Research team:

• Conf. Dr. Dumitru I. Stamate (project director), University of Bucharest, CV
• Conf. Dr. Mircea Cimpoeas, Universitatea Politehnica din Bucuresti & IMAR, CV
• C.S. Dr. Rodica Dinu, "Simion Stoilow" Institute of Mathematics of the Romanian Academy (IMAR), CV
• Asist. Dr. Claudia Ciobanu, Universitatea Politehnica din Bucuresti, CV
• Gabriela Petcu, University of Bucharest, Ph.D. student, CV [hired since 1 August 2022]
• Teodor Grigorescu, Ph.D. student, CV. [hired since 25 April 2023].

Description of the project:
In this project we considered problems in commutative algebra related to Groebner bases, free resolutions and persistence of the invariants through deformations, and also new extensions of the Gorenstein property. On one hand, we extended the theoretical framework, and on the other hand we treated concrete classes of combinatorially defined algebras. The problems we considered are of large interest in current research. The computation of numerical invariants (like Betti numbers, degree, dimension, (s-, h-)depth, etc.) is a challenge for many classes of algebras, hence we consider our work to have a significant impact.

[RO] In acest proiect am studiat probleme de algebra comutativa referitoare la baze Groebner, rezolutii libere, persistenta invariantilor la deformari, si noi extinderi ale proprietatii Gorenstein. Pe de o parte am extins cadrul teoretic, pe de alta parte am lucrat cu clase concrete de algebre definite combinatorial. Problemele tratate sunt de mare interes in cercetarea curenta, si rezultatele obtinute speram sa devina vizibile specialistilor. Calculul invariantilor numerici (numere Betti, grad, dimensiune, (s-, h-)depth, etc.) este o mare provocare pentru multe clase de algebre, si consideram astfel ca lucrarile obtinute sa aiba impactul dorit.

Publications and preprints:

1. Dumitru I. Stamate, On the occurrence of complete intersections in shifted families of numerical semigroups, Research in the Mathematical Sciences vol 9 (2022), Article number: 32 (5 pp). DOI 10.1007/s40687-022-00328-0. Preprint version.
2. Rodica Dinu, Francesco Navarra, Non-simple polyominoes of Konig type and their canonical module, preprint 2024, submitted. 30 pp. Preprint version available at arXiv:2210.12665 [math.AC].
3. Mircea Cimpoeas, On a generalization of monomial groups, Monatshefte fur Mathematik, vol. 202, issue 1, September 2023, pages 53–64. DOI 10.1007/s00605-022-01762-6.
4. Mircea Cimpoeas, Alexandru F. Radu, On supercharacter theoretic generalizations of monomial groups and Artin's conjecture, Czechoslovak Mathematical Journal Vol. 72 (2022), No. 4, pp. 1065-1079. DOI 10.21136/CMJ.2022.0352-21.
5. Mircea Cimpoeas, Alexandru F. Radu, On the monoid algebra associated to the monomial characters of a finite group, Quaestiones Mathematicae Vol. 46 (2023), Issue 10, 2065-2078. DOI 10.2989/16073606.2022.2142862. Preprint version available at arXiv:2205.13597 [math.RT].
6. Juergen Herzog, Shinya Kumashiro, Dumitru I. Stamate, The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one, Journal of Algebra 619 (2023), 626-642. DOI 10.1016/j.jalgebra.2022.12.026.
7. Antonino Ficarra, Juergen Herzog, Dumitru I. Stamate, Vijaylaxmi Trivedi, The canonical trace of determinantal rings, 10 pp. Submitted. Preprint version available at arXiv:2212.00393 [math.AC].
8. Mircea Cimpoeas, Adrian Gabriel Neacsu, Geometrical isomorphisms between categories of fuzzy coverings and fuzzy partitions, Fuzzy Sets and Systems, Volume 461, 15 June 2023, 108493. DOI 10.1016/j.fss.2023.02.014. Preprint version available at arXiv:2210.07948 [GM].
9. Silviu Balanescu, Mircea Cimpoeas, Depth and Stanley depth of powers of the path ideal of a path graph, Preprint 16 pp., submitted. Preprint version available at arXiv:2303.01132 [math.AC] .
10. Silviu Balanescu, Mircea Cimpoeas, Christian Krattenthaller, On the quasi depth of monomial ideals, Preprint 18 pp, submitted. Preprint version available at arXiv:2306.09450 [math.AC]
11. Mircea Cimpoeas, Alexandra Teodor, A note on the restricted k-multipartition function, UPB Scientific Bulletin-Series A, Volume 85, Issue 3, 2023, Page 11-18.
12. Rodica Dinu, Martin Vodicka, Phylogenetic degrees for claw trees, Journal of Combinatorial Theory Series A, Volume 206, August 2024, 105886. DOI 10.1016/j.jcta.2024.105886 . Preprint version available at arXiv:2303.16812 [math.AG].
13. Rodica Dinu, Martin Vodicka, Classification of normal phylogenetic varieties for tripods, Preprint 13 pp. Submitted. Preprint version available at arXiv:2309.05301 [math.AG].
14. Rodica Dinu, Francesco Navarra, On the rook polynomial of grid polyominoes, Preprint 28 pp. Submitted. Preprint version available at arXiv:2309.01818 [math.CO].
15. Rodica Dinu, Nayana Shibu-Deepthi, (S_2)-condition of edge rings for cactus graphs, Preprint 13 pp. Submitted. Preprint version available at arXiv:2402.17413 [math.AC].
16. Rodica Dinu, Martin Vodicka, Phylogenetic degrees for Jukes-Cantor model. Preprint 21 pp. Submitted. Preprint version available at arXiv:2405.20184 [math.AG].
17. Antonino Ficarra, Juergen Herzog, Dumitru I. Stamate, The toric ring of one dimensional simplicial complexes, Preprint 16 pp. Submitted. Preprint version available at arXiv:2306.05020 [math.AC].
18. Silviu Balanescu, Mircea Cimpoeas, Depth and Stanley depth of powers of the path ideal of a cycle graph, Preprint 15 pp. Submitted. Preprint version available at arXiv:2303.15032 [math.AC].
19. Silviu Balanescu, Mircea Cimpoeas, Several combinatorial inequalities related to squarefree monomial ideals, Preprint 8 pp. Submitted. Preprint version available at arXiv:2307.03018 [math.AC].
20. Silviu Balanescu, Mircea Cimpoeas, On the Hilbert depth of the Hilbert function of a finitely generated graded module, Preprint 12 pp. Submitted. Preprint version available at arXiv:2308.14031 [math.AC].
21. Silviu Balanescu, Mircea Cimpoeas, On the arithmetic Hilbert depth, Preprint 17 pp. Submitted. Preprint version available at arXiv:2309.10521 [math.AC].
22. Silviu Balanescu, Mircea Cimpoeas, Remarks on the Hilbert depth of squarefree monomial ideals, Preprint 10 pp. Submitted. Preprint version available at arXiv:2310.12339 [math.AC].
23. Silviu Balanescu, Mircea Cimpoeas, Remarks on the Hilbert depth of powers of the maximal graded ideal, Preprint 11 pp. Submitted. Preprint version available at arXiv:2311.11079 [math.AC].
24. Andreea I. Bordianu, Mircea Cimpoeas, Remarks on the Stanley depth and Hilbert depth of monomial ideals with linear quotients, Preprint 10 pp. Submitted. Preprint version available at arXiv:2102.07196 [math.AC].
25. Silviu Balanescu, Mircea Cimpoeas, Thanh Vu, Betti numbers of powers of path ideals of cycles, Preprint 27 pp. Submitted. Preprint version available at arXiv:2404.17880 [math.AC].
26. Mircea Cimpoeas, Dumitru I. Stamate, On Betti numbers for intersections of ideals associated to numerical semigroup rings. In preparation.

Scientific reports:

• 2022: [pdf]
• 2023: [pdf]
• 2024: [pdf]

Research stages abroad funded by the grant:

2022:
• Rodica Dinu at the University of Duisburg-Essen, Campus Essen, Germany, June 2022 (2 weeks). Host: Prof. Juergen Herzog,
• Gabriela Petcu attended the EMS Summer School on Combinatorial Commutative Algebra, Gebze Technical University, Turkey, 8-12 August 2022.
• Dumitru Stamate at the University of Duisburg-Essen, Campus Essen, Germany, September 2022 (2 weeks). Host: Prof. Juergen Herzog.
2023:
• Dumitru Stamate at the University of Duisburg-Essen, Campus Essen, Germany, May 2023 (2 weeks). Host: Prof. Juergen Herzog.
• Dumitru Stamate at the University of Genova, Italy, September 2023 (2 weeks). Host: Dr. Francesco Strazzanti .
2024:
• Dumitru Stamate at the University of Genova, Italy, January/February 2024 (2 weeks). Host: Dr. Francesco Strazzanti .
• Dumitru Stamate at the University of La Laguna, Tenerife, Spain, April 2024 (1 week). Host: Dr. Ignacio Garcia Marco.

Invited specialists :

• Alexandra Seceleanu from the University of Nebraska, Lincoln, USA visited between 16-19 July 2022. She also gave one lecture on Symmetric shifted monomial ideals in the Commutative Algebra Seminar at the Faculty of Mathematics and Computer Science, University of Bucharest.
• Francesco Navarra from the University of Messina, Italy, visited us between 16 March-3 April 2023
• Alexandra Seceleanu from the University of Nebraska, Lincoln, USA visited between 23-29 June 2023. She gave a series of four lectures on Symbolic powers in algebra and geometry on the occasion of the 28th National School on Algebra-Interactions between Algebra and Geometry in Bucharest, 24-28 June 2023.
• Yairon Cid Ruiz from the Katholieke Universiteit, Leuven, Belgium visited between 23-28 June 2023. He gave four talks gouped under the title Some lectures on multidegrees on the occasion of the 28th National School on Algebra-Interactions between Algebra and Geometry in Bucharest, 24-28 June 2023.
• Antonino Ficarra from the University of Messina, Italy, visited between 23-29 June 2023. He gave the talk The toric ring of 1-dimensional simplicial complexes at the 28th National School on Algebra-Interactions between Algebra and Geometry in Bucharest, 24-28 June 2023.
• Max Kolbl from the University of Osaka, Japan visited between 19-24 August 2023. He gave a lecture on Nearly Gorenstein Polytopes in the joint FMI-IMAR Commutative Algebra Seminar on 22 August 2023.

Dissemination of the results. Conference and seminar talks:

1. Dumitru Stamate, Ulrich elements in normal affine semigroup rings, Algebra Seminar, University of Pisa, 17 May 2022.
2. Dumitru Stamate, Ulrich elements in normal affine semigroup rings, INdAM Workshop: International Meeting on Numerical Semigroups 2022, Rome, Italy, 13-17 June 2022.
3. Mircea Cimpoeas, A note on the monomial characters of a wreath product of groups, C.A.I.M. 29th edition, Universitatea de stat din Tiraspol, 25-28 August 2022, Chisinau, Republic of Moldova.
4. Mircea Cimpoeas, Almost monomial groups, Yearly Conference of SSMR, the Romanian Mathematical Society, Universitatea Politehnica din Bucuresti, 22 October 2022, Bucharest, Romania.
5. Dumitru Stamate, On the occurrence of complete intersections in shifted families of numerical semigroups, FMI-IMAR Commutative Algebra Seminar, 8 November 2022.
6. Mircea Cimpoeas, On the monomial algebra associated to the monomial characters of a finite group, FMI-IMAR Commutative Algebra Seminar, 15 November 2022.
7. Rodica Dinu, Non-simple polyominoes of Koenig type, FMI-IMAR Commutative Algebra Seminar, 10 January 2023.
8. Rodica Dinu, Gorenstein property for phylogenetic trivalent trees, Workshop on Combinatorial and Algebraic Aspects on Lattice Polytopes, Kwansei Gakuin University, Hyogo, Japan, 21 February 2023.
9. Mircea Cimpoeas, Depth and Stanley depth of powers of some classes of monomial ideals, FMI-IMAR Commutative Algebra Seminar, 7 March 2023.
10. Dumitru Stamate, Rings with tiny canonical trace ideals, FMI-IMAR Commutative Algebra Seminar, 25 April 2023.
11. Dumitru Stamate, Progress on canonical trace ideals, Algebra Seminar, University of Pisa, Italy, 17 May 2023.
12. Mircea Cimpoeas, On the monoid algebra associated to the monomial characters of a finite group, Workshop for Young Researchers in Mathematics – 12th Edition, Universitatea “Al.I. Cuza” Iasi, Romania, 19 May 2023.
13. Mircea Cimpoeas, On the quasi depth of monomial ideals, The 28th National School on Algebra: Interactions between Algebra and Geometry in Bucharest, University of Bucharest, Romania, 24 June 2023.
14. Mircea Cimpoeas, On almost monomial groups, 10th Congress of Romanian Mathematicians, University of Pitesti, Romania, 4 July 2023.
15. Rodica Dinu, On varieties coming from phylogenetic group-based modelss, Special Session on Homological Methods in Commutative Algebra, 10th Congress of Romanian Mathematicians, University of Pitesti, Romania, 4 July 2023.
16. Dumitru Stamate, Progress on canonical trace ideals, 10th Congress of Romanian Mathematicians, University of Pitesti, Romania, 4 July 2023.
17. Dumitru Stamate, Canonical trace ideals and Gorenstein related properties, Algebra and Gemetry Seminar, University of Genova, Italy, 19 September 2023.
18. Teodor Grigorescu, Far flung Gorenstein semigroups of small type, FMI-IMAR Commutative Algebra Seminar, 3 October 2023.
19. Mircea Cimpoeas, Quasi depth for monomial ideals, (I), (II), FMI-IMAR Commutative Algebra Seminar, 10, 17 October 2023.
20. Mircea Cimpoeas, On the action of the Hecke group H(2) on subsets of quadratic fields, Number Theory Seminar, IMAR, Romania, 22 November 2023.
21. Dumitru Stamate, Periodicity results for Betti numbers of intersections of ideals, FMI-IMAR Commutative Algebra Seminar, 16 January 2024.
22. Dumitru Stamate, Peridicity for Betti numbers in some families of ideals, Genova - Torino - Milano Seminar, Dipartimento di Matematica, Universita degli Studi di Milano, Italy, 6 February 2024.
23. Rodica Dinu, poster presentation at the Network Meeting of the Alexander von Humboldt-Foundation, University of Konstanz, Germany, 21-23 February 2024.
24. Mircea Cimpoeas, "Busy beavers" and the Riemann hypothesis, Number Theory Seminar, IMAR, Romania, 28 February 2024.
25. Dumitru Stamate, Ulrich elements and slim affine semigroups, 1st Geometry and Algebra in Combinatorics Workshop, Los Silos, Tenerife, Spain, 5 April 2024.
26. Mircea Cimpoeas, Quasi depth is a misnomer for Hilbert depth, FMI-IMAR Commutative Algebra Seminar, 9 April 2024.
27. Rodica Dinu, Algebraic degrees of phylogenetic varieties, FMI-IMAR Commutative Algebra Seminar, 16 April 2024.

Events organized:

• The 11th edition of the Workshop for Young Researchers in Mathematics, 19-20 May 2022, IMAR, Bucharest (hybrid), was co-organized by Cristina Anghel, Rodica Dinu, Alexandra Melnig and Remus Radu. R. Dinu and C. Anghel co-organized the Algebra, Geometry, Number Theory & Topology Section.
• Dumitru Stamate was part of the scientific committee for the EMS Summer School on Combinatorial Commutative Algebra, Gebze Technical University, Turkey, 8-12 August 2022, together with Fatemeh Mohammadi and Mufit Sezer.
• The 12th edition of the Workshop for Young Researchers in Mathematics, 18-19 May 2023, was co-organized at the "Al.I. Cuza" University of Iasi, Romania by Cristina Anghel, Rodica Dinu, Alexandra Melnig and Remus Radu. R. Dinu and C. Anghel co-organized the Algebra, Geometry, Number Theory & Topology Section.
• Dumitru Stamate together with Viviana Ene, Cristodor Ionescu and Marius Vladoiu organized the 28th National School on Algebra-Interactions between Algebra and Geometry in Bucharest, 24-28 June 2023, University of Bucharest, Romania.
• Dumitru Stamate and Dorin Popescu organized the Special Session Homological Methods in Commutative Algebra at the 10th Congress of Romanian Mathematicians, University of Pitesti, Romania, 4 July 2023.