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Tokyo-Nagoya Algebra Seminar
Seminar information archive ～04/30｜Next seminar｜Future seminars 05/01～
| Organizer(s) | Noriyuki Abe, Aaron Chan, Osamu Iyama, Yasuaki Gyoda, Hiroyuki Nakaoka, Ryo Takahashi |
|---|
Seminar information archive
2025/04/15
10:30-12:00 Room # ハイブリッド開催/128 (Graduate School of Math. Sci. Bldg.)
Parth Shimpi (University of Glasgow)
Torsion pairs for McKay quivers (English)
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Parth Shimpi (University of Glasgow)
Torsion pairs for McKay quivers (English)
[ Abstract ]
Classifying torsion classes in the module category has been a problem of much interest in the representation theory of preprojective algebras, owing to its immediate applications in the study of t-structures, bricks, and spherical objects in the derived category. When the preprojective algebra arises from a Dynkin quiver, all such torsion classes must lead to algebraic intermediate hearts— in particular, they arise from tilting modules and therefore admit a finite combinatorial description. Affine ADE quivers, on the other hand, produce infinitely many tilting modules and moreover have geometric hearts arising from the McKay correspondence. By realising the geometric hearts as `limits’ of algebraic ones, I will explain how all torsion pairs for affine preprojective algebras can be described using the above two possibilities; in particular a complete classification is achieved.
Zoom ID 813 0345 0035 Password 706679
[ Reference URL ]Classifying torsion classes in the module category has been a problem of much interest in the representation theory of preprojective algebras, owing to its immediate applications in the study of t-structures, bricks, and spherical objects in the derived category. When the preprojective algebra arises from a Dynkin quiver, all such torsion classes must lead to algebraic intermediate hearts— in particular, they arise from tilting modules and therefore admit a finite combinatorial description. Affine ADE quivers, on the other hand, produce infinitely many tilting modules and moreover have geometric hearts arising from the McKay correspondence. By realising the geometric hearts as `limits’ of algebraic ones, I will explain how all torsion pairs for affine preprojective algebras can be described using the above two possibilities; in particular a complete classification is achieved.
Zoom ID 813 0345 0035 Password 706679
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2025/04/02
10:30-12:00 Online
Koji Matsushita (The University of Tokyo)
因子類群が$\mathbb{Z}^2$であるトーリック環の非可換クレパント特異点解消について (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Koji Matsushita (The University of Tokyo)
因子類群が$\mathbb{Z}^2$であるトーリック環の非可換クレパント特異点解消について (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2025/03/27
10:30-12:00 Online
Ryota Iitsuka (Nagoya University)
Reduction理論における変異が誘導する三角圏構造 (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Ryota Iitsuka (Nagoya University)
Reduction理論における変異が誘導する三角圏構造 (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2025/03/17
14:30-16:00 Online
Junyang Liu (University of Science and Technology of China)
On Amiot's conjecture (English)
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Junyang Liu (University of Science and Technology of China)
On Amiot's conjecture (English)
[ Abstract ]
In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories with cluster-tilting object must come from quivers with potential. This would extend a structure theorem obtained by Keller-Reiten in the case where the endomorphism algebra of the cluster-tilting object is hereditary. Many other classes of examples are also known. We will report on the proof of the conjecture in the general case for categories with *algebraic* 2-Calabi-Yau structure. This result has been obtained in joint work with Bernhard Keller and is based on Van den Bergh's structure theorem for complete Calabi-Yau algebras. We also generalize his structure theorem to the relative case and use it to prove a relative variant of the conjecture.
ミーティング ID: 853 1951 5047
パスコード: 900788
[ Reference URL ]In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories with cluster-tilting object must come from quivers with potential. This would extend a structure theorem obtained by Keller-Reiten in the case where the endomorphism algebra of the cluster-tilting object is hereditary. Many other classes of examples are also known. We will report on the proof of the conjecture in the general case for categories with *algebraic* 2-Calabi-Yau structure. This result has been obtained in joint work with Bernhard Keller and is based on Van den Bergh's structure theorem for complete Calabi-Yau algebras. We also generalize his structure theorem to the relative case and use it to prove a relative variant of the conjecture.
ミーティング ID: 853 1951 5047
パスコード: 900788
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2024/12/20
17:00-18:30 Online
Simon Riche (Université Clermont Auvergne)
Semiinfinite sheaves on affine flag varieties (English)
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Simon Riche (Université Clermont Auvergne)
Semiinfinite sheaves on affine flag varieties (English)
[ Abstract ]
We will explain how, generalizing a construction of Gaitsgory, one can define and study a category of sheaves on the affine flag variety of a complex reductive group that "models" sheaves on the corresponding semiinfinite flag variety, with coefficients in a field of positive characteristic, and which should provide a geometric model for a category of representations of the Langlands dual Lie algebra over the given coefficient field. As an application, we use this construction to compute the dimensions of stalks of the intersection cohomology complex on Drinfeld's compactification, with coefficients in any field of good characteristic. This is joint work with Pramod Achar and Gurbir Dhillon.
ミーティング ID: 882 1561 8969
パスコード: 531394
[ Reference URL ]We will explain how, generalizing a construction of Gaitsgory, one can define and study a category of sheaves on the affine flag variety of a complex reductive group that "models" sheaves on the corresponding semiinfinite flag variety, with coefficients in a field of positive characteristic, and which should provide a geometric model for a category of representations of the Langlands dual Lie algebra over the given coefficient field. As an application, we use this construction to compute the dimensions of stalks of the intersection cohomology complex on Drinfeld's compactification, with coefficients in any field of good characteristic. This is joint work with Pramod Achar and Gurbir Dhillon.
ミーティング ID: 882 1561 8969
パスコード: 531394
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2024/10/23
10:30-12:00 Online
Yasuaki Gyoda (The University of Tokyo)
一般化マルコフ数とそのSL(2,Z)行列化 (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Yasuaki Gyoda (The University of Tokyo)
一般化マルコフ数とそのSL(2,Z)行列化 (Japanese)
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2024/06/28
16:30-18:00 Online
Shunya Saito (The University of Tokyo)
Classifying KE-closed subcategories (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Shunya Saito (The University of Tokyo)
Classifying KE-closed subcategories (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2024/06/21
16:30-18:00 Online
Daigo Ito (UC Berkeley)
松井スペクトラムを用いた復元定理の再解釈 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Daigo Ito (UC Berkeley)
松井スペクトラムを用いた復元定理の再解釈 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2024/06/07
16:30-18:00 Online
Taiki Shibata (Okayama University of Science)
スーパー代数群の表現と奇鏡映について (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Taiki Shibata (Okayama University of Science)
スーパー代数群の表現と奇鏡映について (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2024/03/14
10:30-12:00 Online
Arashi Sakai (Nagoya University)
Lattices of torsion classes in representation theory of finite groups (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Arashi Sakai (Nagoya University)
Lattices of torsion classes in representation theory of finite groups (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2024/01/11
10:30-12:00 Online
Ryo Fujita (RIMS)
量子Grothendieck環とその量子団代数構造について (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Ryo Fujita (RIMS)
量子Grothendieck環とその量子団代数構造について (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/12/26
15:00-16:30 Room #ハイブリッド・002 (Graduate School of Math. Sci. Bldg.)
Ivan Losev (Yale University)
t-structures on the equivariant derived category of the Steinberg scheme (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Ivan Losev (Yale University)
t-structures on the equivariant derived category of the Steinberg scheme (English)
[ Abstract ]
The Steinberg scheme and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in positive characteristics. Based on arXiv:2302.05782.
[ Reference URL ]The Steinberg scheme and the equivariant coherent sheaves on it play a very important role in Geometric Representation theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation theory in positive characteristics. Based on arXiv:2302.05782.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/12/14
10:30-12:00 Online
Xiaofa Chen (University of Science and Technology of China)
On exact dg categories (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Xiaofa Chen (University of Science and Technology of China)
On exact dg categories (English)
[ Abstract ]
In this talk, I will give an introduction to exact dg categories and then explore their application to various correspondences in representation theory. We will generalize the Auslander–Iyama correspondence, the Iyama–Solberg correspondence, and a correspondence considered in a paper by Iyama in 2005 to the setting of exact dg categories. The slogan is that solving correspondence-type problems becomes easier using dg categories, and interesting phenomena emerge when the dg category is concentrated in degree zero or is abelian.
[ Reference URL ]In this talk, I will give an introduction to exact dg categories and then explore their application to various correspondences in representation theory. We will generalize the Auslander–Iyama correspondence, the Iyama–Solberg correspondence, and a correspondence considered in a paper by Iyama in 2005 to the setting of exact dg categories. The slogan is that solving correspondence-type problems becomes easier using dg categories, and interesting phenomena emerge when the dg category is concentrated in degree zero or is abelian.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/10/12
10:30-12:00 Online
Xin Ren (Kansai University)
q-deformed rational numbers, Farey sum and a 2-Calabi-Yau category of A_2 quiver (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Xin Ren (Kansai University)
q-deformed rational numbers, Farey sum and a 2-Calabi-Yau category of A_2 quiver (English)
[ Abstract ]
Let q be a positive real number. The left and right q-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and the right q-deformed rational number is exactly q-deformed rational number considered by Morier-Genoud and Ovsienko, when q is a formal parameter. They gave a homological interpretation for left and right q-deformed rational numbers by considering a special 2-Calabi–Yau category associated to the A_2 quiver.
In this talk, we begin by introducing the above definitions and related results. Then we give a formula for computing the q-deformed Farey sum of the left q-deformed rational numbers based on the negative continued fractions. We combine the homological interpretation of the left and right q-deformed rational numbers and the q-deformed Farey sum, and give a homological interpretation of the q-deformed Farey sum. We also apply the above results to real quadratic irrational numbers with periodic type.
[ Reference URL ]Let q be a positive real number. The left and right q-deformed rational numbers were introduced by Bapat, Becker and Licata via regular continued fractions, and the right q-deformed rational number is exactly q-deformed rational number considered by Morier-Genoud and Ovsienko, when q is a formal parameter. They gave a homological interpretation for left and right q-deformed rational numbers by considering a special 2-Calabi–Yau category associated to the A_2 quiver.
In this talk, we begin by introducing the above definitions and related results. Then we give a formula for computing the q-deformed Farey sum of the left q-deformed rational numbers based on the negative continued fractions. We combine the homological interpretation of the left and right q-deformed rational numbers and the q-deformed Farey sum, and give a homological interpretation of the q-deformed Farey sum. We also apply the above results to real quadratic irrational numbers with periodic type.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/08/24
10:30-12:00 Online
Sota Asai (University of Tokyo)
TF equivalence on the real Grothendieck group (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Sota Asai (University of Tokyo)
TF equivalence on the real Grothendieck group (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/07/14
10:30-12:00 Online
Michael Wemyss (University of Glasgow)
Local Forms of Noncommutative Functions and Applications (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Michael Wemyss (University of Glasgow)
Local Forms of Noncommutative Functions and Applications (English)
[ Abstract ]
This talk will explain how Arnold's results for commutative singularities can be extended into the noncommutative setting, with the main result being a classification of certain Jacobi algebras
arising from (complete) free algebras. This class includes finite dimensional Jacobi algebras, and also Jacobi algebras of GK dimension one, suitably interpreted. The surprising thing is that a classification should exist at all, and it is even more surprising that ADE enters.
I will spend most of my time explaining what the algebras are, what they classify, and how to intrinsically extract ADE information from them. At the end, I'll explain why I'm really interested in this problem, an update including results on different quivers, and the applications of the above classification to curve counting and birational geometry. This is joint work with Gavin Brown.
Meeting ID: 863 9598 8196
passcode: 423160
[ Reference URL ]This talk will explain how Arnold's results for commutative singularities can be extended into the noncommutative setting, with the main result being a classification of certain Jacobi algebras
arising from (complete) free algebras. This class includes finite dimensional Jacobi algebras, and also Jacobi algebras of GK dimension one, suitably interpreted. The surprising thing is that a classification should exist at all, and it is even more surprising that ADE enters.
I will spend most of my time explaining what the algebras are, what they classify, and how to intrinsically extract ADE information from them. At the end, I'll explain why I'm really interested in this problem, an update including results on different quivers, and the applications of the above classification to curve counting and birational geometry. This is joint work with Gavin Brown.
Meeting ID: 863 9598 8196
passcode: 423160
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/07/07
15:00-16:30 Online
Hideto Asashiba (Shizuoka University, Kyoto University, Osaka Metropolitan University)
クイバー表現のパーシステンス加群への応用: 区間加群による近似と分解 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Hideto Asashiba (Shizuoka University, Kyoto University, Osaka Metropolitan University)
クイバー表現のパーシステンス加群への応用: 区間加群による近似と分解 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/05/16
15:00-16:30 Online
Antoine de Saint Germain (University of Hong Kong)
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.
Takumi Otani (Osaka Univeristy)
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.
Kota Murakami (University of Tokyo)
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.
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
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.
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
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.
Martin Kalck (Freiburg University)
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.
Yuki Imamura (Osaka University)
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.
Nicholas Williams (The University of Tokyo)
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
