Broadly speaking, my mathematical interests have revolved around Representation theory of algebras, Cluster algebras, Algebraic combinatorics, and the connections of these areas to Algebraic geometry. More recently, I have also developed interests in the applications of Representation theory in Topological Data Analysis, as well as in Graph Neural Network.
A brief description of some open problems that I am thinking about (for general mathematicians):
A major part of my research in the past few years has been aimed at giving a better understanding of the behavior of bricks and related problems over 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, then 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. It is well-known that bricks play decisive roles in various domains of research, including in stability conditions, wall-chamber structures, τ-tilting theory, lattice theory of torsion classes, wide subcategories, spectrum of algebras, to mention just a few.
Following my work in my PhD studies, and primarily motivated by an open conjecture that I first posed in a preprint in 2019, I have conducted some extensive independent and collaborative research on a systematic treatment of bricks. This 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 modern conjecture is analogous to the celebrated classical Second Brauer-Thrall conjecture (now theorem). As shown recently, 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 summer of 2024, this conjecture is still open in full generality. For further details, please see my recent work here.
A lay description of the mathematics I often think about (for non-mathematicians):
In representation theory of algebras, my main focus of research, roughly speaking one understands the nature of an algebra via its action on some other algebraic structures, known as modules. An analogy I have made and often employ to explain this concept to non-mathematicians is the following. Experts may disagree with this analogy, but this is not for them anyway!
''Suppose you are given a stick A (which we may want to call an algebra) and this stick A is unfamiliar to you. Then, how would you develop some knowledge about it and how can you possibly determine whether the stick A is sturdy?
You probably start by looking at its size, shape and some other physical properties. But, a lot of features of an object become clear only when we analyze its interactions with the other objects. In this case for instance, you may get to know about the unfamiliar stick A if you start hitting it against different objects that you already know and observe its influence on them. Obviously, your stick A cannot act on every arbitrary choice of object that may come to your mind. For example, you cannot hit a dream with it (at least as long as the stick itself is not a dream or part of it)!
So, A can only act on certain objects whose nature is somehow compatible with that of your stick A. From the action of A on each of these objects (which one can call an A-module) you get a better understanding of the strength of the stick A.
To obtain a good knowledge of the sturdiness of A, sometimes one needs to examine A with respect to only finitely many other objects (A-modules) and sometimes one needs to hit infinitely many objects to get to know A more and more! Hence, one cay say some algebras (sticks) are easier to understand, where the complexity could be considered as the number of modules (objects) you need to hit to fully determine the strength of the stick!''