The main focus of my research is Representation theory of algebras. In my work, roughly speaking, one understands the nature of an algebra via its action on some other algebraic structures, known as modules. Below, you can find an analogy that I have come up with, and often find it useful, to explain these concepts to non-mathematicians. Experts may disagree with this analogy, but this is not for them anyway! 

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''Suppose you are given a stick A (which we may want to call an algebra). Suppose this stick A is unfamiliar to you, and you are curious to better understand the nature of A. 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 hit it against different objects that you already know and then observe the influence of this interaction 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 and other properties 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!"