Abstract

The purpose of this proposal is to build up a team in the field of commutative algebra with emphasize on its applications to combinatorics and the development of computer algebra algorithms and software. We want to develop new theoretical and algorithmic methods for studying important topics in the present research and to implement new mathematical software in order to experiment with them. The proposal is based on the interaction between mathematics and computer science, which has generated in the last years many fundamental results. Modern mathematics and, in particular, commutative algebra and combinatorics, substantially benefit from the development of new algorithms and the use of mathematical software. The main subject of this proposal is to investigate the structure of the multigraded Hilbert series associated to finitely generated multigraded modules. Some particular themes which we intend to study are:

- The multigraded Hilbert depth and the relation with Stanley depth,

- Multigraded Boij–Söderberg theory and Hilbert series,

- Algorithms and software for Hilbert series, Hilbert depth and Stanley depth.

Finally, we intend to search to find further applications of the new developed methods in the fields of commutative algebra, combinatorics, combinatorial optimization, algebraic geometry and algebraic statistics.

Abstract Romana

Scopul acestei propuneri este alcatuirea unei echipe de cercetatori in domeniul algebrei comutative cu accentul pus pe aplicatiile sale in combinatorica, in acelasi timp cu dezvoltarea de algoritmi si programe pentru algebra computationala. Vrem sa dezvoltam noi metode teoretice si algoritmice pentru studiul unor probleme importante din cercetarea actuala si sa implementam programe pentru a efectua experimente computationale. Propunerea este bazata pe interactiunea dintre matematica si stiinta calculatoarelor care a generat in ultimii ani multe rezultate fundamentale. Matematica moderna si, in particular, algebra comutativa si combinatorica, beneficiaza substantial din dezvoltarea de noi algoritmi si din folosirea programelor de calcul matematic. Principala tema a acestei propuneri este investigarea structurii seriilor Hilbert multigraduate asociate modulelor finit generate multigraduate. Cateva teme particulare pe care intentionam sa le studiem sunt:

- Hilbert depth multigraduata si legatura cu Stanley depth,

- Teoria Boij–Söderberg multigraduata si legatura cu seriile Hilbert,

- Algoritmi si programe pentru serii Hilbert, Hilbert depth si Stanley depth.

In final, intentionam sa cautam aplicatii ulterioare ale metodelor nou dezvoltate in domeniul algebrei comutative, combinatoricii, optimizarii combinatoriale, geometriei algebrice si statisticii algebrice.

Expected Results

Team:

C.S.III Dr. Bogdan Ichim, Institutul de Matematica "Simion Stoilow" al Academiei Romane.
C.S.I Dr. Dorin-Mihai Popescu, List of publications, Institutul de Matematica "Simion Stoilow" al Academiei Romane.
Lect. Dr. Marius Vladoiu, Universitatea Bucuresti, Institutul de Matematica "Simion Stoilow" al Academiei Romane.
Drd. Andrei Zarojanu, Universitatea Bucuresti, Institutul de Matematica "Simion Stoilow" al Academiei Romane.

Grant budget

Reports

Raport stiintific 2014

Scientific report 2014

Scientific report 2015

Scientific report 2016

Research supported by this grant

a) Articles published in ISI Journals

1. B. Ichim, A. Zarojanu, An algorithm for computing the multigraded Hilbert depth of a module, Experimental Mathematics, 23:3 (2014), 322-331, DOI: 10.1080/10586458.2014.908753

2. B. Ichim, L. Katthän, J. J. Moyano-Fernández, The behavior of Stanley depth under polarization, Journal of Combinatorial Theory, Series A, Volume 135, October 2015, Pages 332–347

3. H. Charalambous, A. Thoma, M. Vladoiu, Binomial fibers and indispensable binomials, J. Symb. Computation 74 (2016), 578-591, doi:10.1016/j.jsc.2015.09.005

4. D. Popescu, Stanley depth of monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie, 58(106) (2015), no 1, 95-101

5. W. Bruns, B. Ichim, C. Soeger, The power of pyramid decomposition in Normaliz, J. Symb. Computation 74 (2016), 513-536, doi:10.1016/j.jsc.2015.09.003

6. B. Ichim, L. Katthän, J. J. Moyano-Fernández, Lcm-lattices and Stanley depth: a first computational approach , Experimental Mathematics, 25:1 (2016), 46-53, DOI: 10.1080/10586458.2015.1005257

b) Articles accepted for publication in ISI Journals

1. B. Ichim, L. Katthän, J. J. Moyano-Fernández, How to compute the Stanley depth of a module , to appear in Mathematics of Computation

c) Submitted articles for publication in ISI Journals

1. B. Ichim, L. Katthän, J. J. Moyano-Fernández, Stanley depth and the lcm-lattice

2. S. Petrovic, A. Thoma, M. Vladoiu, Bouquet algebra of toric ideals

d) Conference Procedings (BDI)

1. B. Ichim, A. Zarojanu, An introduction to Hilbert depth, Proceedings of the Third Conference of Mathematical Society of Moldova IMCS-50, 2014, 86-89

2. D. Popescu, A. Popescu, Four generated, squarefree, monomial ideals Bridging Algebra, Geometry, and Topology, Springer Proceed. Math., and Statistics, 96, 231-248, 2014

e) International oral presentations (selection)

1. M. Vladoiu, 6 noiembrie 2013, University of Ioannina, Asymptotic Behavior Of Algebraic Invariants

2. Bogdan Ichim, 5 noiembrie 2013, University of Osnabrueck, An algorithm for computing the multigraded Hilbert depth of a module

3. A. Zarojanu, 11 iunie 2013, University of Osnabrueck, CoCoA School, Stanley's Conjecture for square-free monomial ideals

4. A. Zarojanu, 20 august 2014, Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova, The Third Conference of Mathematical Society of the Republic of Moldova dedicated to the 50th anniversary of the foundation of the Institute of Mathematics and Computer Science, An introduction to Hilbert depth

5. M. Vladoiu, 12 septembrie 2014, Levico Terme, Italy, Meeting On Combinatorial Commutative Algebra 2014, Markov bases of lattice ideals

6. B. Ichim, 18 septembrie 2014, Instituto Universitario de Matemáticas y Aplicaciones de Castellón, First Mini-Workshop IMAC-Singular in La Plana: Trends in Commutative Algebra, Recent results in Computational Voting Theory

7. B. Ichim, 18 martie 2015, Institut Universitari de Matemàtiques i Aplicacions de Castellón , Ciclo de jóvenes investigadores en el IMAC, How to compute the Stanley depth of a module?

8. M. Vladoiu, 30 martie 2015, Bilkent University, Ankara, , Algebra Seminar

9. B. Ichim, 10 iunie 2015, Porto, Portugal, 2015 International Meeting, How to compute the Stanley depth of a module

10. B. Ichim, 26 iunie 2015, ALEXANDRU IOAN CUZA UNIVERSITY OF IASI, THE EIGHTH CONGRESS OF ROMANIAN MATHEMATICIANS, How to compute the Stanley depth of a module

11. M. Vladoiu, 27 iunie 2015, ALEXANDRU IOAN CUZA UNIVERSITY OF IASI, THE EIGHTH CONGRESS OF ROMANIAN MATHEMATICIANS, Bouquet Algebra of Toric Ideals

12. D. Popescu, 27 iunie 2015, ALEXANDRU IOAN CUZA UNIVERSITY OF IASI, THE EIGHTH CONGRESS OF ROMANIAN MATHEMATICIANS, A theorem of Ploski's type

13. A. Zarojanu, 30 iunie 2015, ALEXANDRU IOAN CUZA UNIVERSITY OF IASI, THE EIGHTH CONGRESS OF ROMANIAN MATHEMATICIANS, On the Stanley Depth

14. B. Ichim, 24 septembrie 2015, Instituto Universitario de Matemáticas y Aplicaciones de Castellón, Second Mini-Workshop IMAC-SINGACOM in La Plana: Topics in Monoid Theory and its applications, On Hilbert decompositions

15. B. Ichim, 6 mai 2016, University of Osnabrueck, Normaliz Workshop 2016, On the score sheets of a round-robin football tournament

16. B. Ichim, 10 mai 2016, University of Frankfurt, Diskrete Mathematik, Geometrie und Optimierung , On the score sheets of a round-robin football tournament

17. B. Ichim, 6 iulie 2016, Levico Terme, Italy, International meeting on numerical semigroups with applications, On the score sheets of a round-robin football tournament

National School on Algebra