Description of the project:
Given R a positively graded algebra and M a finitely generated graded R-module, we are interested in the study of the minimal graded free resolution of M over R. Its invariants (the Betti numbers) are very important in understanding the equations describing M. In many cases of interest, e.g. when R is an algebra of polynomials, and M=I is a monomial or a binomial ideal, there exists some extra structure available on this module, and this is reflected in the homological properties of the modules. Various tools from combinatorics, topology or homological algebra may be used. This is an area of much current research.
When M or R are not graded from the start, a good way to reduce to the graded case is to use the filtration induced by the powers of a maximal ideal m in R and consider the associated graded ring gr_m(R), and similarly for grM. Of course, certain properties may be lost, but we know for sure that the Betti numbers may only increase.
An important class of graded rings R consists of those with R_0 is a field or a semisimple ring and R_0 has a linear resolution as an R-module. If this occurs, we say that R is Koszul. When an algebra is not graded from the very beginning, we may use the idea above to pass to the associated graded ring and study the Koszul property for the latter. This approach has already been exploited by V. Reiner and the PI. There are perspectives to extend its applications.

Publications and preprints:

1. Juergen Herzog, Dumitru I. Stamate, On the defining equations of the tangent cone of a numerical semigroup ring, Journal of Algebra vol 418 (2014), 8-28. DOI 10.1016/j.jalgebra.2014.07.008. Preprint version available at arXiv:1308.4644 [math.AC].
2. Mircea Cimpoeas, Dumitru I. Stamate, On intersections of complete intersection ideals, Journal of Pure and Applied Algebra, vol 220, no. 11 (2016), 3702-3712. DOI 10.1016/j.jpaa.2016.05.008. Preprint 12 pp.
3. Juergen Herzog, Dumitru I. Stamate, Quadratic numerical semigroups and the Koszul property, Kyoto Journal of Mathematics, Volume 57, Number 3 (2017), 585-612. DOI 10.1215/21562261-2017-0007. Preprint version available at arXiv:1510.00935 [math.AC].
4. Dumitru I. Stamate, On the Cohen-Macaulay property for quadratic tangent cones, The Electronic Journal of Combinatorics, Volume 23, Issue 3 (2016), #P3.20. 22 pages. Preprint version available at arXiv:1512.04893 [math.AC].
5. Alexandra Seceleanu, Dumitru I. Stamate, On Sally semigroup rings, in preparation.

Objectives, expected and obtained results: [pdf]

Scientific reports:

• 2013: [pdf]
• 2014: Cumulative report 2013-2014 [pdf/EN] and [pdf/RO].
• 2015: Final report 2013-2015 [pdf/EN] and [pdf/RO].

Research stages abroad:

2013:
• University of Osnabrueck, Germany, June 2013 (2 weeks). Host: Prof. Tim Roemer.
• University of Duisburg-Essen, Campus Essen, Germany, July-August 2013. Host: Prof. Juergen Herzog.
• University of Nebraska, Lincoln, NE, USA, September 2013 (3 weeks). Host: Prof. Roger Wiegand.
• University of Minnesota, Minneapolis, MN, USA, October 2013 (3 weeks). Host: Prof. Victor Reiner.
• University of Missouri, Columbia, MO, USA, 21-24 October 2013. Host: Prof. Hema Srinivasan.
2014:
• University of Nebraska, Lincoln, NE, USA, January 2014 (2 weeks). Host: Prof. Roger Wiegand.
• University of Minnesota, Minneapolis, MN, USA, February 2014 (3 weeks). Host: Prof. Victor Reiner.
• Universite de Montpellier 2, Institut de Mathematiques et de Modelisations de Montpellier, Montpellier, France, May 2014 (1 week). Host: Dr. Ignacio Garcia Marco.
• Universita di Genova, DIMA, Genova, Italy (1 week). Host: Prof. Aldo Conca.
• University of Duisburg-Essen, Campus Essen, Germany, July-August 2014 (5 weeks). Host: Prof. Juergen Herzog.
• University of Nebraska, Lincoln, NE, USA, September 2014 (3 weeks). Host: Dr. Alexandra Seceleanu.
2015:
• University of Sheffield, UK, May 2015 (1 week). Hosts: Prof. Moty Katzman and Dr. Ines Henriques.
• University of Duisburg-Essen, Campus Essen, Germany, September 2015 (1 week). Host: Prof. Juergen Herzog.

Invited specialists :

• Prof. Jorge Ramirez Alfonsin, Universite Montpellier 2, between 2-7 September 2014. He gave a series of 4 lectures on the algebraic properties of matroids at the National Algebra School.
• Prof. Juergen Herzog, Universitaet Duisburg-Essen, Germany between 12-20 June 2015. He lectured at IMAR on Ideals associated to isotone maps between finite posets on 16 June 2015.

Dissemination of the results. Conference and seminar talks:

1. On the CI property of the tangent cone of a toric ring, Workshop for Young Researchers in Mathematics, Universitatea Ovidius Constanta, Romania, 8-10 May 2013.
2. Shifting semigroups, short talk, Workshop: Syzygies in Berlin, Freie Universitaet, Berlin, Germany, 28 May 2013.
3. Shifted semigroup rings, Oberseminar University of Osnabrueck, Germany, 4 June 2013.
4. On the CI property of the tangent cone of a toric ring, Joint International Meeting AMS-RMS, Special Session on Commutative Algebra, Alba Iulia, Romania, 30 June 2013.
5. On the equations of toric rings, University Duisburg-Essen, Essen, 29 August 2013.
6. Tools of Combinatorial Commutative Algebra 2, National Algebra School -- Algebraic methods in Combinatorics, IMAR, Bucharest, 3 September 2013.
7. Matroids and realisability, National Algebra School -- Algebraic methods in Combinatorics, IMAR, Bucharest, 4 September 2013.
8. On the defining equations of the tangent cone of a numerical semigroup ring, Comm. Algebra Seminar talk, University of Nebraska, Lincoln, NE, SUA, 18 September 2013.
9. On the defining equations of the tangent cone of a numerical semigroup ring, Comm. Algebra Seminar talk, University of Minnesota, Minneapolis, MN, SUA, 14 October 2013.
10. On the defining equations of the tangent cone of a numerical semigroup ring, Comm. Algebra Seminar talk, University of Missouri, Columbia, MO, SUA, 22 October 2013.
11. Asymptotic properties of numerical semigroups I, II, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 19 and 26 November 2013.
12. About the structure of Sally rings, Comm. Algebra Seminar talk, University of Minnesota, Minneapolis, MN, SUA, 14 February 2014.
13. On the defining equations of tangent cones of numerical semigroup rings, Workshop for Young Researchers in Mathematics, Universitatea Ovidius Constanta, Romania, 22-23 May 2014.
14. On numerical semigroup rings and their defining relations, Seminaire Algebre et geometrie combinatoires, Universite de Montpellier 2, France, 27 May 2014.
15. On the defining equations of the tangent cone of a numerical semigroup ring, Commutative Algebra Seminar, University of Genova, Italy, 3 June 2014.
16. Asymptotic properties of numerical semigroups, National Algebra School -- Algebraic and Combinatorial Applications of Toric Ideals, IMAR, Bucharest, 3 September 2014.
17. Flavors of Koszul rings, Comm. Algebra Seminar talk, University of Nebraska, Lincoln, NE, SUA, 17 September 2014.
18. On intersections of complete intersection ideals, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 24 February 2015.
19. Koszul filtrations, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 28 April 2015.
20. Koszul filtrations (II). Grobner flags. , Comm. Algebra Seminar IMAR and Univ. Bucuresti, 5 May 2015.
21. Koszul filtrations (III). Strongly Koszul rings and the ungraded version., Comm. Algebra Seminar IMAR and Univ. Bucuresti, 12 May 2015.
22. Koszul rings and the combinatorics of posets., Comm. Algebra Seminar IMAR and Univ. Bucuresti, 19 May 2015.
23. Filtrations for Koszul rings, Workshop for Young Researchers in Mathematics, Universitatea Ovidius Constanta, Romania, 21 May 2015.
24. On the defining equations of tangent cones of numerical semigroup rings, Algebra / Algebraic Geometry seminar, University of Sheffield, United Kingdom, 27 May 2015.
25. Ungraded strongly Koszul rings, The Eighth Congress of Romanian Mathematicians, Iasi, Romania, 26 June-1 July 2015.
26. On the Koszul property for numerical semigroup rings, Syzygies in Algebra and Geometry 2015, Busan, Korea, 26-30 August 2015.
27. Koszul filtrations for ungraded rings, National Algebra School -- Interactions of Computer Algebra with Commutative Algebra, Combinatorics and Algebraic Statistics, IMAR, Bucharest, 3 September 2015.
28. On the Koszul property for numerical semigroup rings (I, II), Comm. Algebra Seminar IMAR and Univ. Bucuresti, 6 and 13 October 2015.
29. Quadratic numerical semigroups and the Koszul property, The first Romanian-Turkish Mathematics Colloquium, Constanta, Romania, 15-16 October 2015.
30. The structure of quadratic CI semigroups, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 10 November 2015.

Other seminar talks :

• Mihai Cipu, Algebraic tools for discrete tomography, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 18 February 2014.
• Mihai Cipu, Quantic Ehrhart polynomials, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 1 April 2014.
• Dumitru Stamate, On the subadditivity problem for maximal shifts in free resolutions, after Herzog et al, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 6 May 2014.
• Mihai Cipu, A conjectural characterization for complete intersection numerical semigroups, Comm. Algebra Seminar IMAR and Univ. Bucuresti, 4 November 2014.

"S. Stoilow" Institute of Mathematics of the Romanian Academy (IMAR)
Facultatea de Matematica si Informatica, Universitatea din Bucuresti
Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii, Romania (UEFISCDI)