedly by our participation in the calls of H2020, both at partner and coordination. For the duration of the Seventh Framework Programme, the UGR has obtained a total of 66 projects, with total funding of 17.97 million euros, and for H2020, until 2015, more than 25 projects with total funding of more than 6 million euros. Our more than 3,000 researchers are grouped into 365 research groups covering all scientific fields and disciplines.

Brief description of the Centre/Research Group

Our group works on commutative monoids. These include affine and numerical semigroups. The main interests of our group can be extracted from the web page of the Monoids and Applications Network (http://www.ugr.es/~semigrupos/MyA/index-en.html).

Both numerical and affine semigroups were initially studied in association to the solutions of linear Diophantine equations. After monomial curves became a very productive source of examples for commutative algebra and algebraic geometry, their study experimented a revitalization in the last thirty years.

Factorization in terms of irreducible elements in integral domains has been carried out without using addition as a binary operation. This allows to endow domains with the monoid structure with respect to the product. As for monomial curves, the treatment of affine and numerical semigroups (as well as the tools which were developed for them) revealed to be very useful for general monoids.

The study of the properties of algebraic-geometry codes in terms of the Weierstrass semigroup associated with one point of a curve pushed forward even more the research on the properties of the Weierstrass semigroups.

Applications to combinatorial configurations of lines and points, as well as the connection between certain families of numerical semigroups and cyclotomic polynomials, provided even more motivation for the investigation of commutative monoids.

Information about the members of the group can be obtained in http://semigrupos.ugr.es/miembros. A graphical interaction with other researchers: http://www.ugr.es/~semigrupos/MyA/nodos-en.html.

Project description

The researcher will work mainly in applications of commutative monoids. These applications will be the study of nonunique factorization invariants, applications of monoids to coding theory, and if the candidate has some basic programming skills, maintenance and enhancement of the GAP package numericalsgps (see https://www.gap-system.org/Packages/numericalsgps.html and https://gap-packages.github.io/numericalsgps/).

There are several nonunique factorization invariants that measure how far is a monoid from being factorial (unique factorizations) or half-factorial (all factorizations have the same length). This is an abstraction of the well known concept of unique factorization domain, where addition is removed to obtain a monoid. Our research group has promoted several algorithms for the computation, and thus study, of some of these invariants: Delta sets, catenary degrees, elasticity, tame degree, omega primality. Apart from the calculation, there are other interesting problems related to this study, such as realization problems: finding monoids with presecribed invariants.

In the last decades numerical semigroups became a tool to measure properties of one-point algebro-geometric codes. We have produced a series of papers computing Feng-Rao distances and numbers. These allow to upper bound for instance the number of errors that such codes may correct. Or build towers of codes for efficient decoding. Only for few families of numerical semigroups these distances are known.

A tool to produce batteries of examples is essential in many cases in research. Producing examples may help to understand whether or not a conjecture holds. To this end we developed the GAP package numericalsgps. The package was accepted in 2015, but we keep enhancing it.

Research Area

• Physics and Mathematics (PHY-MAT)

For a correct evaluation of your candidature, please send the documents below to Professor Pedro A. García Sánchez (pedro@ugr.es):

• CV
• Letter of recommendation (optional)