For an 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 division algebra. It is well-known that bricks play important roles in various areas of research. Motivated by some new applications of bricks in different classical and modern topics in representation theory of algebras, this important family of indecomposable modules has received a lot of attention in the past 10 years. More specifically, it is shown that a good knowledge of bricks provides insights into the study of torsion classes and wide subcategories, stability conditions and wall-chamber structures, tilting and τ-tilting theory, geometry of representation varieties, to mention just a few.

In this series, I will primarily focus on a conjecture that I first posed in 2019, originally to establish a conceptual dictionary between τ-tilting finiteness and geometry of the irreducible components of representation varieties. This conjecture is nowadays called the Second Brick Brauer-Thrall (2nd bBT): A is brick-infinite if and only if, for a positive integer d, there are infinitely many bricks of dimension d. Although this conjecture is settled for some important families of algebras, it is still open in full generality. Thanks to a series of work in the past 5 years, the 2nd bBT conjecture is now reduced to a special family of algebras, and furthermore some interesting results have emerged. After a brief recollection of the necessary algebraic and geometric background and rudiments, I will describe our systematic treatment of the 2nd bBT conjecture, explain the connections between the 2nd bBT with some other open conjectures, and prove the 2nd bBT and several other conjectures for some new families of algebras.

This event consists of three online 50-minute lectures.

Venue link:

https://zoom.us/j/92537690776?pwd=KaiyvvJlacGY4pH1qVE1aysdbDAaAo.1

Meeting ID: 925 3769 0776
Passcode: 2222

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