Hipolito Treffinger

Bricks, walls and chambers

In general, given a finite dimensional algebra, one is interested on the properties and invariants of its module category. Among the variety of techniques available, its is well-known that the stability conditions induce a geometrical invariant of the algebra known as the wall and chamber structure of the algebra. In this talk I will present results showing that some homological properties, known as the $\tau$-tilting theory of the algebra, determines a big part of the wall and chamber structure. In particular, these results imply that the wall and chamber structure of an algebra is a well-behaved simplicial fan.

It is worth noticing that walls are determined by some special modules called "bricks". After explaining this phenomenon, I will give a characterisation of the bricks induced from $\tau$-tilting theory of the algebra. Time permitting I will show how to use the wall and chamber structure of an algebra to calculate its $\tau$-tilting theory.

This is joint work with Thomas Brüstle and David Smith.