Publications, preprints and research projects

(Ultima modifica il 1/9/2026)

I describe here the main lines of my research. The publications number refer to the list below.

Cluster algebras and their categorifications My research began with the PhD thesis problem given to me by Prof. Andrei Zelevinsky, concerning the study of bases with good positivity properties in rank-three cluster algebras. During my PhD period, spent at Northeastern University in Boston between 2005 and 2008, I came into contact with Prof. Gordana Todorov, who introduced me to Auslander-Reiten theory for the study of representations of finite-dimensional Artin algebras. Back in Padua, I had the good fortune to attend a PhD course by Prof. Lidia Angeleri-H"ugel on Auslander-Reiten theory, and since then the representation theory of Artin algebras has been central to my research. The additive categorification of cluster algebras, discovered by Caldero and Chapoton and later extended in full generality by Keller’s school, through the fundamental works of Derksen, Weyman and Zelevinsky, uses the Euler characteristic of certain projective varieties called quiver Grassmannians, which are associated with finite-dimensional representations of Artin algebras. Publications no.~18, 22, 23, 24, 25, 27, 28 and 29.

Quiver Grassmannians My first and most productive research activity concerns the study of quiver Grassmannians of Dynkin, affine, or rigid type. These varieties may be arbitrarily complicated, but their geometry can be studied using representation theory. In this respect, the paper with Prof. Esposito from Padua, written during my last year as a PhD student and my first postdoctoral year, on quiver Grassmannians associated with representations of the Kronecker quiver, was of fundamental importance because it opened the world of algebraic geometry applied to quiver Grassmannians to me. The most recent work that I posted on arXiv is in collaboration with Markus Reineke and Evgeny Feigin and deals with particular Dynkin-type quiver Grassmannians called PrIncipal, because they are associated with the sum of a projective and an injective representation: we discovered that these varieties are extremal within the family to which they belong and that they bound the irreducible varieties of minimal dimension in the family, generalizing special cases of type A. Publications no.~1, 5, 9, 11, 14, 15, 20, 23, 27 and 28.

Linear degenerations of flag varieties Soon afterwards I spent a research semester at HIM in Bonn and attended Evgeny Feigin’s seminar, where he spoke about the degenerate flag varieties that he had recently discovered. Immediately after the seminar, while commenting with Markus Reineke, I said: ``did you notice that Feigin’s varieties are type A quiver Grassmannians?''. This is how this fruitful line of research on linear degenerations of flag varieties began, leading me to publish about 10 papers with people from different research areas, including Peter Littelmann, Martina Lanini, Ghislain Fourier and Xin Fang. Publications no.~1, 5, 7, 10, 11, 13, 16, 17, 21, 26.

Symmetric quivers I have worked on the problem of generalizing linear degenerations of flag varieties to the orthogonal and symplectic cases: this requires understanding the degeneration order for representations of finite-type symmetric quivers. This problem had been open for many years and I solved it, with great effort, in 2021 with the help of Magdalena Boos, who was then a postdoctoral fellow visiting me for one year. This research line also includes a project with Prof. Jerzy Weyman (Krakow) concerning connections with commutative algebra, in particular perfect ideals and self-dual resolutions of length four. Publications no.~6, 8, 12.

Representation varieties and orbit closures I have worked on the long-standing problem of understanding whether the closures of orbits of Dynkin-type representations are normal. I have an ongoing project on this with Markus Reineke, Grzegorz Zwara and Marco Trevisiol, which I presented at Oberwolfach in 2020.

Algebras with polynomial identities In the last year I have worked on algebras with polynomial identities and on the problem of classifying path algebras of quivers that satisfy polynomial identities. This project resulted in two 2026 papers in collaboration with my PhD student Javier De Loera Chavez and my postdoctoral fellow Elena Pascucci, one of them with the contribution of the eminent Allan Berele, one of the leading experts in PI algebras. Publications no.~3 and 4.

Knots and quivers In the last year I have worked on the relation between knots and quivers discovered by Ralf Schiffler and V'eronique Bazier-Matte. Together with Domenico Fiorenza and his students, we found a homological interpretation of this relation that allowed us to give a new and elegant proof of Kauffman’s Clock Theorem and to describe, in a more intrinsic way, the relationship between Kauffman states and certain representations of the quiver with potential naturally associated with the knot and, more generally, with a planar graph. This work was uploaded to the archive in May 2026 and submitted to a journal shortly afterwards; we are currently awaiting the referee report. Publication no.~2.

Publications and preprints:

  1. ``Extremality of PrIncipal quiver Grassmannians'' G.~Cerulli~Irelli, E.~Feigin, M.~Reineke arXiv: 2607.20336, (2026).
  2. ``A categorification of Kauffman states for planar graphs'' G.~Cerulli~Irelli, D.~Fiorenza, E.~Landi, M.~Matteucci arXiv: 2605.19872, (2026).
  3. ``Polynomial identities for quivers via incidence algebras'' A.~Berele, G.~Cerulli~Irelli, J.~De~Loera~Chavez, E.~Pascucci Bulletin of the London Mathematical Society 58 (2026), no.~5.
  4. ``Quivers with Polynomial Identities'' G.~Cerulli~Irelli, J.~De~Loera~Chavez, E.~Pascucci Annals of Representation Theory 3, no. 2, (2026), 165-177.
  5. ``Specialization map for quiver Grassmannians'' G. Cerulli Irelli, F. Esposito, X. Fang, G. Fourier Algebras and Representation Theory (2026).
  6. ``On degenerations and extensions of symplectic and orthogonal quiver representations'' M.~Boos, G.~Cerulli Irelli Arkiv för mathematik, 63 (2025), 61–115.
  7. ``Motzkin combinatorics in linear degenerations of the flag variety'' G. Cerulli Irelli, F. Esposito, M. Marietti IMRN 2023 (2023), no.~22, 19184-19204.
  8. ``Symmetric degenerations are not in general induced by type A degenerations'' M.~Boos, G.~Cerulli Irelli Rendiconti di Matematica e delle sue applicazioni (7), 43 (2022), no.~2, 133-149.
  9. ``Cell decompositions and algebraicity of cohomology for quiver Grassmannians'' G.~Cerulli~Irelli, F.~Esposito, H.~Franzen, M.~Reineke Advances in Mathematics, 379 (2021).
  10. ``Linear degenerations of flag varieties: partial flags, defining equations, and group actions'' G.~Cerulli~Irelli, X.~Fang, E.~Feigin, G.~Fourier, M.~Reineke Mathematische~Zeitschrift 296 (2020), no~1-2, 453-477.
  11. ``Three lectures on quiver Grassmannians'' G.~Cerulli~Irelli. Representation theory and beyond. Workshop and 18th International Conference on Representations of Algebras (ICRA 2018) August 8-17, 2018. Prague, Czech Republic. Contemporary Mathematics 758 (2020).
  12. ``Parabolic orbits of 2-nilpotent elements for classical groups'' M.~Boos, G.~Cerulli~Irelli, F.~Esposito Journal of Lie Theory 29 (2019), no.~4, 969-996.
  13. ``Linear degenerations of flag varieties'' G.~Cerulli~Irelli, X.~Fang, E.~Feigin, G.~Fourier, M.~Reineke Mathematische~Zeitschrift 287 (2017), no~1-2, 615-654.
  14. ``Geometry of quiver Grassmannians of Dynkin type with applications to cluster algebras''. G.~Cerulli Irelli. Representation theory-current trends and perspectives, 13-45. European Mathematical Society, Series of Congress Reports (2017).
  15. ``Schubert Quiver Grassmannians''. G.~Cerulli Irelli, E.~Feigin, M.~Reineke. Algebras and Representation Theory 20 (2017), no.~1, 147-161.
  16. ``Degenerate flag varieties and Schubert varieties: a characteristic free approach''. G.~Cerulli~Irelli, M.~Lanini, P.~Littelmann Pacific Journal of Mathematics 284 (2016), no.~2, 283-308.
  17. ``Degenerate flag varieties of type A and C are Schubert varieties''. G.~Cerulli~Irelli, M.~Lanini. International Mathematics Research Notices 2015 (2015), no~15, 6353-6374.
  18. ``Caldero–Chapoton algebras''. G.~Cerulli Irelli, D. Labardini Fragoso, J. Schr"oer. Transactions of the American Mathematical Society 367 (2015), 2787–2822.
  19. ``Homological approach to the Hernandez–Leclerc construction and quiver varieties''. G.~Cerulli Irelli, E.~Feigin, M.~Reineke. Representation Theory of the American Mathematical Society 18 (2014), 1–14.
  20. ``Desingularization of quiver Grassmannians associated with Dynkin quivers''. G.~Cerulli Irelli, E.~Feigin, M. ~Reineke. Advances in Mathematics 245 (2013), 182–207.
  21. ``Degenerate flag varieties: moment graphs and Schr"oder numbers''. G.~Cerulli Irelli, E. Feigin, M. Reineke. Journal of Algebraic Combinatorics 38 (2013), no. 1, 159–189.
  22. ``Linear independence of cluster monomials for skew–symmetric cluster algebras''. G.~Cerulli~Irelli, B.~Keller, D.~Labardini~Fragoso, P.-G.~Plamondon. Compositio Mathematica 149 (2013), 1753–1764.
  23. ``A homological interpretation of transverse quiver Grassmannians''. G.~Cerulli~Irelli, G.~Dupont, F.~Esposito. Algebras and Representation Theory 16 (2013), no. 2, 437-444.
  24. ``Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials''. G.~Cerulli~Irelli, D.~Labardini~Fragoso. Compositio Mathematica 148 (2012), 1833–1866.
  25. ``Cluster algebras of type $A_{2**^{(1)**$''. G.~Cerulli~Irelli Algebras and Representation Theory **15** (2012), no. 5, 977-1021.
  26. ``Quiver Grassmannians and degenerate flag varieties''. G.~Cerulli~Irelli, E. Feigin, M. Reineke. Algebra and Number Theory 6 (2012), no. 1, 165-194.
  27. ``Quiver Grassmannians associated with string modules''. G.~Cerulli~Irelli. Journal of Algebraic Combinatorics 33 (2011), 259–276.
  28. ``Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras''. G.~Cerulli~Irelli, F. Esposito. Algebra and Number Theory 5 (2011), no. 6, 777-801.
  29. Tesi di dottorato: ``Structural theory of rank three cluster algebras of affine type''. G.~Cerulli~Irelli. Universit'a di Padova (2008).