Department of Mathematics, Faculty of Mathematics and Computer Science, "Babeș-Bolyai" University, Cluj-Napoca

Research Project: "Contributions to Silting Theory"

(Versiune în Română)


Code: PN-III-P4-ID-PCE-2020-0454
Contract: 75/04.01.2021
Period: January 2021-December 2023
Financed by: Unitatea Executivă pentru Finanțarea Învățământului Superior, a Cercetării, Dezvoltării și Inovării UEFISCDI



  1. Team
  2. Abstract
  3. Results
  4. Research visits, conferences, workshops
  5. Contact


Team:



Abstract:

The main aim is to study (co)silting objects which can be associated to some categories, as module categories over some particular rings, Grothendieck or triangulated categories.

  1. SCIENTIFIC OBJECTIVES:

    1. Contributions to silting theory. The main topics will be the following:
      1. Silting and cosilting complexes in the derived categories. Structures related to these mathematical objects.
      2. The transfer of the (co)silting properties via functors.
    2. Special classes and objects. The topics are the following:
      1. Approximations and definable classes. We will study preenveloping and precovering classes induced by (co)silting objects.
      2. Pure-injective objects in Grothendieck and triangulated categories. We will try to use some triangulated versions for the classical notion of p-functor in order to obtain new information about the class of pure-injective objects.

  2. ADMINISTRATIVE OBJECTIVES:

    1. Completion of the research infrastructure by assuring the bibliographical background (books, journals) and the material background of the research (computers, office consumables etc.)
    2. Improving the level of knowledge of the team members and other young researchers and students.

  3. RESEARCH SEMINARS:

       (S1) Silting objects in module theory (silting, cosilting and t-tilting modules) 
       (S2) Silting objects in triangulated categories 
       (S3) Splitting properties and the structure of (pure) projective and (pure) injective modules.
       (S4) Identities in rings and approximations

Reports of our activities: 2021 , 2022 , 2021-2023 , 2021-2023 (in Romanian)



Results:

Summary: Our results concern several settings: the general theory of silting objects in triangulated categories as well as some particular situations (derived categories over commutative rings or dg-algebras, or $\tau$-tilting theory for some generalizations of group algebras) which require specific approach and special techniques. Our team members investigations also provided results on module categories over finite dimensional algebras (like group algebras or path algebras over some special quivers) or on some methods to associate affine structures to module categories.

  1. Conference: Trends in Representation Theory and Related Topics


  2. Papers submmited or accepted for publication:

    1. S. Breaz: A characterization of (co)silting objects, arXiv
    2. S. Breaz, G.C. Modoi: Derived equivalences induced by good silting complexes, in Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods, editor Alexander Martsinkovsky, book series Springer Proceedings in Mathematics and Statistics
    3. A. Marcus, V.A. Minuta: An exact sequence for the graded Picent, Journal of Group Theory, arXiv
    4. I. Simion, D. Testerman: On minimal epimorphic subgroups in simple algebraic groups of rank 2, arXiv
    5. S. Breaz, T. Brzeziński, B. Rybołowicz, P. Saracco: Heaps of modules: Categorical aspects, arXiv
    6. S. Breaz, M. Hrbek, G.C. Modoi: Silting, cosilting, and extensions of commutative rings, arXiv
    7. S. Breaz, A. Marcus, C.-G. Modoi: Support τ-tilting modules and semibricks over group graded algebras, J. Algebra 637 (2024), 90--111 arXiv
    8. S. Breaz, T. Brzeziński, B. Rybołowicz, P. Saracco: Heaps of modules and affine spaces; Annali di Matematica Pura ed Applicata (1923--). https://doi.org/10.1007/s10231-023-01369-0 (2023) arXiv
    9. S. Breaz, C. Rafiliu: Decompositions of matrices by using commutators; Linear Algebra and its Applications, 662 (2023), 39--48; arXiv
    10. Cs. Szanto, I. Szollosi: On some Ringel-Hall polynomials associated to tame indecomposable modules, Journal of Pure and Applied Algebra<\i>, paper
    11. S. Breaz: On a theorem of Stelzer for some classes of mixed groups; Mediterr. J. Math., 19 (4) nr. articol 159 (2022), 14 pagini
    12. Cs. Szanto, I. Szollosi: Ringel-Hall polynomials associated to a quiver of type $\tilde D_4$, Period Math Hung (2023), paper
    13. S. Breaz, Y. Zhou: When is every non central-unit a sum of two nilpotents?; va apărea în Contemporary Math. 785 (2023), 47--55. arXiv
    14. Cs. Szanto, I. Szollosi: Combinatorial methods in the representation theory of finite dimensional tame algebras, monografie 2023.


  3. Visits, conferences, workshops

    1. S. Breaz: Homological Methods in Representation Theory. A conference in honour of Lidia Angeleri Hugel; 03 - 08 October 2021; Fraueninsel (Chiemsee), Abtei Frauenworth, Germany; Talk: Transfer of homological properties along some canonical functors.
    2. C.-G. Modoi: Homological Methods in Representation Theory. A conference in honour of Lidia Angeleri Hugel; 03 - 08 October 2021; Fraueninsel (Chiemsee), Abtei Frauenworth, Germany.
    3. Cs. Szanto: Algebra Seminar of Renyi Institute; 07 March 2022; Budapest, Hungary; Talk: Ringel-Hall polynomials associated to a quiver of type $\tilde{D}_{4}$.
    4. S. Breaz: Malga Seminar Padova-Verona, Universita di Verona; 24 May 2022; Talk: Silting complexes and extensions of commutative rings.
    5. S. Breaz: Functor Categories, Model Theory, and Constructive Category Theory, Universidad de Almeria; 11-15 July 2022; Talk: Change of scalars functors and silting complexes.
    6. C.-G. Modoi: Functor Categories, Model Theory, and Constructive Category Theory, Universidad de Almeria; 11-15 July 2022; Talk: Not necessarily compact approximability via silting theory.
    7. S. Breaz: Hopf algebras, monoidal categories and related topics; 27-29 July 2022; IMAR, Bucharest, Romania; Talk: Heaps of Modules.
    8. A. Marcus: Hopf algebras, monoidal categories and related topics; 27-29 July 2022; IMAR, Bucharest, Romania; Talk: Tilting complexes over a G-graded G-algebra.
    9. T. Micu: Young Researchers' Conference on Non-Archimedean and Tropical Geometry; 01-05 August 2022; Universität Regensburg; Talk: The structure of special fibers through valuations.
    10. T. Micu: Representation Theory and Triangulated Categories; 26-30 September 2022; Paderborn, Germany.
    11. V.-A. Minuță: Representation Theory and Triangulated Categories; 26-30 September 2022; Paderborn, Germany.
    12. C.-G. Modoi: Representation Theory and Triangulated Categories; 26-30 September 2022; Paderborn, Germany.
    13. S. Breaz: Algebra Seminar of Charles University; 10-12 October 2022; Prague, The Czech Republic; Talk: The Baer-Kaplansky Theorem and Heaps of Modules.
    14. S. Breaz: The Tenth Congress of Romanian Mathematicians, June 30-July 5, 2023, Pitesti, Romania; Talk: Change of scalars and bounded silting complexes.
    15. G.C. Modoi: The Tenth Congress of Romanian Mathematicians, June 30-July 5, 2023, Pitesti, Romania; Talk: Migration of silting-like properties via adjoint pairs.
    16. V.-A. Minuta: The Tenth Congress of Romanian Mathematicians, June 30-July 5, 2023, Pitesti, Romania; Talk: An exact sequence for the graded Picent group.
    17. I.I. Simion: The Tenth Congress of Romanian Mathematicians, June 30-July 5, 2023, Pitesti, Romania; Talk: On the product expansion of normal subsets in simple groups.
    18. S. Breaz: Trends in Representation Theory and Related Topics, 12-16 Septembrie 2023, Babe\c s-Bolyai University Cluj-Napoca, Romania, \url{https://math.ubbcluj.ro/trtrt2023}. Talk: On a characterization of (co)silting objects.
    19. Cs. Szanto: Summer school, Quiver Representations, Quiver Varieties and Combinatorics (BIP Blended Intensive Program), 22-26 Mai 2023, Bologna University, Italia, \url{https://eventi.unibo.it/bip-quiver/summer-school}. Talk: Ringel-Hall polynomials associated to tame quivers.
    20. Cs. Szanto: Trends in Representation Theory and Related Topics, 12-16 Septembrie 2023, Babe\c s-Bolyai University Cluj-Napoca, Romania, \url{https://math.ubbcluj.ro/trtrt2023}. Talk: Ringel-Hall polynomials associated to tame quivers.
    21. A. Marcus: 17th International Conference on Applied Mathematics and Computer Science, 11-13 Iulie 2023, Technical University Cluj-Napoca. Talk: Support tau-tilting modules and semibricks over group graded algebras.
    22. V.-A. Minuta: 17th International Conference on Applied Mathematics and Computer Science, 11-13 July 2023, Cluj-Napoca, Romania. Talk: An exact sequence for the graded Picent group
    23. I.-I. Simion: Trends in Representation Theory and Related Topics, September 12-16, 2023, Cluj-Napoca, Romania, \url{https://math.ubbcluj.ro/trtrt2023}. Talk: Epimorphic subgroups in simple algebraic groups.
    24. V.-A. Minuta: Trends in Representation Theory and Related Topics, 12-16 September 2023, Cluj-Napoca, Romania, \url{https://math.ubbcluj.ro/trtrt2023}. Talk: Relations between module triples, and derived equivalences for wreath products
    25. S. Breaz: Homological Algebra and Representation Theory, 10-14 July 2023, Karlovasi, Samos, Greece.
    26. G. C. Modoi: Homological Algebra and Representation Theory, 10-14 July 2023, Karlovasi, Samos, Greece.
    27. G. C. Modoi: Trends in Representation Theory and Related Topics, 12-16 Septembrie 2023, Babe\c s-Bolyai University Cluj-Napoca.
    28. A. Marcus: The 10-th Congress of Romanian Mathematicians, 30.06-05.07 2023, Pitesti, Romania.
    29. A. Marcus: Trends in Representation Theory and Related Topics, 12-16 Septembrie 2023, Babe\c s-Bolyai University Cluj-Napoca.
    30. T. Micu: Trends in Representation Theory and Related Topics, 12-16 Septembrie 2023, Babe\c s-Bolyai University Cluj-Napoca.
    31. C. Pelea: Trends in Representation Theory and Related Topics, 12-16 Septembrie 2023, Babe\c s-Bolyai University Cluj-Napoca.
    32. T. Micu: - Summer School on Higher Structures in Algebra and Geometry; 13-18 August 2023; Nordfjordeid, Norway.
    33. T. Micu: Tau-Tilting Research School; 5-8 September 2023; Koln, Germany.
    34. T. Micu: Representation Theory and Triangulated Categories; 26-30 September 2022; Paderborn, Germany.
    35. V.-A. Minuta: Representation Theory and Triangulated Categories; 26-30 September 2022; Paderborn, Germany.


    Contact

    e-mail:
    username: bodo    server: math.ubbcluj.ro
    username: ssbreaz    server: gmail.com
    fax:
        +40 264 691906
    adresa:
        Simion Breaz
        "Babeș-Bolyai" University
        Faculty of Mathematics and Computer Science
        Str. Mihail Kogălniceanu nr. 1
        RO-400084 Cluj-Napoca
        Romania