Tokyo-Nagoya Algebra Special Seminar

Monday 10:30 ~ 12:00 at A328

Monday 14:45 ~ 16:15 at B114

Rigidity of bricks and brick-Brauer-Thrall conjectures

In these talks, I try to accomplish two goals: First, to give a summary of some new developments in the study of bricks and their connections to several open problems/conjectures in $\tau$-tilting theory, stability conditions and related topics. Second, to share some concrete problems that can be viewed as the middle steps towards a systematic treatment of these conjectures. These talks will be primarily based on my earlier work in my PhD thesis and a series of recent joint papers with Charles Paquette. Our treatment of bricks relies on the fruitful interplays between the algebraic and geometric aspects of representation theory of finite dimensional algebras. While doing so, we revisit some classical notions and problems (such as generic modules and the celebrated Brauer-Thrall conjectures) and then use some new results on the behavior of bricks to give new characterizations of some classical families of algebras and study their modern counterparts.

Wednesday 09:30 ~ 11:00 at A358

Wednesday 11:30 ~ 13:00 at A358

Semistable torsion classes and canonical decompositions in Grothendieck groups

We study two classes of torsion classes which generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen-Fei. More strongly, for E-tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. If time permits, we will answer a question by Derksen-Fei negatively by giving examples of algebras which do not satisfy the ray condition, and also give an explicit description of TF equivalence classes of preprojective algebras of affine type A.

Wednesday 14:30 ~ 16:00 at GSM 552

Wednesday 16:15 ~ 17:45 at GSM 552

Interval neighborhoods of silting cones

In the study of 2-term (pre)silting complexes for a finite-dimensional algebra, silting cones and the wall-chamber structure given by stability conditions in the real Grothendieck group are strong tools. To use them more efficiently, I defined the interval neighborhood of each silting cone, and proved that the local wall-chamber structure inside the interval neighborhood of each silting cone recovers the whole wall-chamber structure for the algebra appearing in the $\tau$-tilting reduction of Jasso at the corresponding 2-term presilting complex. In joint work with Osamu Iyama, we investigated interval neighborhoods more deeply. I would like to explain some of our representation-theoretical and combinatorial results on interval neighborhoods in my talks.