| 研究実績の概要 |
This year, I have been finishing and submitting the article in collaboration with Osamu Iyama, Nathan Reading, Idun Reiten and Hugh Thomas about the lattice theory of torsion classes. It turned out that we got better results than we expected as most results we get about torsion classes are true even when there are infinitely many torsion classes. In particular, we extend the notion of congruence uniform lattice to the case of complete infinite lattices. This result is surprising both from an algebraic point of view and from a lattice theoretical point of view, as it shows that tors A has rather unique properties, compared to classical and well studied lattices. In particular, we show that its Hasse quiver contains a big part of its information, even though it is an infinite lattice.
In parallel, I started with Aaron Chan a project of classification of torsion classes over Brauer graph algebras. We expect them to be classified by systems of non-crossing, unbounded curves on some compact surfaces up to homotopy. In this way, it extends in a natural way the classical notion of laminations, extensively studied by William Thurston. Moreover, we have in mind a purely combinatorial description of the lattice structure of these torsion classes.
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