## Tokyo-Nagoya Algebra Seminar

Seminar information archive ～06/07｜Next seminar｜Future seminars 06/08～

Organizer(s) | Noriyuki Abe, Aaron Chan, Erik Darpoe, Osamu Iyama, Tsutomu Nakamura, Hiroyuki Nakaoka, Ryo Takahashi |
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**Seminar information archive**

### 2023/05/16

15:00-16:30 Online

Cluster-additive functions and frieze patterns with coefficients (English)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

**Antoine de Saint Germain**(University of Hong Kong)Cluster-additive functions and frieze patterns with coefficients (English)

[ Abstract ]

In his study of combinatorial features of cluster categories and cluster-tilted algebras, Ringel introduced an analogue of additive functions of stable translation quivers called cluster-additive functions.

In the first part of this talk, we will define cluster-additive functions associated to any acyclic mutation matrix, relate them to mutations of the cluster X variety, and realise their values as certain compatibility degrees between functions on the cluster A variety associated to the Langlands dual mutation matrix (in accordance with the philosophy of Fock-Goncharov). This is based on joint work with Peigen Cao and Jiang-Hua Lu. In the second part of this talk, we will introduce the notion of frieze patterns with coefficients based on joint work with Min Huang and Jiang-Hua Lu. We will then discuss their connection with cluster-additive functions.

ミーティングID: 815 4247 1556

パスコード: 742240

[ Reference URL ]In his study of combinatorial features of cluster categories and cluster-tilted algebras, Ringel introduced an analogue of additive functions of stable translation quivers called cluster-additive functions.

In the first part of this talk, we will define cluster-additive functions associated to any acyclic mutation matrix, relate them to mutations of the cluster X variety, and realise their values as certain compatibility degrees between functions on the cluster A variety associated to the Langlands dual mutation matrix (in accordance with the philosophy of Fock-Goncharov). This is based on joint work with Peigen Cao and Jiang-Hua Lu. In the second part of this talk, we will introduce the notion of frieze patterns with coefficients based on joint work with Min Huang and Jiang-Hua Lu. We will then discuss their connection with cluster-additive functions.

ミーティングID: 815 4247 1556

パスコード: 742240

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2023/04/28

13:00-14:30 Online

Please see the reference URL for details on the online seminar.

Full exceptional collections associated with Bridgeland stability conditions (Japanese)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Takumi Otani**(Osaka Univeristy)Full exceptional collections associated with Bridgeland stability conditions (Japanese)

[ Abstract ]

The space of Bridgeland stability conditions on a triangulated category is important in mirror symmetry and many people develop various techniques to study it. In order to study the homotopy type of the space of stability conditions, Macri studied stability conditions associated with full exceptional collections. Based on his work, Dimitrov-Katzarkov introduced the notion of a full σ-exceptional collection for a stability condition σ.

In this talk, I will explain the relationship between full exceptional collections and stability conditions and some properties. I will also talk about the existence of full σ-exceptional collections for the derived category of an acyclic quiver.

[ Reference URL ]The space of Bridgeland stability conditions on a triangulated category is important in mirror symmetry and many people develop various techniques to study it. In order to study the homotopy type of the space of stability conditions, Macri studied stability conditions associated with full exceptional collections. Based on his work, Dimitrov-Katzarkov introduced the notion of a full σ-exceptional collection for a stability condition σ.

In this talk, I will explain the relationship between full exceptional collections and stability conditions and some properties. I will also talk about the existence of full σ-exceptional collections for the derived category of an acyclic quiver.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2023/04/21

13:00-14:30 Online

Please see the reference URL for details on the online seminar.

Categorifications of deformed Cartan matrices (Japanese)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Kota Murakami**(University of Tokyo)Categorifications of deformed Cartan matrices (Japanese)

[ Abstract ]

In a series of works of Gei\ss-Leclerc-Schr\″oer, they introduced a version of preprojective algebra associated with a symmetrizable generalized Cartan matrix and its symmetrizer. For finite type, it can be regarded as an un-graded analogue of Jacobian algebra of certain quiver with potential appeared in the theory of (monoidal) categorification of cluster algebras.

In this talk, we will present an interpretation of graded structures of the preprojective algebra of general type, in terms of a multi-parameter deformation of generalized Cartan matrix and relevant combinatorics motivated from several contexts in the theory of quantum loop algebras or quiver $\mathcal{W}$-algebras. From the vantage point of the representation theory of preprojective algebra, we will prove several purely combinatorial properties of these concepts. This talk is based on a joint work with Ryo Fujita (RIMS).

[ Reference URL ]In a series of works of Gei\ss-Leclerc-Schr\″oer, they introduced a version of preprojective algebra associated with a symmetrizable generalized Cartan matrix and its symmetrizer. For finite type, it can be regarded as an un-graded analogue of Jacobian algebra of certain quiver with potential appeared in the theory of (monoidal) categorification of cluster algebras.

In this talk, we will present an interpretation of graded structures of the preprojective algebra of general type, in terms of a multi-parameter deformation of generalized Cartan matrix and relevant combinatorics motivated from several contexts in the theory of quantum loop algebras or quiver $\mathcal{W}$-algebras. From the vantage point of the representation theory of preprojective algebra, we will prove several purely combinatorial properties of these concepts. This talk is based on a joint work with Ryo Fujita (RIMS).

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2023/02/10

17:00-18:30 Online

Please see the reference URL for details on the online seminar.

Silting discrete代数上のsemibrick複体とspherical objects (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Wahei Hara**(University of Glasgow)Silting discrete代数上のsemibrick複体とspherical objects (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2023/01/20

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

Tropical cluster transformations and train track splittings (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Shunsuke Kano**(Tohoku University)Tropical cluster transformations and train track splittings (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/10/20

16:40-18:10 Online

Please see the reference URL for details on the online seminar.

A surface and a threefold with equivalent singularity categories (English)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Martin Kalck**(Freiburg University)A surface and a threefold with equivalent singularity categories (English)

[ Abstract ]

We discuss a triangle equivalence between singularity categories of an

affine surface and an affine threefold.

Both are isolated cyclic quotient singularities.

This seems to be the first (non-trivial) example of a singular

equivalence involving varieties of even and odd Krull dimension.

The same approach recovers a result of Dong Yang showing a singular

equivalence between certain cyclic quotient singularities in dimension

2 and certain finite dimensional commutative algebras.

This talk is based on https://arxiv.org/pdf/2103.06584.pdf

[ Reference URL ]We discuss a triangle equivalence between singularity categories of an

affine surface and an affine threefold.

Both are isolated cyclic quotient singularities.

This seems to be the first (non-trivial) example of a singular

equivalence involving varieties of even and odd Krull dimension.

The same approach recovers a result of Dong Yang showing a singular

equivalence between certain cyclic quotient singularities in dimension

2 and certain finite dimensional commutative algebras.

This talk is based on https://arxiv.org/pdf/2103.06584.pdf

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/07/20

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

Grothendieck enriched categories (Japanese)

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Yuki Imamura**(Osaka University)Grothendieck enriched categories (Japanese)

[ Abstract ]

Grothendieck圏は、入射的余生成子の存在や随伴関手定理の成立など、アーベル圏の中でも特に良い性質を持つことで知られる。通常Grothendieck圏は、生成子を持つ余完備なアーベル圏であって、フィルター余極限を取る関手が完全関手になるような圏として内在的な性質で以て定義されるが、加群圏の"良い部分圏"として実現できるという外在的な特徴づけ(Gabriel-Popescuの定理)も存在する。アーベル圏が自然なプレ加法圏(アーベル群の圏Ab上の豊穣圏)の構造を持つことから、Gabriel-Popescuの定理はAb-豊穣圏に対する定理だと思うことができる。本講演では、より一般のGrothendieckモノイダル圏V上の豊穣圏に対してGabriel-Popescuの定理の一般化を定式化し証明する。特にVとしてアーベル群の複体の圏Chを取ることによりGrothendieck圏のdg圏類似とそのGabriel-Popescuの定理が得られることも確認する。

[ Reference URL ]Grothendieck圏は、入射的余生成子の存在や随伴関手定理の成立など、アーベル圏の中でも特に良い性質を持つことで知られる。通常Grothendieck圏は、生成子を持つ余完備なアーベル圏であって、フィルター余極限を取る関手が完全関手になるような圏として内在的な性質で以て定義されるが、加群圏の"良い部分圏"として実現できるという外在的な特徴づけ(Gabriel-Popescuの定理)も存在する。アーベル圏が自然なプレ加法圏(アーベル群の圏Ab上の豊穣圏)の構造を持つことから、Gabriel-Popescuの定理はAb-豊穣圏に対する定理だと思うことができる。本講演では、より一般のGrothendieckモノイダル圏V上の豊穣圏に対してGabriel-Popescuの定理の一般化を定式化し証明する。特にVとしてアーベル群の複体の圏Chを取ることによりGrothendieck圏のdg圏類似とそのGabriel-Popescuの定理が得られることも確認する。

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/07/06

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

Cyclic polytopes and higher Auslander--Reiten theory 3 (English)

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Nicholas Williams**(The University of Tokyo)Cyclic polytopes and higher Auslander--Reiten theory 3 (English)

[ Abstract ]

This continues part 2. In the third talk, we consider the relationship between the objects from the first two talks. We explain how triangulations of even-dimensional cyclic polytopes may be interpreted in terms of tilting modules, cluster-tilting objects, or d-silting complexes. We then proceed in the d-silting framework, and show how the higher Stasheff--Tamari orders may be interpreted algebraically for even dimensions. We explain how this allows one to interpret odd-dimensional triangulations algebraically, namely, as equivalence classes of d-maximal green sequences. We briefly digress to consider the issue of equivalence of maximal green sequences itself. We then show how one can interpret the higher Stasheff--Tamari orders on equivalence classes of d-maximal green sequences. We finish by drawing out some consequences of this algebraic interpretation of the higher Stasheff--Tamari orders.

[ Reference URL ]This continues part 2. In the third talk, we consider the relationship between the objects from the first two talks. We explain how triangulations of even-dimensional cyclic polytopes may be interpreted in terms of tilting modules, cluster-tilting objects, or d-silting complexes. We then proceed in the d-silting framework, and show how the higher Stasheff--Tamari orders may be interpreted algebraically for even dimensions. We explain how this allows one to interpret odd-dimensional triangulations algebraically, namely, as equivalence classes of d-maximal green sequences. We briefly digress to consider the issue of equivalence of maximal green sequences itself. We then show how one can interpret the higher Stasheff--Tamari orders on equivalence classes of d-maximal green sequences. We finish by drawing out some consequences of this algebraic interpretation of the higher Stasheff--Tamari orders.

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/06/29

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

Cyclic polytopes and higher Auslander--Reiten theory 2 (English)

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Nicholas Williams**(The University of Tokyo)Cyclic polytopes and higher Auslander--Reiten theory 2 (English)

[ Abstract ]

This continues part 1. In the second talk, we focus on higher Auslander--Reiten theory. We survey the basic setting of this theory, starting with d-cluster-tilting subcategories of module categories. We then move on to d-cluster-tilting subcategories of derived categories in the case of d-representation-finite d-hereditary algebras. We explain how one can construct (d + 2)-angulated cluster categories for such algebras, generalising classical cluster categories. We finally consider the d-almost positive category, which is the higher generalisation of the category of two-term complexes. Throughout, we illustrate the results using the higher Auslander algebras of type A, and explain how the different categories can be interpreted combinatorially for these algebras.

[ Reference URL ]This continues part 1. In the second talk, we focus on higher Auslander--Reiten theory. We survey the basic setting of this theory, starting with d-cluster-tilting subcategories of module categories. We then move on to d-cluster-tilting subcategories of derived categories in the case of d-representation-finite d-hereditary algebras. We explain how one can construct (d + 2)-angulated cluster categories for such algebras, generalising classical cluster categories. We finally consider the d-almost positive category, which is the higher generalisation of the category of two-term complexes. Throughout, we illustrate the results using the higher Auslander algebras of type A, and explain how the different categories can be interpreted combinatorially for these algebras.

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/06/22

17:00-18:30 Online

Please see the reference URL for details on the online seminar.

Update on singular equivalences between commutative rings (English)

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Martin Kalck**(Freiburg University)Update on singular equivalences between commutative rings (English)

[ Abstract ]

We will start with an introduction to singularity categories, which were first studied by Buchweitz and later rediscovered by Orlov. Then we will explain what is known about triangle equivalences between singularity categories of commutative rings, recalling results of Knörrer, D. Yang (based on our joint works on relative singularity categories. This result also follows from work of Kawamata and was generalized in a joint work with Karmazyn) and a new equivalence obtained in arXiv:2103.06584.

In the remainder of the talk, we will focus on the case of Gorenstein isolated singularities and especially hypersurfaces, where we give a complete description of quasi-equivalence classes of dg enhancements of singularity categories, answering a question of Keller & Shinder. This is based on arXiv:2108.03292.

[ Reference URL ]We will start with an introduction to singularity categories, which were first studied by Buchweitz and later rediscovered by Orlov. Then we will explain what is known about triangle equivalences between singularity categories of commutative rings, recalling results of Knörrer, D. Yang (based on our joint works on relative singularity categories. This result also follows from work of Kawamata and was generalized in a joint work with Karmazyn) and a new equivalence obtained in arXiv:2103.06584.

In the remainder of the talk, we will focus on the case of Gorenstein isolated singularities and especially hypersurfaces, where we give a complete description of quasi-equivalence classes of dg enhancements of singularity categories, answering a question of Keller & Shinder. This is based on arXiv:2108.03292.

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/06/15

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

Cyclic polytopes and higher Auslander--Reiten theory 1 (English)

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Nicholas Williams**(The University of Tokyo)Cyclic polytopes and higher Auslander--Reiten theory 1 (English)

[ Abstract ]

In this series of three talks, we expand upon the previous talk given at the seminar and study the relationship between cyclic polytopes and higher Auslander--Reiten theory in more detail.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNA/2021/Williams-Cyclic_polytopes_and_higher_AR.pdf

In the first talk, we focus on cyclic polytopes. We survey important properties of cyclic polytopes, such as different ways to construct them, the Upper Bound Theorem, and their Ramsey-theoretic properties. We then move on to triangulations of cyclic polytopes. We give efficient combinatorial descriptions of triangulations of even-dimensional and odd-dimensional cyclic polytopes, which we will use in subsequent talks. We finally define the higher Stasheff--Tamari orders on triangulations of cyclic polytopes. We give important results on the orders, including Rambau's Theorem, and the equality of the two orders.

[ Reference URL ]In this series of three talks, we expand upon the previous talk given at the seminar and study the relationship between cyclic polytopes and higher Auslander--Reiten theory in more detail.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNA/2021/Williams-Cyclic_polytopes_and_higher_AR.pdf

In the first talk, we focus on cyclic polytopes. We survey important properties of cyclic polytopes, such as different ways to construct them, the Upper Bound Theorem, and their Ramsey-theoretic properties. We then move on to triangulations of cyclic polytopes. We give efficient combinatorial descriptions of triangulations of even-dimensional and odd-dimensional cyclic polytopes, which we will use in subsequent talks. We finally define the higher Stasheff--Tamari orders on triangulations of cyclic polytopes. We give important results on the orders, including Rambau's Theorem, and the equality of the two orders.

https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/06/08

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

超平面配置の特性準多項式 II (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Masahiko Yoshinaga**(Osaka University)超平面配置の特性準多項式 II (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/06/01

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

超平面配置の特性準多項式 I (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Masahiko Yoshinaga**(Osaka University)超平面配置の特性準多項式 I (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/04/13

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

Tilting ideals of deformed preprojective algebras

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Yuta Kimura**(Osaka Metropolitan University)Tilting ideals of deformed preprojective algebras

[ Abstract ]

Let $K$ be a field and $Q$ a finite quiver. For a weight $\lambda \in K^{|Q_0|}$, the deformed preprojective algebra $\Pi^{\lambda}$ was introduced by Crawley-Boevey and Holland to study deformations of Kleinian singularities. If $\lambda = 0$, then $\Pi^{0}$ is the preprojective algebra introduced by Gelfand-Ponomarev, and appears many areas of mathematics. Among interesting properties of $\Pi^{0}$, the classification of tilting ideals of $\Pi^{0}$, shown by Buan-Iyama-Reiten-Scott, is fundamental and important. They constructed a bijection between the set of tilting ideals of $\Pi^{0}$ and the Coxeter group $W_Q$ of $Q$.

In this talk, when $Q$ is non-Dynkin, we see that $\Pi^{\lambda}$ is a $2$-Calabi-Yau algebra, and show that there exists a bijection between tilting ideals and a Coxeter group. However $W_Q$ does not appear, since $\Pi^{\lambda}$ is not necessary basic. Instead of $W_Q$, we consider the Ext-quiver of rigid simple modules, and use its Coxeter group. When $Q$ is an extended Dynkin quiver, we see that the Ext-quiver is finite and this has an information of singularities of a representation space of semisimple modules.

This is joint work with William Crawley-Boevey.

[ Reference URL ]Let $K$ be a field and $Q$ a finite quiver. For a weight $\lambda \in K^{|Q_0|}$, the deformed preprojective algebra $\Pi^{\lambda}$ was introduced by Crawley-Boevey and Holland to study deformations of Kleinian singularities. If $\lambda = 0$, then $\Pi^{0}$ is the preprojective algebra introduced by Gelfand-Ponomarev, and appears many areas of mathematics. Among interesting properties of $\Pi^{0}$, the classification of tilting ideals of $\Pi^{0}$, shown by Buan-Iyama-Reiten-Scott, is fundamental and important. They constructed a bijection between the set of tilting ideals of $\Pi^{0}$ and the Coxeter group $W_Q$ of $Q$.

In this talk, when $Q$ is non-Dynkin, we see that $\Pi^{\lambda}$ is a $2$-Calabi-Yau algebra, and show that there exists a bijection between tilting ideals and a Coxeter group. However $W_Q$ does not appear, since $\Pi^{\lambda}$ is not necessary basic. Instead of $W_Q$, we consider the Ext-quiver of rigid simple modules, and use its Coxeter group. When $Q$ is an extended Dynkin quiver, we see that the Ext-quiver is finite and this has an information of singularities of a representation space of semisimple modules.

This is joint work with William Crawley-Boevey.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/03/11

13:00-14:30 Online

Please see the URL below for details on the online seminar.

Modular representation theory of finite groups – local versus global II (English)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Shigeo Koshitani**(Chiba University)Modular representation theory of finite groups – local versus global II (English)

[ Abstract ]

We are going to talk about representation theory of finite groups. In the 1st part it will be on "Equivalences of categories ” showing up for block theory in modular representation theory, and it should be kind of introductory lecture/talk. So the audience is supposed to have knowledge only of definitions of groups, rings, fields, modules, and so on. In the 2nd part we will discuss kind of local—global conjectures in modular representation theory of finite groups, that originally and essentially are due to Richard Brauer (1901–77).

[ Reference URL ]We are going to talk about representation theory of finite groups. In the 1st part it will be on "Equivalences of categories ” showing up for block theory in modular representation theory, and it should be kind of introductory lecture/talk. So the audience is supposed to have knowledge only of definitions of groups, rings, fields, modules, and so on. In the 2nd part we will discuss kind of local—global conjectures in modular representation theory of finite groups, that originally and essentially are due to Richard Brauer (1901–77).

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/03/09

13:00-14:30 Online

Please see the URL below for details on the online seminar.

Modular representation theory of finite groups – local versus global I (English)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Shigeo Koshitani**(Chiba University)Modular representation theory of finite groups – local versus global I (English)

[ Abstract ]

We are going to talk about representation theory of finite groups. In the 1st part it will be on "Equivalences of categories ” showing up for block theory in modular representation theory, and it should be kind of introductory lecture/talk. So the audience is supposed to have knowledge only of definitions of groups, rings, fields, modules, and so on. In the 2nd part we will discuss kind of local—global conjectures in modular representation theory of finite groups, that originally and essentially are due to Richard Brauer (1901–77).

[ Reference URL ]We are going to talk about representation theory of finite groups. In the 1st part it will be on "Equivalences of categories ” showing up for block theory in modular representation theory, and it should be kind of introductory lecture/talk. So the audience is supposed to have knowledge only of definitions of groups, rings, fields, modules, and so on. In the 2nd part we will discuss kind of local—global conjectures in modular representation theory of finite groups, that originally and essentially are due to Richard Brauer (1901–77).

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/01/21

16:45-18:15 Online

Please see the URL below for details on the online seminar.

Exact-categorical properties of subcategories of abelian categories 2 (Japanese)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Haruhisa Enomoto**(Osaka Prefecture University)Exact-categorical properties of subcategories of abelian categories 2 (Japanese)

[ Abstract ]

Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.

In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).

In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).

If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).

[ Reference URL ]Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.

In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).

In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).

If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2022/01/18

15:00-16:30 Online

Please see the URL below for details on the online seminar.

Exact-categorical properties of subcategories of abelian categories 1 (Japanese)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Haruhisa Enomoto**(Osaka Prefecture University)Exact-categorical properties of subcategories of abelian categories 1 (Japanese)

[ Abstract ]

Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.

In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).

In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).

If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).

[ Reference URL ]Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.

In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).

In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).

If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/12/16

16:45-18:15 Online

Please see the URL below for details on the online seminar.

Cyclic polytopes and higher Auslander-Reiten theory (English)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Nicholas Williams**(University of Cologne)Cyclic polytopes and higher Auslander-Reiten theory (English)

[ Abstract ]

Oppermann and Thomas show that tilting modules over Iyama’s higher Auslander algebras of type A are in bijection with triangulations of even-dimensional cyclic polytopes. Triangulations of cyclic polytopes are partially ordered in two natural ways known as the higher Stasheff-Tamari orders, which were introduced in the 1990s by Kapranov, Voevodsky, Edelman, and Reiner as higher-dimensional generalisations of the Tamari lattice. These two partial orders were conjectured to be equal in 1996 by Edelman and Reiner, and we prove that this conjecture is true. We further show how the higher Stasheff-Tamari orders correspond in even dimensions to natural orders on tilting modules which were studied by Riedtmann, Schofield, Happel, and Unger. This then allows us to complete the picture of Oppermann and Thomas by showing that triangulations of odd-dimensional cyclic polytopes correspond to equivalence classes of d-maximal green sequences, which we introduce as higher-dimensional analogues of Keller’s maximal green sequences. We show that the higher Stasheff-Tamari orders correspond to natural orders on equivalence classes of d-maximal green sequences, which relate to the no-gap conjecture of Brustle, Dupont, and Perotin. The equality of the higher Stasheff-Tamari orders then implies that these algebraic orders on tilting modules and d-maximal green sequences are equal. If time permits, we will also discuss some results on mutation of cluster-tilting objects and triangulations.

[ Reference URL ]Oppermann and Thomas show that tilting modules over Iyama’s higher Auslander algebras of type A are in bijection with triangulations of even-dimensional cyclic polytopes. Triangulations of cyclic polytopes are partially ordered in two natural ways known as the higher Stasheff-Tamari orders, which were introduced in the 1990s by Kapranov, Voevodsky, Edelman, and Reiner as higher-dimensional generalisations of the Tamari lattice. These two partial orders were conjectured to be equal in 1996 by Edelman and Reiner, and we prove that this conjecture is true. We further show how the higher Stasheff-Tamari orders correspond in even dimensions to natural orders on tilting modules which were studied by Riedtmann, Schofield, Happel, and Unger. This then allows us to complete the picture of Oppermann and Thomas by showing that triangulations of odd-dimensional cyclic polytopes correspond to equivalence classes of d-maximal green sequences, which we introduce as higher-dimensional analogues of Keller’s maximal green sequences. We show that the higher Stasheff-Tamari orders correspond to natural orders on equivalence classes of d-maximal green sequences, which relate to the no-gap conjecture of Brustle, Dupont, and Perotin. The equality of the higher Stasheff-Tamari orders then implies that these algebraic orders on tilting modules and d-maximal green sequences are equal. If time permits, we will also discuss some results on mutation of cluster-tilting objects and triangulations.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/11/19

17:00-18:30 Online

Please see the URL below for details on the online seminar.

有限群のブロック上の$\tau$-傾理論 (Japanese) (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Yuta Kozakai**(Tokyo University of Science)有限群のブロック上の$\tau$-傾理論 (Japanese) (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/07/08

16:00-17:30 Online

Please see the URL below for details on the online seminar.

Sign-stable mutation loops and pseudo-Anosov mapping classes (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Tsukasa Ishibashi**(RIMS, Kyoto University)Sign-stable mutation loops and pseudo-Anosov mapping classes (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/06/24

16:00-17:30 Online

Please see the URL below for details on the online seminar.

Rank 2 free subgroups in autoequivalence groups of Calabi-Yau categories

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Kohei Kikuta**(Chuo University)Rank 2 free subgroups in autoequivalence groups of Calabi-Yau categories

[ Abstract ]

Via homological mirror symmetry, there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds.

In this talk, as an analogue of mapping class groups of closed oriented surfaces, we study autoequivalence groups of Calabi-Yau triangulated categories. In particular, we consider embeddings of rank 2 (non-commutative) free groups generated by spherical twists. It is interesting that the proof of main results is almost similar to that of corresponding results in the theory of mapping class groups.

[ Reference URL ]Via homological mirror symmetry, there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds.

In this talk, as an analogue of mapping class groups of closed oriented surfaces, we study autoequivalence groups of Calabi-Yau triangulated categories. In particular, we consider embeddings of rank 2 (non-commutative) free groups generated by spherical twists. It is interesting that the proof of main results is almost similar to that of corresponding results in the theory of mapping class groups.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/06/02

16:00-17:30 Online

Please see the URL below for details on the online seminar.

An equivariant Hochster's formula for $S_n$-invariant monomial ideals (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Satoshi Murai**(Waseda University)An equivariant Hochster's formula for $S_n$-invariant monomial ideals (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/05/20

16:00-17:30 Online

Please see the URL below for details on the online seminar.

This talk is based on joint work with Tsutomu Nakamura. For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable flat cotorsion right modules and the isoclasses of indecomposable injective left modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality between Ziegler spectra.

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Ryo Kanda**(Osaka city University)This talk is based on joint work with Tsutomu Nakamura. For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable flat cotorsion right modules and the isoclasses of indecomposable injective left modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality between Ziegler spectra.

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/05/06

16:00-17:30 Online

Please see the URL below for details on the online seminar.

Derived quotients of Cohen-Macaulay rings (English)

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Liran Shaul**(Charles University)Derived quotients of Cohen-Macaulay rings (English)

[ Abstract ]

It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. Several applications of this result are given, including a generalization of Hironaka's miracle flatness theorem to derived algebraic geometry.

[ Reference URL ]It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. Several applications of this result are given, including a generalization of Hironaka's miracle flatness theorem to derived algebraic geometry.

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html