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!