静岡大学解析セミナー 日　時： 2013年11月20日（水）16:30--18:00 場　所： 静岡大学理学部 C 棟 309 号室 講演者： Prof. Dr. Reinhard Farwig (Technische Universität Darmstadt, Germany) 講演タイトル： Optimal initial values and regularity conditions of Besov space type for weak solutions to the Navier-Stokes system

(+ := @ipc.shizuoka.ac.jp)

その場合，1コマ目に遅刻する可能性があれば，事前に連絡しておいてもらった方が無難です．
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### プログラム

#### 12月13日（金）

13:45 - 14:45 三内顕義（東京大学，数理科学研究科）
A numerical invariant and singularities in positive characteristic, I

15:00 - 16:00 Luo Xueyu（名古屋大学，多元数理科学研究科）
Ice Quivers with Potential associated with Triangulations and Cohen-Macaulay Modules over Orders, I

16:15 - 17:15 相原琢磨（名古屋大学，多元数理科学研究科）
Flipping Brauer graphs, I

18:30 - 懇親会

#### 12月14日（土）

09:30 - 10:30 Luo Xueyu（名古屋大学，多元数理科学研究科）
Ice Quivers with Potential associated with Triangulations and Cohen-Macaulay Modules over Orders, II

10:45 - 11:45 三内顕義（東京大学，数理科学研究科）
A numerical invariant and singularities in positive characteristic, II

13:15 - 14:15 相原琢磨（名古屋大学，多元数理科学研究科）
Flipping Brauer graphs, II

14:30 - 15:30 浅芝 秀人（静岡大学，理学研究科）
Gluing of derived equivalences along bimodules

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### 講演概要

• #### 三内 顕義

A numerical invariant and singularities in positive characteristic
Huneke, Leuschkeによって定義された不変量であるF-signatureを加群の不変量として再定義し、F-rational, Gorenstein, regularなどの特徴付けを行う。またこの不変量はMCMの圏の構造と密接な関係にあり、それについても解説する。
• #### Xueyu Luo

Ice Quivers with Potential associated with Triangulations and Cohen-Macaulay Modules over Orders
This is a joint work with Laurent Demonet. A quiver together with a linear combination of cyclic paths in the quiver is called a quiver with potential (QP for short). For each triangulation of a bordered surface with marked points, a QP was introduced by Labardini-Fragoso. In the first talk, we extend Labardini's construction and associate an ice QP to each triangulation of a polygon with $n$ vertices by adding a set of $n$ frozen vertices corresponding to the edges of the polygon and certain arrows. We study the associated frozen Jacobian algebra and show that it has the structure of a special kind of $K[x]$-order $\Lambda$ (i.e. a $K[x]$-algebra which is finitely generated free as $K[x]$-module). In the second talk, we will construct a bijection between the set of the isomorphism classes of all indecomposable Cohen-Macaulay $\Lambda$-modules (i.e. all the $\Lambda$-modules which are also finitely generated free as $K[x]$-module) and the set of all sides and diagonals of the polygon. We show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is $2$-Calabi-Yau and is triangle-equivalent to the cluster category of type $A_{n-3}$.
• #### 相原 琢磨

Flipping Brauer graphs
The class of tilting complexes is one of the most important classes in representation theory of algebras, since tilting complexes give derived equivalences. By tilting mutation we can get a new tilting complex from a given one. A problem is to explicitly describe the endomorphism algebra of a tilting complex given by tilting mutation. In this talk, we give a complete answer to this problem for Brauer graph algebras.
• #### 浅芝 秀人

Gluing of derived equivalences along bimodules
Let k be a commutative ring, and consider the bicategory k-Cat^b of small k-categories C, bimodules over them and bimodule morphisms (the 1-morphisms C --> C' are the C'-C-bimodules and the composite C --> C' --> C'' is given by the tensor over C'). We give a definition of the Grothendieck construction Gr(X) of a lax functor X from a small category I to the bicategory k-Cat^b, which extends the corresponding one of a colax functor from I to the 2-category k-Cat of small k-categories, k-functors and natural transformations, and which enables us to construct new k-categories by gluing k-categories together along bimodules, in particular, triangular matrix algebras and the tensor algebras of k-species. When k is a field, we can construct a derived equivalence between Grothendieck constructions Gr(X) and Gr(X') of lax functors X and X': I --> k-Cat^b by gluing derived equivalences between X(i) and X'(i) (i is an object of I) together along bimodules X(a) and X'(a) (a is a morphism of I) if X and X' are "derived equivalent".

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