In recent years, a major focus of my research has been to deepen our understanding of the behavior of bricks and related phenomena in the context of finite-dimensional algebras. For a finite dimensional algebra A over an algebraically closed field k, a (left) A-module M is called a brick if the endomorphism algebra of M is a skew field; that is, every non-zero A-homomorphism f:M--->M is invertible. Provided that M is a finitely generated A-module, M is a brick if and only if its endomorphism algebra is the underlying ground field k. In this case, bricks are also known as Schur representations. Bricks have long been recognized for their central role in various areas of research, including their diverse classical and modern applications in stability conditions, wall-chamber structures, τ-tilting theory, lattice theory of torsion classes, wide subcategories, spectrum of algebras, to mention just a few. For a summary of some recent treatment of a series of modern problems on bricks, see this survey!
Building on my earlier work conducted during my doctoral studies, and primarily motivated by two open conjectures that I first posed in my preprint in 2019 (Conjecture 6.6), I have carried out extensive work on a systematic study of bricks. In fact, my stronger conjecture, nowadays called the Second brick-Brauer-Thrall conjecture (2nd bBT), is concerned with the distribution of bricks over those algebras which admit infinitely many non-isomorphic bricks. This phenomenon can be viewed as the modern analogue of the celebrated classical Second Brauer-Thrall conjecture (now theorem). For some remarks on the 2nd bBT and related problems, see Section 2 of this paper!
As shown in my independent and collaborative work, the 2nd bBT conjecture establishes some novel conceptual connections between several classical and modern aspects of representation theory, including the geometry of the representation varieties, families of stable modules, components of the Auslander-Reiten quivers, the g-vector fans of algebras, and also behavior of infinite dimensional modules, particularly in connection with the generic modules. Furthermore, the 2nd bBT conjecture relates to, and in some cases implies, some other open conjectures in representation theory of algebras. Although we have settled the 2nd bBT conjecture for some families of algebras, as of Fall of 2025, this conjecture remains open! For further details, recent developments, and future goals, please see my Research Statement!
Preprints and Publications
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"On the bricks (Schur representations) of finite dimensional algebras", (arXiv:2508.11789),
Coauthor: C. Paquette. -
"Brick-splitting torsion classes and Trim lattices", (arXiv:2506.13602),
Coauthors: S. Asai, O. Iyama, C. Paquette. -
"Geometric interactions between bricks and τ-rigidity", (arXiv:2311.14863),
Coauthor: C. Paquette. -
"A Continuous Associahedron of Type A", (arXiv:2108.12927) [to appear in Mathematische Zeitschrift],
Coauthors: M. Kulkarni, J. Matherne, J. Rock. -
"Hom-Orthogonal modules and brick-Brauer-Thrall conjectures", (arXiv & Journal) [to appear in Journal of Algebra], Coauthor: C. Paquette.
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"Biserial algebras and generic bricks", (arXiv & Journal), Mathematische Zeitschrift (2025),
Coauthor: C. Paquette. -
"ABHY associahedra and Newton polytopes of F-ploynomials for cluster algebras of simply laced finite type", (arXiv & Journal), Journal of the London Mathematical Society (2024),
Coauthors: V. Bazier-Matte, N. Chapelier-Laget, G. Douville, H. Thomas, E. Yıldırım. -
"Minimal (τ-)tilting infinite algebras", (arXiv & Journal), Nagoya Mathematical Journal (2023)
Coauthor: C. Paquette. -
"τ-Tilting finiteness of biserial algebras", (arXiv & Journal), Algebras and Representation Theory (2023).
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"τ-Tilting finiteness of non-distributive algebras and their module varieties", (arXiv & Journal), Journal of Algebra (2022).
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"On the combinatorics of gentle algebras", (arXiv & Journal), Canadian Journal of Mathematics (2020),
Coauthors: T. Brüstle, G. Douville, H. Thomas, E. Yıldırım. -
"A categorification of biclosed sets of strings", (arXiv & Journal), Journal of Algebra (2020),
Coauthors: A. Garver and T. McConville. -
"On Some Properties of Toeplitz Matrices", (Journal) Cogent Mathematics (2016),
Coauthors: D. Kucerovsky and A. Sarraf.
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PhD Dissertation: "τ-tilting finiteness of minimal representation-infinite algebras", Defended in June (2020),
at Université du Québec à Montréal, under the supervision of Prof. Hugh Thomas.
Not for publication (yet!)
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"Extended Abstract: Distribution of bricks and an open algebro-geometric conjecture (click to read the pdf. file).
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"Strict Dihedral Algebras are tau-tilting finite" (click to read the pdf. file).
Posters (click to see the content)
Academic events (co)-organization
In the future, I am co-organizing the following event(s):
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Spectrums in Representation Theory of Algebras and Related Topics, Osaka, Japan (December 16th-19th, 2025):
Website: https://pabloocal.github.io/SRTART2025/
Previously, I have co-organized the following events:
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Thematic Program: TDA PARTI - Topological Data Analysis, Persistence And Representation Theory Intertwined
Okinawa Institute of Science and Technology (OIST), Japan (June 23rd-August 9th, 2025)
Website: https://www.oist.jp/visiting-program/tp25td
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Workshop on Bricks and Endofinite Representations,
Bielefeld, Germany (March 3rd-5th, 2025)
Website: https://www.math.uni-bielefeld.de/birep/meetings/bricks2025/
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Winter School on Large Modules and Endofiniteness,
Stuttgart, Germany (February 24th-28th, 2025)
Website: https://sites.google.com/view/winterschoolconference-germany/home
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Autumn School & Conference on New Developments in Representation Theory of Algebras,
Okinawa, Japan (November 18th-30th, 2024)
Website: https://sites.google.com/view/autumn-oist-school-conference/home
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Summer Research School on Applications of Representation Theory in Topological Data Analysis and Geometric Invariant Theory, Montreal, Canada (June 3rd-7th, 2024)
Website: https://sites.google.com/view/montreal-rep-summer-school2024/home?authuser=0
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Spring Workshop on the Representation Theory of Algebras and related areas,
Okinawa, Japan (April 22nd-26th, 2024)
Website: https://sites.google.com/view/oist-rep-theory-springworkshop/home?authuser=0
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Advances in Representation Theory of Algebras (ARTA IX),
Kingston, Canada (June 12th-16th, 2023)
Website: https://sites.google.com/view/arta9/home
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Representation Theory of Algebras, A special Session at Canadian Mathematical Society Winter Meeting 2022
Toronto, Canada (December 2nd-5th, 2022)
Website: https://winter22.cms.math.ca/
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Mutations: From Cluster Algebras to Representation Theory, An ISM Discovery research school
Montreal, Canada (July 4th-8th, 2022)
Website: https://repmutationschool.wixsite.com/website
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Representation Theory and Geometry, An online conference during Covid period
virtually hosted by Queen's University, Canada (February 14th-16th, 2022)
Website: https://reptheogeometry202.wixsite.com/representationtheory