Abstract: 

For an algebra A, let brick(A) denote the set of all isomorphism classes of bricks over A. That is, for every M in brick(A), all nonzero A-homomorphism from M to M are invertible. Bricks are known to play pivotal roles in the combinatorial, lattice theoretical, homological and geometric aspects of representation theory of A. In this talk, we discuss an open conjecture on the behavior of bricks and primarily focus on tame representation-infinite algebras. To this end, we define the notions of brick discrete vs. brick continuous, and after a brief recollection of some fundamental properties, we introduce the notion of "generic brick". The aforementioned notions closely relate to the algebro-geometric study of bricks by Chinrdris-Kinser-Weyman, as well as the work of Carroll-Chindris. This new language allows us to state a stronger version of our conjecture over tame algebras. For (special) biserial algebras, we verify the stronger conjecture through the explicit classification of minimal brick infinite algebras of this type. Finally, we leave some remarks on the more general setting.

This is a report on joint work in progress with Charles Paquette.