3November 11-15, 201in Nagoya University

The 13th International ConferenceGraduate School of Mathematics, Nagoya University第13回 名古屋国際数学コンファレンス

Perspectives ofRepresentationTheory of Algebrascelebrating Kunio Yamagata's 65th birthday

The 13th International Conference,Graduate School of Mathematics, Nagoya University

Perspectives of Representation Theory of Algebras— celebrating Kunio Yamagata’s 65th birthday —

Period: November 11–15, 2013

Venue: Sakata–Hirata Hall, Science South Bldg. (Nov. 11–14),ES Hall, E&S Bldg. (Nov. 15),Nagoya University, Nagoya, Japan

Organizers:

Hideto Asashiba (Shizuoka University),Osamu Iyama (Nagoya University),Jun-ichi Miyachi (Tokyo Gakugei University),Izuru Mori (Shizuoka University),Masahisa Sato (Yamanashi University),Andrzej Skowronski (Nicolaus Copernicus University),Morio Uematsu (Jobu University),Yuji Yoshino (Okayama University) Prof. Kunio Yamagata

TIME TABLE

13:00–

09:30–10:20

11:00–11:30

11:40–12:30

14:00–14:50

15:40–16:30

16:40–17:30

18:00–

Coffee Break

Lunch Break

Coffee Break Coffee Break

Lunch Break

Banquet

Excursion

R. Kase*Y. Mizuno*

G. Jasso M. Błaszkiewicz T. Itagaki R. Kanda

H. Koga*M. Yoshiwaki*

K. Ueyama

T. Aihara

L. Demonet

Y. Kimura

J. Białkowski

P. Malicki

A. Skowyrski

R. Takahashi A. Takahashi

H. Minamoto M. Wemyss

I. Reiten A. Skowronski

C. Xi

K. Erdmann

O. Kerner

Y. Yoshino

J. Miyachi

D. Zacharia

H. Lenzing

C. M. Ringel

Time Nov. 11 (Mon) Nov. 12 (Tue) Nov. 13 (Wed) Nov. 14 (Thu) Nov. 15 (Fri)

• Monday: 14:00–14:30 Kase, 14:35–15:05 Mizuno.• Thursday: 14:00–14:30 Yoshiwaki, 14:35–15:05 Koga.• There will be a Conference Excursion on November 13th (Wed) afternoon.• We will take a group photo just after morning session on November 14th (Thu).• A banquet is planned on November 14th (Thu) from 18:00 at Mei-dining. (See Figure C)

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PROGRAM

November 11th (Mon) (at Sakata–Hirata Hall)

9:00– 9:30 — Registration —

9:30–10:20 Idun Reiten (Norwegian University of Science and Technology)

Lattice structure of torsion classes

— Coffee Break —

11:00–11:30 Gustavo Jasso (Nagoya University)

On the simplicial complex associated with support τ-tilting modules

11:40–12:30 Takuma Aihara (Nagoya University)

Tilting mutation theory

— Lunch Break —

14:00–14:30 Ryoichi Kase (Osaka University)

On the poset of pre-projective tilting modules over path algebras

14:35–15:05 Yuya Mizuno (Nagoya University)

τ-tilting modules over preprojective algebras of Dynkin type

15:40–16:30 Laurent Demonet (Nagoya University)Ice quivers with potential associated with triangulations and Cohen-Macaulay modulesover orders (case An and Dn)

16:40–17:30 Otto Kerner (University of Dusseldorf )

From wild hereditary to tilted algebras, and back

November 12th (Tue) (at Sakata–Hirata Hall)

9:30–10:20 Andrzej Skowronski (Nicolaus Copernicus University)

Selfinjective algebras with deforming ideals

— Coffee Break —

11:00–11:30 Marta Błaszkiewicz (Nicolaus Copernicus University)

Selfinjective algebras of finite representation type with maximal almost split sequences

11:40–12:30 Jerzy Białkowski (Nicolaus Copernicus University)

Periodicity of selfinjective algebras of polynomial growth

— Lunch Break —

14:00–14:50 Piotr Malicki (Nicolaus Copernicus University)

Finite cycles of indecomposable modules

— Coffee Break —

15:40–16:30 Adam Skowyrski (Nicolaus Copernicus University)

Homological problems for cycle-finite algebras

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16:40–17:30 Changchang Xi (Capital Normal University)

Higher dimensional tilting modules and recollements

November 13th (Wed) (at Sakata–Hirata Hall)

9:30–10:20 Karin Erdmann (University of Oxford)

Support varieties via Hochschild cohomology: a necessary condition

— Coffee Break —

11:00–11:30 Tomohiro Itagaki (Tokyo University of Science)Cyclic homology of truncated quiver algebras and notes on the no loops conjecture forHochschild homology

11:40–12:30 Dan Zacharia (Syracuse University)

A characterization of the graded center of a Koszul algebra

13:00– — Excursion —

November 14th (Thu) (at Sakata–Hirata Hall)

9:30–10:20 Yuji Yoshino (Okayama University)

Dependence of total reflexivity conditions

— Coffee Break —

11:00–11:30 Ryo Kanda (Nagoya University)

Specialization orders on atom spectra of Grothendieck categories

11:40–12:30 Ryo Takahashi (Nagoya University)

Existence of cohomology annihilators and strong generation of derived categories

— Group Photo —

— Lunch Break —

14:00–14:30 Michio Yoshiwaki (Osaka City University)

Dimensions of triangulated categories with respect to subcategories

14:35–15:05 Hirotaka Koga (University of Tsukuba)

Clifford extension

— Coffee Break —

15:40–16:30 Hiroyuki Minamoto (Osaka Prefecture University)

Derived Gabriel topology, localization and completion of dg-algebras

16:40–17:30 Jun-ichi Miyachi (Tokyo Gakugei University)

Researches on the representation theory of algebras at University of Tsukuba

18:00– — Banquet —

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November 15th (Fri) (at ES Hall)

9:30–10:20 Helmut Lenzing (University of Paderborn)

The ubiquity of the equation x2+ y3+ z5 = 0

— Coffee Break —

11:00–11:30 Kenta Ueyama (Shizuoka University)Ample Group Actions on AS-regular Algebras and Noncommutative Graded IsolatedSingularities

11:40–12:30 Atsushi Takahashi (Osaka University)

Weyl groups and Artin groups associated to weighted projective lines

— Lunch Break —

14:00–14:50 Yoshiyuki Kimura (Osaka City University)

Quiver varieties and Quantum cluster algebras

— Coffee Break —

15:40–16:30 Michael Wemyss (University of Edinburgh)

Some new examples of self–injective algebras

16:40–17:30 Claus Michael Ringel (Bielefeld University)

The root posets

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ABSTRACTS

Idun Reiten (Norwegian Univ. of Sci. and Tech.) . . . . . . . November 11th (Mon), 9:30–10:20

Lattice structure of torsion classesThis lecture is based on work with Iyama, Thomas and Todorov. A general problem for a finite

dimensional algebra A over an algebraically closed field k is to investigate when the partially orderedset of functorially finite torsion classes are a lattice. Some work by Mizuno gives a motivation forconsidering this problem. We give the answer when A is the path algebra kQ for a finite acyclic quiverQ.

Gustavo Jasso (Nagoya University) . . . . . . . . . . . . . . . . . . November 11th (Mon), 11:00–11:30

On the simplicial complex associated with support τ-tilting modulesThe class of support τ-tilting modules was introduced recently by Adachi–Iyama–Reiten so as to

provide a completion of the class of tilting modules from the point of view of mutation. Let A be afinite dimensional algebra with n simple modules. In this talk, I will explain how to construct an abstractsimplical complex ∆(A) whose maximal faces are in bijection with the isomorphism classes of basicsupport τ-tilting modules. Using established results, we will describe the combinatorial properties of∆(A). In particular, if ∆(A) is finite, then the geometric realization of ∆(A) is homeomorphic to a (n−1)-dimensional sphere. This work is part of a joint project with Osamu Iyama.

Takuma Aihara (Nagoya University) . . . . . . . . . . . . . . . . November 11th (Mon), 11:40–12:30

Tilting mutation theoryIn representation theory of algebras, the notion of mutation plays crucial roles. Three kinds of mu-

tation are well-known: silting mutation [AI], quiver-mutation [FZ] and cluster tilting mutation [BMRRT,IY]. Moreover these are closely related with each other, that is, in hereditary case there are one-to-onecorrespondences compatible with mutation, among silting objects, clusters and cluster tilting objects[AIR].

In this talk, we focus on silting mutation, which is a generalization of tilting mutation studied first byBernstein–Gelfand–Ponomarev and Auslander–Platzeck–Reiten, it was investigated also by Riedtmann–Schofield and Happel–Unger. We note that silting mutation is always possible, but tilting mutation isnot.

We mainly discuss silting quivers to observe the behavior of silting mutation. A finite dimensionalalgebra over a field is said to be silting-connected if its silting quiver is connected. Now we raise thefollowing question:

Question. When is a finite dimensional algebra over a field silting-connected?

Our goal of this talk is to give a partial answer of this question.Now we state the main theorem.

Main Theorem. Any of the following algebras is silting-connected:

(1) local algebras [AI];

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(2) hereditary algebras [AI];(3) representation-finite symmetric algebras [A];(4) Brauer graph algebras of type odd [AAC].

In particular, I will explain the strategy of the proof of (3) and (4) in this theorem.

References

[AAC] T. Adachi; T. Aihara; A. Chan, On tilting complexes of Brauer graph algebras I: combina-torics arising from two-term tilting complexes and mutation quivers of tilting complexes. inpreparation.

[AIR] T. Adachi; O. Iyama; I. Reiten, τ-tilting theory. to appear in Compos. Math.[A] T. Aihara, Tilting-connected symmetric algebras. Algebr. Represent. Theory 16 (2013), no. 3,

873–894.[AI] T. Aihara; O. Iyama, Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85

(2012), no. 3, 633–668.[BMRRT] A. B. Buan; R. Marsh; M. Reineke; I. Reiten; G. Todorov, Tilting theory and cluster combi-

natorics. Adv. Math. 204 (2006), no. 2, 572–618.[FZ] S. Fomin; A. Zelevinsky, Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002),

no. 2, 497–529.[IY] O. Iyama; Y. Yoshino, Mutation in triangulated categories and rigid Cohen–Macaulay mod-

ules. Invent. Math. 172 (2008), no. 1, 117–168.

Ryoichi Kase (Osaka University) . . . . . . . . . . . . . . . . . . . . November 11th (Mon), 14:00–14:30

On the poset of pre-projective tilting modules over path algebras

To classify tilting modules is an important problem of the representation theory of finite dimensionalalgebras. Theory of tilting-mutation introduced by Riedtmann and Schofield is one of the approach tothis problem. Riedtmann and Schofield defined the tilting quiver related with tilting-mutation. Happeland Unger defined the partial order on the set of (isomorphic classes of) basic tilting modules and showedthat tilting quiver is coincided with Hasse quiver of this poset.

In this talk we consider the poset of pre-projective tilting modules over path algebras. First wegive an equivalent condition for the poset of pre-projective tilting modules to be a distributive lattice.Moreover we realize the poset of pre-projective tilting modules as an ideal-poset.

Yuya Mizuno (Nagoya University) . . . . . . . . . . . . . . . . . . . November 11th (Mon), 14:35–15:05

τ-tilting modules over preprojective algebras of Dynkin type

Recently, the notion of (support) τ-tilting modules was introduced by Adachi–Iyama–Reiten. Thisgives a generalization of tilting modules and it also has several nice properties. In this talk, we discusssupport τ-tilting modules over preprojective algebras of Dynkin type. In particular, we classify all supportτ-tilting modules by giving a bijection with elements in the corresponding Weyl groups.

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Laurent Demonet (Nagoya University) . . . . . . . . . . . . . . November 11th (Mon), 15:40–16:30

Ice quivers with potential associated with triangulations and Cohen–Macaulay mod-ules over orders (case An and Dn)

(joint with Xueyu Luo)

In this talk, we attach an ice quiver with potential (Qσ,Wσ,F) to each triangulation σ of a polygon(resp. a polygon with one puncture) where the set F of frozen vertices correspond to the sides of thepolygon. This quiver extends the one introduced by Labardini–Fragoso and Cerulli Irelli. Thus, weconsider the (non-completed) frozen Jacobian algebra Γσ = P(Qσ,Wσ,F). One of the main result is thatΓσ as the structure of an order over K[x], that Λ = eFΓσeF is a Gorenstein order independent of σ (eF isthe sum of idempotents at the frozen vertices of σ). Moreover, the cluster tilting objects in CM(Λ) arethe modules eFΓσ for all triangulations σ. More precisely, its stable category CM(Λ) is equivalent to acluster category of type A (resp. type D).

Otto Kerner (University of Dusseldorf ) . . . . . . . . . . . . . . November 11th (Mon), 16:40–17:30

From wild hereditary to tilted algebras, and back

Let H be a finite dimensional basic connected wild hereditary algebra over an algebraically closedfield K, and H-mod the category of finite dimensional left H-modules. The Auslander–Reiten quiverΓ(H) of H-mod consists of a unique preprojective and preinjective component and infinitely many regularcomponents of type ZA∞. The modules at the boarder of such a regular component are called quasi-simple. The full subcategory of regular H-modules is denoted by H-reg. This category is closed underextensions and images, but not closed under kernels and cokernels. Let n be the number of pairwisenonisomorphic simple H-modules.

For n ≥ 3 there always exist indecomposable regular quasi-simple modules without self-extensions.Take such a module X. Using Bongartz’s construction, one gets a squarefree tilting module T = M ⊕Xwith HomH(X,M) = 0. Thus M is in the right perpendicular category X⊥ of X, and it is a minimalprojective generator there. Hence X⊥ C-mod, where C = EndH(M), and C is connected wild hereditarywith n− 1 simple modules. We will identify X⊥ with C-mod via the functor HomH(M,−). If 0 →τHX → Z → X → 0 is the Auslander–Reiten sequence ending in X, then Z ∈ C-reg is a quasi-simplebrick. The tilted algebra B = EndH(T ) C[Z] is the one-point extension of C by Z. With these notationsthe following holds:

Theorem (Crawley-Boevey, K.) There exists a full and dense functor

F : C-reg→ H-reg

with FτC τHF. One has F(U) = 0 if and only if U ∈ addτiCZ|i ∈ Z.

For U ∈C-reg one gets F(U) = τ−rH τ

rT τ

sT τ−sC U, for r, s≫ 0; here τT denotes the relative Auslander–

Reiten translation in the tilting torsion class T of H-modules generated by T . For the proof direct andinverse limits are used.

It follows from this theorem that any two connected wild hereditary K-algebras H and H′ are almostequivalent. This means that there exist regular H-modules U and regular H′-modules U′, such that the

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factor categories of H-reg, respectively H′-reg, by the τH orbit of U, respectively the τH′-orbit of U′, areequivalent.

I also will give another description of this functor, using minimal approximations instead of limits,and apply this description to deduce some consequences. For example, it follows that the module Z isorbital elementary. This means that the τC-orbit of the module Z has the following property: Each shortexact sequence 0→ U → Z → V → 0 with Z ∈ addτiCZ|i ∈ Z splits, provided U and V are regular C-modules. Moreover, most wild hereditary algebras H have filtration closed regular components. Recallthat a regular component C is called filtration closed, if for each short exact sequence 0→ U → A→V → 0 with A ∈ addC also U and V are in addC, provided they are regular H-modules.

Andrzej Skowronski (Nicolaus Copernicus University) . . . November 12th (Tue), 9:30–10:20

Selfinjective algebras with deforming ideals

In the representation theory of selfinjective artin algebras a prominent role is played by the orbitalgebras B/G, where B is the repetitive category of an artin algebra B and G is an admissible infinitecyclic group of automorphisms of B. The major effort of my joint research with Kunio Yamagata duringthe last 20 years concerned criteria for a selfinjective artin algebra A to be isomorphic (respectively, socleequivalent, stably equivalent) to an orbit algebra of the form B/(φνB), where B is an artin algebra, νB theNakayama automorphism of B, and φ a positive automorphism of B. This was strongly motivated by theproblem of describing the structure of selfinjective artin algebras for which the Auslander–Reiten quiveradmits a generalized standard component.

The aim of the talk is to present the main results achieved in this direction.

Marta Błaszkiewicz (Nicolaus Copernicus University) . . . November 12th (Tue), 11:00–11:30

Selfinjective algebras of finite representation type with maximal almost split sequences

This is report on joint work with Andrzej Skowronski.

Let A be a finite dimensional K-algebra (associative, with an identity) over an arbitrary field K, andmod A the category of finite dimensional right A-modules. For a nonprojective indecomposable moduleX in mod A, there is an almost split sequence

0 −→ τAX −→ Y −→ X −→ 0,

where τAX is the Auslander–Reiten translation of X. Then we may associate to X the numerical invariantα(X) being the number of summands in a decomposition Y = Y1 ⊕ . . .⊕ Yr of Y into a direct sum ofindecomposable modules in mod A, which measures the complexity of homomorphisms in mod Awith domain τAX and codomain X. It has been proved by R.Bautista and S. Brenner (1981) that, if A isof finite representation type and X is a nonprojective indecomposable module in mod A, then α(X) ≤ 4,and if α(X) = 4, then the middle Y of an almost split sequence in mod A with the right term X admitsan indecomposable projective-injective direct summand. An almost split sequence in a module categorymod A of an algebra A of finite representation type with the middle term being a direct sum of fourindecomposable modules is called a maximal almost split sequence in mod A.

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We will discuss the structure of basic, indecomposable, finite dimensional selfinjective K-algebrasA over a field K for which mod A admits a maximal almost split sequence.

Jerzy Białkowski (Nicolaus Copernicus University) . . . . November 12th (Tue), 11:40–12:30

Periodicity of selfinjective algebras of polynomial growthThis is report on joint work with K. Erdmann and A. Skowronski

Let A be a finite dimensional K-algebra over an algebraically closed field K. Denote by ΩA thesyzygy operator on the category mod A of finite dimensional right A-modules, which assigns to a moduleM in mod A the kernel ΩA(M) of a minimal projective cover PA(M)→ M of M in mod A. A module Min mod A is said to be periodic if Ωn

A(M) M for some n ≥ 1. Then A is said to be a periodic algebra ifA is periodic in the module category mod Ae of the enveloping algebra Ae = Aop ⊗K A, that is, periodicas an A-A-bimodule. The periodic algebras A are selfinjective and their module categories mod A areperiodic (all modules in mod A without projective direct summands are periodic). The periodicity of analgebra A is related with the periodicity of its Hochschild cohomology algebra HH∗(A) and is invariantunder equivalences of the derived category Db(mod A) of bounded complexes over mod A. One of theexciting open problems in the representation theory of selfinjective algebras is to determine the Moritaequivalence classes of periodic algebras.

It has been proved by Dugas that every selfinjective algebra of finite representation type, withoutsemisimple summands, is a periodic algebra. During the talk we will present a description of all basic,indecomposable, representation-infinite periodic algebras of polynomial growth.

Piotr Malicki (Nicolaus Copernicus University) . . . . . . . November 12th (Tue), 14:00–14:50

Finite cycles of indecomposable modulesThis is report on joint work with J. A. de la Pena and A. Skowronski.

Let A be a basic indecomposable artin algebra over a commutative artin ring K, mod A the categoryof finitely generated right A-modules and ind A the full subcategory of mod A formed by the indecom-posable modules. Denote by rad∞A the infinite radical of mod A, being the intersection of all powersradi

A, i ≥ 1, of the radical radA of mod A. By a result of Auslander, rad∞A = 0 if and only if A is of finiterepresentation type. On the other hand, if A is of infinite representation type then (rad∞A )2 , 0, by a re-sult due to Coelho–Marcos–Merklen–Skowronski. For a module M in mod A, consider a decompositionA = PM ⊕QM of A in mod A such that the simple summands of the semisimple module PM/ rad PM areexactly the simple composition factors of M, and the ideal tA(M) in A generated by the images of allhomomorphisms from QM to A in mod A. Then Supp(M) = A/tA(M) is called the support algebra of M.A cycle in ind A is a sequence

M0f1−−→ M1→ ·· · → Mn−1

fn−−→ Mn = M0

of nonzero nonisomorphisms in ind A, and such a cycle is said to be finite if the homomorphisms f1, . . . , fndo not belong to rad∞A . Following Ringel, a module M in ind A which does not lie on a cycle in ind Ais called directing. A nondirecting module M in ind A is said to be cycle-finite if every cycle in ind Apassing through M is finite.

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In 1984 Ringel proved that the support algebra of a directing module is a tilted algebra. The aim ofthe talk is to present solution of the long standing open problem concerning the structure of the supportalgebras of cycle-finite indecomposable modules (being a natural extension of the problem considered byRingel). This is achieved by a conceptual description of the support algebras of cycle-finite componentsin the cyclic Auslander–Reiten quiver cΓA of an algebra A. A prominent role in this description is playedby the generalized multicoil algebras (defined by Malicki–Skowronski) and the generalized double tiltedalgebras (defined by Reiten–Skowronski).

Adam Skowyrski (Nicolaus Copernicus University) . . . . November 12th (Tue), 15:40–16:30

Homological problems for cycle-finite algebras

By an algebra we mean an artin K-algebra, where K is a commutative artin ring. Given an algebraA, we denote by mod A the category of finitely generated right A-modules, by ind A the full subcategoryof mod A consisting of indecomposable modules, by rad∞A the infinite Jacobson radical of mod A, andby τA = DTr the Auslander–Reiten translation.

Following Ringel, a cycle in mod A is a sequence

X0f1 // X1

f2 // . . .fr // Xr = X0

of nonzero nonisomorphisms in ind A, and such a cycle is called finite provided that all homomorphismsf1, . . . , fr do not belong to rad∞A . There is an important and wide class of algebras introduced by Assemand Skowronski, called cycle-finite algebras. Recall that an algebra A is said to be a cycle-finite algebraif and only if all cycles in mod A are finite.

A prominent role in the representation theory is played by quasitilted algebras and generalized dou-ble tilted algebras. We are concerned with the following three open problems formulated by Skowronski.

(1) An algebra A is a generalized double tilted algebra or a quasitilted algebra if and only if, for all butfinitely many isomorphism classes of modules X in ind A, we have pdA X ⩽ 1 or idA X ⩽ 1.

(2) An algebra A is a generalized double tilted algebra or a quasitilted algebra if and only if, for allbut finitely many isomorphism classes of modules X in ind A, we have HomA(D(A),X) = 0 orHomA(X,A) = 0.

(3) An algebra A is a generalized double tilted algebra if and only if mod A admits a faithful mod-ule M, such that, for all but finitely many isomorphism classes of modules X in ind A, we haveHomA(M, τAX) = 0 or HomA(X,M) = 0.

The aim of the talk is to present solutions of these problems for cycle-finite algebras.

Changchang Xi (Capital Normal University) . . . . . . . . . . November 12th (Tue), 16:40–17:30

Higher dimensional tilting modules and recollements

Infinitely generated tilting modules behave very differently from finitely generated tilting modules.In this talk, we shall present some recent results on infinitely generated tilting modules which is closelyrelated to noncommutative localizations, homological ring epimorphisms, recollements, Mittag–Lefflerconditions and coproducts of rings. Our study will be carried out under the framework of Ringel modules,

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and focused on the derived categories of the endomorphism rings of infinitely (that is, not necessarilyfinitely) generated tilting modules of projective dimension at least 2. This approach allows us to dealwith infinitely generated cotilting modules in a uniform way.

The contents of this talk are mainly taken from a joint preprint “Higher dimensional tilting modulesand homological subcategories” with H. X. Chen.

Karin Erdmann (University of Oxford) . . . . . . . . . . . . . . . November 13th (Wed), 9:30–10:20

Support varieties via Hochschild cohomology: a necessary conditionLet A be a finite-dimensional self-injective algebra over a field K, and let HH∗(A) be the Hochschild

cohomology algebra of A, this acts on Ext∗A(M,M) for any A-module M. If HH∗(A) is noetherian andthe Ext algebra of A is finitely generated over HH∗(A) then A-modules have supports defined via thisaction, which share many properties of supports defined via group cohomology. Unfortunately, thesefinite generation properties are difficult to verify. However, if the algebra has a module with complexity= 1 which is not Ω periodic (called ‘criminal’) then it follows that such support cannot exist. We showthat it is very common for socle deformations of self-injective algebras to have criminals. We also showthat the criminals we construct are counterexamples to the generalized Auslander–Reiten condition.

Tomohiro Itagaki (Tokyo University of Science) . . . . . . . November 13th (Wed), 11:00–11:30

Cyclic homology of truncated quiver algebras and notes on the no loops conjecture forHochschild homology

(Joint work with Katsunori Sanada)

In this talk, we show the dimension formula of the cyclic homology of truncated quiver algebrasover an arbitrary field, and we extend the 2-truncated cycles version of the no loops conjecture (cf. [2])to the m-truncated cycles version for a class of finite dimesional algebras over an algebraically closedfield.

In [6], for a truncated quiver algebra A over a commutative ring, Skoldberg gives a left Ae-projectiveresolution of A and computes the Hochschild homology HHn(A). By means of this result and a theoremin Loday’s book(1992), Taillefer [7] gives a dimension formula of the cyclic homology of truncatedquiver algebras over a field of characteristic zero.

We compute the dimension formula of the cyclic homology of truncated quiver algebras over anarbitrary field by means of chain maps in [1] and a spectral sequence. Our result generalizes the aboveresult by Taillefer.

Moreover, we show that the m-truncated cycles version of the no loops conjecture holds for a classof bound quiver algebras over an algebraically closed field as an application of the chain map from Cibils’projective resolution (cf. [3]) to Skoldberg’s projective resolution given in [1].

References[1] G. Ames, L. Cagliero, P. Tirao, Comparison morphisms and the Hochschild cohomology ring of

truncated quiver algebras, J. Algebra 322(5)(2009), 1466–1497.[2] P. A. Bergh, Y. Han, D. Madsen, Hochschild homology and truncated cycles, Proc. Amer. Math. Soc.

(2012), no. 4, 1133–1139.

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[3] C. Cibils, Cohomology of incidence algebras and simplicial complexes, J. Pure Appl. Algebra 56(3)(1989), 221–232.

[4] T. Itagaki, K. Sanada, The dimension formula of the cyclic homology of truncated quiver algebrasover a field of positive characteristic, submitted.

[5] T. Itagaki, K. Sanada, Notes on the Hochschild homology dimension and truncated cycles, in prepa-ration.

[6] E. Skoldberg, Hochschild homology of truncated and quadratic monomial algebras, J. Lond. Math.Soc. (2) 59 (1999), 76–86.

[7] R. Taillefer, Cyclic homology of Hopf algebras, K-Theory 24 (2001), 69–85.

Dan Zacharia (Syracuse University) . . . . . . . . . . . . . . . . . November 13th (Wed), 11:40–12:30

A characterization of the graded center of a Koszul algebraI will talk on joint work with Ed Green and Nicole Snashall. Let k be a field and let S be a graded

k-algebra. The graded center of S is is the graded subring Zgr(S ) generated by all the homogeneouselements u of S such that uv = (−1)|u||v|vu for every homogeneous element v of S , where |x| denotes thedegree of the homogeneous element x. We will present a characterization of Zgr(S ) in the case when Sis a Koszul algebra and some applications of the ideas involved.

Yuji Yoshino (Okayama University) . . . . . . . . . . . . . . . . . . . November 14th (Thu), 9:30–10:20

Dependence of total reflexivity conditionsLet R be a commutative noetherian ring. Then the following theorem is proved by using the chro-

matic tower theorem of Neeman.

Theorem 0.1. Let W ∈ D±f g(R) and X ∈ D(R). Then, RHomR(W,X) = 0 if and only if W L⊗RX = 0.

In my talk I will show how I proved this theorem and I will present some of its application. Inparticular this can be presumably applied to prove the following result, but not completely verified yet.

Question 0.2. Let R be a generically Gorenstein ring, i.e. Rp is Gorenstein for all p ∈ Ass(R). If afinitely generated R-module M satisfies ExtiR(M,R) = 0 for all i > 0, then M is totally reflexive.

Ryo Kanda (Nagoya University) . . . . . . . . . . . . . . . . . . . . . November 14th (Thu), 11:00–11:30

Specialization orders on atom spectra of Grothendieck categoriesThe notion of the atom spectra of Grothendieck categories is a generalization of the prime spectra of

commutative rings. We develop a theory of the specialization orders on the atom spectra of Grothendieckcategories and introduce systematic methods to construct Grothendieck categories from colored quivers.We show that any partially ordered set is realized as the atom spectrum of some Grothendieck category,which is an analog of Hochster’s result in commutative ring theory.

Ryo Takahashi (Nagoya University) . . . . . . . . . . . . . . . . . . November 14th (Thu), 11:40–12:30

Existence of cohomology annihilators and strong generation of derived categoriesThis talk is based on joint work with Srikanth Iyengar.

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Let Λ be a Noether algebra, i.e., a Noetherian ring that is module-finite over its center Λc. We calla nonzerodivisor x of Λc a cohomology annihilator of Λ if there exists a positive integer n such that

x ·ExtnΛ(M,N) = 0

for all finitely generated Λ-modules M and N. Studies on such cohomology annihilation were startedby Dieterich [2], Popescu and Roczen [3] and Yoshino [5] in the 1980s in relation to the Brauer–Thrallconjectures for Cohen–Macaulay modules over Cohen–Macaulay rings. In the 1990s, Wang [4] provedthat every complete equicharacteristic local domain with perfect residue field possesses a cohomologyannihilator. Recently, Buchweitz and Flenner explored cohomology annihilation over Gorenstein orders.

In this talk, we consider how to proceed in the general case. We also relate cohomology annihilationwith strong generation of the bounded derived category of finitely generated Λ-modules in the sense ofBondal and Van den Bergh [1].

References[1] A. Bondal; M. Van den Bergh, Generators and representability of functors in commutative and

noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258.[2] E. Dieterich, Reduction of isolated singularities, Comment. Math. Helv. 62 (1987), no. 4, 654–676.[3] D. Popescu; M. Roczen, Indecomposable Cohen–Macaulay modules and irreducible maps, Compos.

Math. 76 (1990), no. 1-2, 277–294.[4] H.-J. Wang, On the Fitting ideals in free resolutions, Michigan Math. J. 41 (1994), no. 3, 587–608.[5] Y. Yoshino, Brauer–Thrall type theorem for maximal Cohen–Macaulay modules, J. Math. Soc. Japan

39 (1987), no. 4, 719–739.

Michio Yoshiwaki (Osaka City University) . . . . . . . . . . . . November 14th (Thu), 14:00–14:30

Dimensions of triangulated categories with respect to subcategoriesThis talk is based on joint work [1] with Takuma Aihara, Tokuji Araya, Osamu Iyama and Ryo Takahashi.

The notion of dimension of a triangulated category was introduced by Rouquier [3] based on workof Bondal and van den Bergh [2] on Brown representability. It measures how many extensions are neededto build the triangulated category out of a single object, up to finite direct sum, direct summand and shift.

It is still a hard problem in general to give a precise value of the dimension of a given triangulatedcategory. Our aim is to provide new information on this problem.

In this talk, we will introduce a concept of dimension of a triangulated category with respect to afixed full subcategory. Then we will give upper bounds of our relative dimensions of derived categoriesin terms of global dimensions. Namely, our main result is the following.

Theorem 1. Let A be an abelian category and X a contravariantly finite subcategory that generatesA. Then

X- tri.dimDb(A) ≤ gl.dim(modX).

Our methods not only recover some known results on the dimensions of derived categories in thesense of Rouquier, but also apply to various commutative and non-commutative noetherian rings. In fact,we obtain the following consequence of Theorem 1.

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Corollary 2. Let Λ be a noetherian ring and T a cotilting module. Then

XT - tri.dimDb(modΛ) ≤max1, inj.dimT ,

where XT = X ∈modΛ | ExtiΛ

(X,T ) = 0 for any i > 0.

References[1] T. Aihara, T. Araya, O. Iyama, R. Takahashi and M. Yoshiwaki, Dimensions of triangulated cate-

gories with respect to subcategories, to appear in J. Algebra (arXiv:1204.6421).[2] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and

noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258.[3] R. Rouquier, Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2, 193–256 and

errata, 257–258.

Hirotaka Koga (University of Tsukuba) . . . . . . . . . . . . . . . November 14th (Thu), 14:35–15:05

Clifford extension(Joint work with Mitsuo Hoshino and Noritsugu Kameyama)

Auslander–Gorenstein rings appear in various fields of current research in mathematics (see [1],[2], [3] and [5]). However, little is known about constructions of Auslander–Gorenstein rings. We havealready known that Frobenius extensions of Auslander–Gorenstein rings are Auslander–Gorenstein rings.In this note, formulating the construction of Clifford algebras (see e.g. [4]) we introduce the notion ofClifford extensions and show that Clifford extensions are Frobenius extensions. Consequently Cliffordextensions of Auslander–Gorenstein rings are Auslander–Gorenstein.

Let n ≥ 2 be an integer. We fix a set of integers I = 0,1, . . . ,n−1 and a ring R. We use the notationA/R to denote that a ring A contains R as a subring. First, we will construct a Frobenius extension Λ/Rusing a certain pair (σ,c) of σ ∈ Aut(R) and c ∈ R. Namely, we will define an appropriate multiplicationon a free right R-module Λ with a basis vii∈I . Then we restrict ourselves to the case where n = 2 in orderto recover the construction of Clifford algebras. For m ≥ 1 we construct ring extensions Λm/R which wecall Clifford extensions using the following data: a sequence of elements c1,c2, · · · in Z(R) and signsε(i, j) for 1 ≤ i, j ≤ m. Namely, we will define an appropriate multiplication on a free right R-moduleΛm with a basis vxx∈Im . We show that Λm is obtained by iterating the construction above m times, thatΛm/R is a Frobenius extension, and that if ci ∈ rad(R) for 1 ≤ i ≤ m then R/rad(R)

∼−→ Λm/rad(Λm).

References[1] M. Artin, J. Tate and M. Van den Bergh, Modules over regular algebras of dimension 3, Invent. Math.

106 (1991), no. 2, 335–388.[2] J.-E. Bjork, Rings of differential operators, North-Holland Mathematical Library, 21. North-Holland

Publishing Co., Amsterdam–New York, 1979.[3] J.-E. Bjork, The Auslander condition on noetherian rings, in: Seminaire d′Algebre Paul Dubreil et

Marie-Paul Malliavin, 39eme Annee (Paris, 1987/1988), 137–173, Lecture Notes in Math., 1404,Springer, Berlin, 1989.

[4] D. J. H. Garling, Clifford algebras: an introduction, London Mathematical Society Student Texts,78. Cambridge University Press, Cambridge, 2011, viii+200pp.

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[5] J. Tate and M. Van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124(1996), no. 1-3, 619–647.

Hiroyuki Minamoto (Osaka Prefecture University) . . . . November 14th (Thu), 15:40–16:30

Derived Gabriel topology, localization and completion of dg-algebras

Gabriel topology is a special class of linear topology on rings, which plays an important role in thetheory of localization of (not necessary commutative) rings. Several evidences have suggested that thereshould be a corresponding notion for dg-algebras. In this talk I introduce a notion of Gabriel topologyon dg-algebras, derived Gabriel topology, and show its basic properties. In the same way we give thedefinition of topological dg-modules over a dg-algebra equipped with derived Gabriel topology.

We show that derived bi-duality module is obtained by tautological homotopy limit. From the viewpoint of derived Gabriel topology, this is a derived version of Lambek’s theorem about localization andcompletion of rings. We see that this gives generalizations and conceptual proof of several results aboutdg-algebras.

Jun-ichi Miyachi (Tokyo Gakugei University) . . . . . . . . . November 14th (Thu), 16:40–17:30

Researches on the representation theory of algebras at University of Tsukuba

We survey researches on the representation theory of algebras at University of Tsukuba from 1980sto 1990s.

Helmut Lenzing (University of Paderborn) . . . . . . . . . . . . . November 15th (Fri), 9:30–10:20

The ubiquity of the equation x2+ y3+ z5 = 0

This is an expository talk dealing with the ubiquity of the equation x2 + y3 + z5 = 0. Among otherswe will deal with a challenging interrelationship between commutative and non-commutative algebra,by considering solution sets in a commutative respectively a non-commutative environment (fields resp.matrices).

The equation x2 + y3 + z5 = 0 has an interesting history relating to Plato’s classification of regularsolids, via Felix Klein’s study of the symmetry group of the icosahedron and its associated invariant the-ory. It thus links to what is now known as McKay theory. The solutions of the equation x2+y3+z5 = 0 incomplex 3-space form a geometric object, known as the E8-singularity, an object related to many furthermathematical subjects. I will additionally consider solutions of the equation which are (composable se-quences of) matrices. This immediately yields to the representation theory of the canonical algebras, asintroduced by Ringel, yielding a canonical algebra Λ with arm lengths (2,3,5), and thus to the represen-tation theory of a tame quiver of extended Dynkin type E8. Conversely, we discuss how to recover theequation from the representation theory of Λ. By using coherent sheaves and stable categories of Cohen–Macaulay modules we link the commutative and the non-commutative solutions of x2+ y3+ z5 = 0, thusshowing a remarkable unity of mathematics. This fact relates to a theorem of Orlov (2009).

Further appearances of the equation concern preprojective and orbit algebras, the classification offactorial rings, fundamental groups of 2-orbifolds and the invariant subspace (more precisely subflags)

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problem for nilpotent operators, a subject treated by Simson (2007), Ringel–Schmidmeier (2008) andKussin–Meltzer–Lenzing (2011-2013).

Kenta Ueyama (Shizuoka University) . . . . . . . . . . . . . . . . . November 15th (Fri), 11:00–11:30

Ample Group Actions on AS-regular Algebras and Noncommutative Graded IsolatedSingularities

This is a report on joint work with Izuru Mori. AS-regular algebras are the most important classof algebras studied in noncommutative algebraic geometry. They are noncommutative analogues ofthe polynomial algebra. In this talk, we introduce a notion of ampleness of a group action on a rightnoetherian graded algebra, and show that, for an AS-regular algebra S , the ampleness of G on S isstrongly related to the notion of S G to be a noncommutative graded isolated singularity. Moreover, wegive a relationship between S G and the skew group algebra ∇S ∗G of the quantum Beilinson algebra ∇S .

Atsushi Takahashi (Osaka University) . . . . . . . . . . . . . . . . November 15th (Fri), 11:40–12:30

Weyl groups and Artin groups associated to weighted projective lines

We report on our recent study on a correspondence among weighted projective lines, cusp singular-ities and cuspidal root systems. Our purpose is to study the fundamental group of the complement of thediscriminant of the Frobenius manifold constructed from the Gromov–Witten theory for a weighted pro-jective line. If the number of orbifold points is at most three, then the Frobenius manifold is isomorphic,by mirror symmetry, to the base space of a universal unfolding of a cusp singularity, which is a quotientspace of a product of the complexified Tits cone and a complex line by the extended cuspidal Weyl groupassociated to the correponding cuspidal root system. We start from a weighted projective line and thenassociate to it a cuspidal Weyl group and a cuspidal Artin group by generaters and relations followingSaito–Takebayashi and Yamada. We show that the cuspidal Artin group is isomorphic to the fundamen-tal group of the regular orbit space of the complexified Tits cone under the action of the cuspidal Weylgroup.

Yoshiyuki Kimura (Osaka City University) . . . . . . . . . . . . November 15th (Fri), 14:00–14:50

Quiver varieties and Quantum cluster algebras

Let Av(n(w)) be the quantum coordinate ring of unipotent subgroup associated with a Weyl groupelement w of symmetric Kac–Moody Lie algebra g. It is shown that Av(n(w)) has the dual canonicalbasis which is compatible with Kashiwara–Lusztig’s dual canonical basis in [2]. Geiss–Leclerc–Schroer[1] have shown that Av(n(w)) has a structure of quantum cluster algebra which is induced by the pre-projective algebra Λ. It is conjectured that the dual canonical basis of Av(n(w)) contains the quantumcluster monomial of that. Let Q be an acyclic quiver, cQ be the corresponding acyclic Coxeter word andw = c2

Q with ℓ(w) = 2|Q|. In [3], we identify the (twisted) quantum Grothendieck ring of the correspond-ing graded quiver variety and Av(n(c2

Q)) and identify the basis of simple perverse sheaves with the dualcanonical basis. Using this isomorphism, it can be shown that the set of quantum cluster monomials iscontained in the dual canonical basis of Av(n(c2

Q)). This is a joint work with Fan Qin.

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References[1] Christof Geiß, Bernard Leclerc, and Jan Schroer. Cluster structures on quantum coordinate rings. to

appear in Selecta Math., 2012. e-print arxiv http://arxiv.org/abs/1104.0531.[2] Yoshiyuki Kimura. Quantum unipotent subgroup and dual canonical basis. Kyoto J. Math.,

52(2):277–331, 2012.[3] Yoshiyuki Kimura and Fan Qin. Graded quiver varieties, quantum cluster algebras and dual canonical

basis. e-print arxiv http://arxiv.org/abs/1205.2066v2, 2012.

Michael Wemyss (University of Edinburgh) . . . . . . . . . . . . November 15th (Fri), 15:40–16:30

Some new examples of self–injective algebrasI will talk about my recent work with Will Donovan (arXiv:1309.0698), which gives new invariants

of curves inside 3-folds X. In the minimal model program, certain surgeries called flips and flops appear,and the basic idea is that to each of these we can, using noncommutative deformation theory, associate anot-necessarily-commutative finite dimensional algebra Λcon, together with a functor

Db(modΛcon)→ Db(coh X).

In the flops setting, the algebra Λcon turns out to be self–injective, and control much of the homologicalalgebra.

Roughly speaking, our original curve (drawn in bold in the pictures below) deforms in our ambientvariety, and by tracking how it deforms we obtain a finite dimensional algebra. The left hand pictureillustrates the commutative deformations, the right hand picture the noncommutative ones.

Surprisingly (at least to us), most of the finite dimensional self-injective algebras that arise in thisway turn out to be new, and so I will spend lots of time explicitly presenting these algebras, since it wouldbe very interesting if they appeared in different contexts. Also, there are many examples of self-injectivealgebras (even given by a quiver with potential) that we cannot yet explicitly present, and I will explainhow the geometry helps to gain information regarding their properties. I will also discuss some openproblems.

Claus Michael Ringel (Bielefeld University) . . . . . . . . . . . November 15th (Fri), 16:40–17:30

The root posetsGiven a (finite) root system, the choice of a root basis divides it into the positive and the negative

ones, it also yields an ordering on the set of positive roots. The set of positive roots with respect to this

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ordering is called a root poset. The root posets have attracted a lot of interest in recent years: The setA of antichains in a root poset (with a suitable ordering) turns out to be a lattice, it is called lattice of(generalized) non-crossing partitions. Non-crossing partitions play an interesting role in several parts ofmathematics: not only in algebra and geometry, but even in free probability theory. The maximal chainsin A correspond bijectively to the factorizations of a Coxeter element in the Weyl group using reflections,the number of maximal chains was determined already in 1974 by Deligne–Tits–Zagier.

If R is a Dynkin algebra, then the iso classes of the indecomposable R-modules form a poset withrespect to the subfactor ordering: in a joint paper with Dlab (1979) we have shown that one obtains inthis way all root posets, thus one can use the representation theory of R-modules in order to study the rootposets and their antichain lattices. According to Ingalls and Thomas (2009), the lattice is isomorphic tothe lattice of thick subcategories of the category of R-modules, and, as observed by Krause and Hubery(2013), the maximal chains in the antichain lattice correspond bijectively to the complete exceptionalsequences of R-modules.

The lecture will provide a survey of some properties of the root posets, in particular, we will focusthe attention to the role of the exponents.

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INFORMATION

Welcome to Nagoya. The following information will be useful in helping you enjoy your stayduring the conference. Please feel free to get in touch with the organizers (Prof. Iyama) and the secretary(Ms. Kozaki) if any questions arise or if you encounter unexpected trouble.

Conference VenueConference mainly takes place at Sakata–Hirata Hall in Science South Building on campus (Fig-

ure A). However, please note that the talks on Friday (15th) will be held at ES Hall in E&S Building(Figure B).

Registration DeskFrom November 11th to 14th, the registration desk will be set in the lobby of Sakata–Hirata Hall,

where the conference takes place. You can pick up your conference package. The desk operates from9:00 to 16:30 through Monday to Thursday.

If you need to make copies, please ask at the registration desk. Those who want to send messagesby fax can ask at the desk as well.

Coffee BreaksCoffee will be served in the lobby of Sakata–Hirata Hall. All participants are requested not to bring

food into the conference hall.

Internet ConnectionThe Wi-Fi access is available in and around conference hall. Please ask for the detailed information

at the registration desk. You can choose to connect to eduroam as well.

RestaurantsFor lunch and dinner, there are some restaurants in the campus, or outside of the campus. You will

find them in Figure F.

Group photoWe will take a group photo of all participants in the Conference Hall. We are planning this on

November 14th (Thu) just after the morning session.

Conference ExcursionAn English guided bus tour is planned on the afternoon of November 13th (Wed). We will visit

Nagoya Castle, Noritake Garden where we will observe (and even experience) a manufacturing processof high-quality ceramic tableware. Weather permitting, we will also visit Atsuta Shrine the one of themajor shrines in Japan. All the participants are welcome, and the fee is 3,500 JPY which includes a smalllunch in the bus. Please sign up for it by 12:00 on November 12th (Tue) at the registration desk of theconference. We will stop accepting applications once all the seats are taken.

The chartered bus will depart at no delay of 13:00 from the front of the Toyoda Auditorium. (SeeFigure C) We require you to be on time.

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BanquetThere will be a conference banquet on November 14th (Thu) from 18:00 at Mei-dining, which is

situated west part on campus (Figure C). The banquet costs 5,000 JPY per person.If you want to join the banquet, please come to the registration desk by 13:00 on November 12th.

LibraryThe participants may use the Mathematics Library.

Location: Science Building A, 1st Floor (Figure C)Hours: 9:00–17:00

Services: The browsing, reference are available.

Please ask the librarians for details.

Money ExchangeYou will find a Post Office authorized to conduct foreign money exchange on campus (Figure C).

Japan Post Bank provides cash withdrawal services for credit cards and cash cards issued by overseasfinancial institutions. Cards bearing

VISA, VISAELECTRON, VISA PLUS, MasterCard, Maestro, Cirrus, AmericanExpress, Diners Club, JCB, China Unionpay and DISCOVER

can make withdrawals at Japan Post Bank ATMs. The ATMs on campus (Figure C) and in ChunichiBuilding (Figure E) are most convenient for you. The cards Master, VISA and PLUS can be used at theCitiBank (Figure E) for 24 hours.

SightseeingPlease consult the brochures you received at the registration desk and the following webpages for

other sight seeing spots in and around Nagoya.

http://www.ncvb.or.jp/index_e.html

is a good place to start.

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EV

Conference Hall(Sakata–Hirata Hall)

Conference Hall(Sakata–Hirata Hall)

Registration DeskRegistration Desk

ConferenceOfce

Lounge

Figure A: Science South Bldg.(Nov. 11–14)

EV

EV

Central Library of Graduate School of EngineeringRestaurantchez Jiroud

Entrance Hall

To 2FTo 2F

To 2FTo 2F

Conference Hall(ES Hall)

Conference Hall(ES Hall)

Figure B: E&S Bldg. (Nov. 15)

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Graduate School ofMathematicsGraduate School ofMathematics

Cafeteria “Mei-dining”(Banquet)Cafeteria “Mei-dining”(Banquet)

Science South Bldg.(Conference, Nov. 11–14)Science South Bldg.(Conference, Nov. 11–14)

E&S Bldg.(Conference, Nov. 15)

E&S Bldg.(Conference, Nov. 15)

Bus Stop(for Excursion)Bus Stop(for Excursion)

Toyoda AuditoriumToyoda Auditorium

Nagoya Daigaku Sta.Nagoya Daigaku Sta.

to Motoyama Sta.

to Kanayama Sta.

Subway Meijo Line

Subway Meijo Line

Post Ofce / Japan Post Bank (ATM)Post Ofce / Japan Post Bank (ATM)Post Ofce / Japan Post Bank (ATM)

Science LibraryScience LibraryScience Library

Figure C: Campus Map

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Subway Sakura-dori Line

Subway Kamiiida Line

Subway Meijō Line

Subway Meiko Line

Subway Higashiyama Line

Subway Tsurumai Line

16–24min

24min

Sakae Sta.

Kanayama Sta.

Nagoya Daigaku Sta.

Motoyama Sta.Motoyama Sta.

to Tokyō

to Kyōto / Shin-Ōsaka

JR Tokaido Shinkansen

to Chubu International Airport

Meitetsu Line

Nagoya Sta.Nagoya Sta. Fujigaoka Sta.

Takabata Sta.

Ōzone Sta.

Figure D: Subway Map with some Railways

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Undergrdound MallUndergrdound Mall

Hotel Trusty Nagoya SakaeHotel Trusty Nagoya Sakae

Sakae Sta. Subway Higashiyama Line

Post Office / Japan Post Bank (ATM)(in Chunichi Bldg., 1F)Post Office / Japan Post Bank (ATM)(in Chunichi Bldg., 1F)

CitiBank (ATM)(in Sakae Parkside Place, 1F)

CitiBank (ATM)(in Sakae Parkside Place, 1F)

Su

bw

ay Meijo

Line

to Motoyama Sta.to Motoyama Sta.to Nagoya Sta.to Nagoya Sta.

to Kanayama Sta.to Kanayama Sta.

to Ōzone Sta.to Ōzone Sta.

Undergrdound MallUndergrdound Mall

Ferris Wheel(on Sunshine Sakae, Roof Top)

Ferris Wheel(on Sunshine Sakae, Roof Top)

Bus Terminal(in Oasys 21)Bus Terminal(in Oasys 21)

Bus TerminalBus Terminal

21 3

678

9

16

10

11

5

15 14

13

12

4

10A

7

7

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10 2

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9A

9B

Figure E: Map of Sakae (around Hotel Trusty Nagoya Sakae)

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0m 100 200 300 400 500m 1km

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Graduate School of MathematicsGraduate School of Mathematics

Nagoya UniversityNagoya University

Haloki (hamburg, steak)Lawson (CVS)

Bento-Man (take out lunch)Botan-tei (Chinese)

Pion (Korean BBQ)Xiang Lan Lou (Chinese)

Gran Piatto (Italian)

Nanzan University

Kazukian (soba)

Pastel (sweets, cafe)

Cafe Downey (cafe)

Ichiemon Dining (tavern)

CircleK (CVS)

Eas Salon (curry)

Phonon Cafe (cafe)

Haruten (Japanese)IZUMI (French)

LAWSON100 (CVS)Ankuru (cafe)

Marina (western)

Ampio Fiume (Italian)Kaisenkan (Chinese)

Sukiya(Japanese fast food)

Midori Sushi (sushi)

Hisaya (Japanese)Richiru (cafe)Ikatsu (okonomiyaku)

Miroku (steak)Cha Raku (cafe)

Nakau(Japanese fast food)McDonald’s (hamburger)

Gusto(Japanese, western)Tori-Kin (chikin)

Yorimichi (tavern)Sapporo-Tei (Chinese noodles)

Kurashiki (Japanese)SUNKS (CVS)Bon (cafe)

Mister Donuts (sweets, cafe)The Don (Japanese)SevenEleven (CVS)

Mos Burger (hamburger)

Cafe Nishihara (cafe)Kakure-ga (tavern)

CoCo! Ichiban-ya (curry)Gaya (tavern)KFC (fast food)

Sakanayama (Japanese)Ashitaba (udon, soba)Furaipan (western)Kangaroo Pocket (western)

Deli Deli (CVS)Retrocalm (cafe)

Doutor (cafe)Za·Watami (tavern)

Eegaya (curry)

Men-ya Ryōma (Chinese noodles)Tori-Tori-Tei (tavern)

Machikadoya (Japanese fast food)

For You (rice in omelet)Roppa (tavern)

Paragon (Asian)

Le Sourire(French)

SevenEleven (CVS)

Bai Toong (Asian)

Tai-Sho-Ken(Chineese noodle)

Shoro Zushi (sushi)

Masaruya Café Rest B2 (western)

Bunmeikan (tavern) Bun Boo Lassi (tavern)Motoyama55 (Chinese noodles)

Hamakinu (tavern)

Kochab (French)

Dorako(Korean BBQ)

Wakabayashi (Japanese)

En (Chinese)La Pallete (western)

Moe (cafe)

SevenEleven (CVS)

SevenEleven (CVS)

Shokuin Shokudō (cafeteria)

Friendly Nanbu (cafeteria)

Nanbu Shokudō (cafeteria)

Nanamitei (cafeteria)

Hokubu Shokudō (cafeteria)

Hananoki (Japanese)

Dinning Forest (cafeteria)Café Fronte (cafe)

Universal Club (cafeteria)

FamilyMart(CVS)

FamilyMart(CVS)

SevenEleven (CVS)

CircleK (CVS)

Rike-Shop (CSV)

Craig’s Cafe (cafe)

Chez Jiroud (French)

Starbacks (cafe) I-B Cafe (cafe)

Palu (cafe)

Drawing (cafe & bar)

Yakusou Labo Toge(cafe)

Miura (cafe)

Hoja Nasreddin (curry)Kobalele Cafe (cafe)

Figure F: Lunch Map

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