Ring Theory and Representation Theory Seminar in 2019

This seminar is coorganized by Osamu Iyama, Ryo Takahashi and Erik Darpö.
Previous seminars: 2018, 2017, 2015-16, 2014, 2013--.


11 March (Monday) 13:00-14:30
Room: 309 math.
Speaker: Atsushi Takahashi (Osaka University)
Title: Maximally graded matrix factorizations for an invertible polynomial of chain type

Abstract: In 1977, Orlik-Randell proposed a conjecture on the existence of certain distinguished basis of vanishing cycles in the Milnor fiber associated to an invertible polynomial of chain type. Under the homological mirror symmetry, it is expected from their conjecture that the triangulated category of maximally-graded matrix factorizations for the Berglund--H\"{u}bsch transpose admits a full exceptional collection with a nice numerical property. The purpose of this talk is to prove this algebraic analogue of Orlik-Randell conjecture.
6 March (Wednesday) 13:00-14:30
Room: 409 math.
Speaker: Shunsuke Kano (Tokyo Institute of Technology)
Title: 曲面の擬Anosov写像類と圏論的エントロピー

Abstract: 曲面の写像類のうち、最も一般的な種類として擬Anosov写像類と呼ばれるものがある。擬Anosov写像類の不変量として位相的エントロピーが有名である。Dimitrov--Haiden--Katzarkov--Kontsevich は位相的エントロピーの圏化として、三角圏の自己関手の圏論的エントロピーを定義した。また、曲面の三角形分割 T から三角圏 D(T) を構成する方法が知られている。今回は擬Anosov写像類 f を D(T) 上の同値関手に持ち上げ、その圏論的エントロピーが元の f の位相的エントロピーと一致することを解説する。この同値関手の構成は、三角圏 D(T) の安定性条件への作用や、クラスター多様体の境界への作用と密接に関連するため、これらについても時間が許せば話したい。
1 March (Friday) 10:30-12:00
Room: 552 math.
Speaker: Sota Asai (Nagoya University)
Title: The wall-chamber structures of the real-valued Grothendieck groups

Abstract: We consider a finite-dimensional algebra $A$ over a field and the real-valued Grothendieck group of the category of finite-dimensional projective $A$-modules. The real-valued Grothendieck group can be identified with a Euclidean space, and Br\"{u}stle--Smith--Treffinger defined a wall-chamber structure of the real-valued Grothendieck group via the semistability conditions by King. In this talk, I will introduce my new combinatorial algorithm to obtain the wall-chamber structure in the case $A$ is a path algebra. I will also explain my result that the chambers of the wall-chamber structure bijectively correspond to the 2-term silting objects of the perfect derived category, by using the numerical torsion(-free) classes defined by Baumann--Kamnitzer--Tingley.
1 March (Friday) 13:30-15:00
Room: 552 math.
Speaker: Hiroki Matsui (Nagoya University)
Title: On the equivariant smash nilpotence theorem

Abstract: One of the important approaches to understand the structure of a given triangulated category is to classify its thick subcategories. Originally, such an approach is considered by Devinatz, Hopkins, and Smith in their studies of stable homotopy theory. After that Hopkins, Neeman, and Thomason gave a corresponding classification result of the thick tensor ideals of the perfect derived category of a noetherian scheme. On the other hand, Benson, Carlson, Iyengar et. al. classified the thick tensor ideals of the bounded derived category of finitely generated representations of a finite group. In this talk, we consider schemes X admitting an action of a finite group G and assume |G| acts as a unit on X. Under this assumption, we classify thick tensor ideals of the category of equivariant perfect complexes D_{perf}(X)^G. If we take G to be the trivial group, then this result recovers Thomason's result and if we consider X to be the spectrum of a commutative ring R, then we obtain the classification of thick tensor ideals of the perfect derived category of a skew group algebra R*G. The key point to prove the theorem is to establish the equivariant version of the smash nilpotence theorem.
1 March (Friday) 15:30-17:00
Room: 552 math.
Speaker: Laurent Demonet (Nagoya University)
Title: Torsion class over Brauer graph algebras and gentle algebras

Abstract: [jt. work in progress with A. Chan] Applying techniques introduced in a joint work with Iyama, Reading, Reiten and Thomas [DIRRT], we classify torsion classes over Brauer graph algebras and finite dimensional gentle algebras. More specifically, we give a combinatorial-geometric realization of the complete lattice of torsion classes. After giving several examples for concrete algebras, we explain the strategy of the proof, recalling some important results of [DIRRT].