13th June 10:0010:10
10:1011:00
Akira Ishii (Hiroshima)  Introduction to the McKay correspondence and ArtinVerdier theory 
Abstract  For a finite subgroup G of SL(2, C), the quotient singularity C^2/G has a resolution whose dual graph is a Dynkin graph of type ADE. In the late 70’s, McKay observed that one can obtain the affine Dynkin graph of the same type by using the representation theory of G. Here, the vertices correspond to irreducible representations of G and thus the existence of a natural correspondence between exceptional curves and representations was expected. In 1984, Artin and Verdier gave a conceptual proof of the existence of such a correspondence. Moreover, as an appendix to the ArtinVerdier paper, Esnault and Knorrer gave an explanation for the correspondence of edges. I will talk about these old results and explain how they can be used to describe the correspondence between derived categories. 
References  [1] J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure
Math., vol. 37, Amer. Math. Soc., Providence, R. I., 1980, pp. 183186. [2] Artin, M. and Verdier, J.L., Reflexive modules over rational double points, Math. Ann. 270 (1985), 79–82 [3] Esnault, H. and Knörrer, H., Reflexive modules over rational double points, Math. Ann. 272 (1985), 545–548. [4] Esnault, H., Reflexive modules on quotient surface singularities, J. reine angew. Math. 362 (1985), 63–71. [5] Wunram, J., Reflexive modules on quotient surface singularities, Math. Ann. 279 (1988), 583–598. [6] Ishii, A., On the McKay correspondence for a finite small subgroup of GL(2,ℂ). J. Reine Angew. Math. 549 (2002), 221–233. [7] M. Van den Bergh, Threedimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455. 
11:1512:15
Yujiro Kawamata (Tokyo)  Derived McKay correspondence for finite abelian groups and derived toric MMP 
Abstract  We review the connection between the toric MMP (minimal model program) and the corresponding sequence of SOD (semiorthogonal decompositions) of the derived categories, yielding the derived McKay correspondence for finite abelian groups. 
References  [1] Dequivalence and Kequivalence, math.AG/0205287, J. Diff. Geom. 61 (2002), 147171.
[2] Derived categories of toric varieties, Michigan Math. J. 54 (2006), 517535.
[3] Derived categories of toric varieties II, Michigan Math. J. 622 (2013), 353363.
[4] Derived categories of toric varieties III, European Journal of Mathematics,
2 (2016), 196207.

13:4514:45
Kazushi Ueda (Tokyo)  Dimer models 
Abstract  We give an introduction to the theory of dimer models, with emphasis
on noncommutative crepant resolutions and derived categories of coherent sheaves. 
References  [1] Nathan Broomhead, Dimer models and CalabiYau algebras, Memoirs of
the American Mathematical Society, Volume 215, Number 1011, viii+86 pages,
2012, arXiv:0901.4662 [2] Raf Bocklandt, A Dimer ABC, arXiv:1510.04242 
15:0016:00
Alastair Craw (Bath)  The McKay correspondence by variation of GIT quotient 
Abstract  For a finite subgroup G of SL(3,C), Bridgeland, King and Reid's proof of the McKay correspondence appeared to single out Nakamura's Hilbert scheme of Gclusters as a preferred minimal resolution of the singularity C^3/G. I'll talk about an old paper with Akira Ishii where we demonstrated, at least for G abelian, that every projective minimal resolution can be constructed as a fine moduli space of certain stable modules over the skew group algebra of G (called Gconstellations). Recently, Ishii and Ueda extended these results from C^3/G to any Gorenstein affine toric 3fold. 
References 
[1] Alastair Craw and Akira Ishii, Flops of GHilb and equivalences of
derived categories by variation of GIT quotient, Duke Math. J. 124 (2004),
no.2, 259307.
[2] Akira Ishii and Kazushi Ueda, Dimer models and crepant resolutions, arXiv:1303.4028. 
16:1517:15
Hokuto Uehara (TMU)  A trichotomy for the autoequivalence groups of derived categories on surfaces 
Abstract  We study the group of autoequivalences of the derived categories on smooth projective surfaces, and show a trichotomy of types of the groups, that is, of K3 type, of elliptic surface type and of general type. We also pose a conjecture on the description of each type of the groups, and prove it in some special cases. 
References  (in preparation) 
14th June 9:3010:15
Ryo Takahashi (Nagoya)  Introduction to CohenMacaulay representation theory 
Abstract  The main subject of CohenMacaulay representation theory is to study the structure of (maximal) CohenMacaulay modules over a given CohenMacaulay ring. In this talk, I will roughly explain the socalled "classical period" of CohenMacaulay representation theory, which concerns CohenMacaulay local commutative rings and has been established by 1990. If time permits, I would like to mention a celebrated result of Auslander on isolated singularities and speak about a recent progress. 
References  [1] M. Auslander, Isolated singularities and existence of almost split
sequences, Representation theory, II (Ottawa, Ont., 1984), 194242, Lecture
Notes in Math., 1178, Springer, Berlin, 1986. [2] H. Dao; R. Takahashi, The dimension of a subcategory of modules, Forum Math. Sigma 3 (2015), e19, 31 pp. [3] Y. Yoshino, CohenMacaulay modules over CohenMacaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990. 
10:3011:30 / 13:0014:00
Michel Van den Bergh (Hasselt)  Noncommutative crepant resolutions and modules of covariants I & II 
Abstract  The CohenMacaulay modules that enter in the classical McKay correspondence
are constructed from the representations of a finite group. They are known
as "modules of covariants". In the lectures I will discuss modules of covariants for reductive groups. I will start by explaining some of my own old work which gives a sufficient combinatorial condition for a module of covariants to be CohenMacaulay. Then I will discuss recent joint work with Špela Špenko in which we show that such modules of covariants may be used to construct noncommutative (crepant) resolutions for quotient singularities. Finally I will discuss the the interesting special case of determinantal varieties for (skew) symmetric matrices. 
References  [1] R. Buchweitz, G. J. Leuschke, and M. Van den Bergh, Noncommutative
desingularization of determinantal varieties I, Invent. Math. 182 (2010),
47115. [2] R. Buchweitz, G. J. Leuschke, and M. Van den Bergh, Noncommutative desingularization of determinantal varieties, II, to appear in Int. Math. Res. Not. [3] Michel Brion, Sur les modules de covariants, Annales scientifiques de l'École Normale Supérieure 26 (1993), 121 [4] S. Spenko and M. Van den Bergh, Noncommutative resolutions of quotient singularities, submitted. [5] S. Spenko and M. Van den Bergh, Comparing the commutative and noncommutative resolutions for determinantal varieties of skew symmetric and symmetric matrices, submitted. [6] M. Van den Bergh, CohenMacaulayness of modules of covariants, Invent. Math. 106 (1991), 389409. 
14:1515:15
Hai Long Dao (Kansas)  On rigid MCM modules 
Abstract  Let R be a CohenMacaulay local ring. A maximal CohenMacaulay Rmodule M is called rigid if Ext^1(M,M)=0. Rigid MCM modules appear naturally in many contexts, especially divisor class groups and noncommutative resolutions of dimension 3 and higher singularities. However, even their existence remains rather mysterious. In this talk, I will describe recent progress in understanding rigid MCM modules. Some of the materials come from joint work with Ian Shipman. 
References 
[1] H. Dao, Remarks on noncommutative crepant resolutions of complete intersections. Advances in Mathematics, 224, 10211030.
[2] H. Dao, Picard groups of punctured spectra of dimension three local hypersurfaces
are torsionfree. Compositio Mathematica, 148, 145152.
[3] H. Dao, I. Shipman, Representation schemes and rigid maximal Cohen Macaulay modules. Selecta Mathematica, to appear.
[4] O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid CohenMacaulay modules, Invent. Math. 172 (2008) 117168.

15:2516:25
Osamu Iyama (Nagoya)  Higher preprojective algebras and CohenMacaulay representations 
Abstract  It is wellknown that simple singularities R in dimension two and the path
algebras kQ of (extended) Dynkin quivers have similar representation theory
[1,2,3]. This is explained by using preprojective algebras, which contain
both R and kQ as subalgebras. Recently this classical result was generalized by using higher preprojective algebras and cluster tilting modules [4,5]. If time permits, I will explain some of basic results for cluster tilting based on [5]. 
References  [1] M. Auslander, I. Reiten, Almost split sequences for rational double points. Trans. Amer. Math. Soc. 302 (1987), no. 1, 87–97. [2] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory of finitedimensional algebras. Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265–297, Lecture Notes in Math., 1273, Springer, Berlin, 1987. [3] H. Kajiura, K. Saito, A.Takahashi, Matrix factorization and representations of quivers. II. Type ADE case. Adv. Math. 211 (2007), no. 1, 327–362. [4] C. Amiot, O. Iyama, I. Reiten, Stable categories of CohenMacaulay modules and cluster categories, Amer. J. Math. 137 (2015), no. 3, 813–857. [5] L. de Thanhoffer de V\"olcsey and M. Van den Bergh, Explicit models for some stable categories of maximal CohenMacaulay modules, to appear in Mathematical Research Letters, arXiv:1006.2021. [6] O. Iyama, M. Wemyss, Maximal modifications and AuslanderReiten duality for nonisolated singularities, Invent. Math. 197 (2014), no. 3, 521–586. 
15th June 9:3010:30
Izuru Mori (Shizuoka)  McKay correspondence and Beilinson correspondence for ASregular algebras 
Abstract  McKay correspondence claims that the minimal resolution of a quotient singularity is derived equivalent to the preprojective algebra of a McKay quiver, and Beilinson correspondence claims that the projective space is derived equivalent to the path algebra of a Beilinson quiver with commutative relations. In this talk, by integrating these ideas, we will show that a certain combination of McKay correspondence and Beilinson correspondence holds for ASregular algebras, which are noncommutative generalizations of polynomial algebras. 
References 
[1] I. Mori, McKaytype correspondence for ASregular algebras, J. Lond.
Math. Soc. 88 (2013) 97117. 
10:4511:45
Shinnosuke Okawa (Osaka)  On noncommutative Hirzebruch surfaces 
Abstract 
Hirzebruch surfaces form a basic and important class of algebraic surfaces
in commutative algebraic geometry. By definition, they are projective bundles
of fiber dimension 1 over the projective line. It is well known that isomorphism
classes of such surfaces are classified by nonnegative integers d and
that those with the same parity are connected by unobstructed deformations.
Flat deformations of the abelian category of coherent sheaves on Hirzebruch
surface, or noncommutative deformations, have been studied by several people.
Contrary to the case of del Pezzo surfaces, nc eformations of the dth
Hirzebruch surface are obstructed if d > 3. On the other hand, Michel
Van den Bergh introduced the notion of sheaf Zalgebras and proved that
any noncommutative deformation of a Hirzebruch surface over a complete
Noetherian local ring is obtained from a sheaf Zalgebra associated to
a locally sheaf bimodule of rank 2 on the projective line.
In this talk, I will give some introduction to this subject and explain our result on the geometric classification of those locally sheaf bimodules. I will also explain some results from the point of view of derived categories.
The talk will be based on my joint work with Izuru Mori and Kazushi Ueda.

References  [1] Michel Van den Bergh, Noncommutative P1bundles over commutative
schemes, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6279–6313.
[2] Michel Van den Bergh, Noncommutative quadrics, International Mathematics
Research Notices. IMRN (2011), no. 17, 3983–4026.
[3] Tarig Abdelgadir, Shinnosuke Okawa, and Kazushi Ueda, Compact moduli of noncommutative projective planes, arXiv:1411.7770. 
16th June 9:3011:00
11:1512:00
Yusuke Nakajima (Nagoya)  Ulrich modules over quotient surface singularities 
Abstract  An Ulrich module is a maximal CohenMacaulay module which has the
maximum number of generators. In this talk, I will discuss Ulrich modules
over quotient surface singularities. In the process of finding them, the
notion of special CohenMacaulay modules (see [Wun1,Wun2]) plays a crucial
role. Especially, I will observe how to classify Ulrich modules over cyclic
quotient surface singularities by using special ones. This talk is based on a joint work with Kenichi Yoshida [NY]. 
References  [NY] Y.Nakajima and K. Yoshida, Ulrich modules over cyclic quotient
surface singularities, arXiv:1504.07688 [Wun1] J. Wunram, Reflexive modules on cyclic quotient surface singularities, Lecture Notes in Mathematics, SpringerVerlag 1273 (1987), 221231. [Wun2] J. Wunram, Reflexive modules on quotient surface singularities, Math. Ann. 279 (1988), no. 4, 583598. 
13:3014:30
Robin Hartshorne (UC Berkeley) 
Ulrich Bundles on Cubic Hypersurfaces 
Abstract  I will review definitions, properties and existence questions for ACM bundles and Ulrich bundles on projective varieties. Then I will discuss in some detail the Ulrich bundles on cubic hypersurfaces in P^3 and P^4. 
References  [1] Casanellas, Hartshorne, ACM bundles on cubic surfaces, J Eur. math. Soc. 13, 2004 709731
[2] Casanellas, Hartshorne, Geiss, Schreyer, Stable Ulrich Bundles, , Internat.
J. Math. 23, 2012, no. 8, 50 pp.

14:4515:45
Iku Nakamura (Hokkaido)  Global McKay correspondence for quotient surface singulariteis (based on a joint work with A. Ishii) 
Abstract  This is a report of our joint work in progress. The GHilbert scheme here
is a scheme which parametrizes dimensional Ginvariant subschemes (each
called a Gcluster) of C^2 of length the order of G. Two dimensional McKay
correspondence for quotient singularities (in terms od GHilbert schemes
[ItoNakamura99] and [Ishii02]) is reformulated on the Ghilbert scheme
as a global isomorphism of certain universal coherent sheaf with the sume
of O_E(1) for E all irreducible components of the exceptional sent. The
quiver structure in the structure sheaf of each cluster is discussed. 
References  [ItoNakamura99] Hilbert Schemes and simple singularities, New Trends in Algebraic Geometry, Proc. of EuroConference on Algebraic Geometry, Warwick 1996, ed. by K. Hulek et al., CUP, (1999), 151233. [Ishii02] Ishii, A., On the McKay correspondence for a finite small subgroup of GL(2,C). J. Reine Angew. Math. 549 (2002), 221–233. [IshiiNakamura] in preparation 
16:0017:00
Poster session  Posters will be shown from Wednesday to Friday at the room 110 
Presenters  Alvaro Nolla de Celis (Rey Juan Carlos): Group actions and dimer models, SeungJo Jung (KIAS): On the CrawIshii conjecture, Joseph Karmazyn (Bath): Noncommutative Knörrer periodicity for cyclic quotient surface singularities, Antoine Caradot (Lyon): Inhomogeneous Kleinian singularities and quivers, Tomoaki Shirato (Nagoya): On Frobenius split Abelian fiber spaces over curves, Kenta Sato (Tokyo): Stability of test ideals of divisors with small multiplicity, Chihiro Enomoto & KenIchi Yoshida (Nihon): Cleannes of CohenMacaulay monomial ideal generated by five elements, Wahei Hara (Waseda): Rouquier Dimension and Orlov Spectrum of Singular Varieties, Ben Wormleighton (Berkeley): GHilb for trihedral groups, Genki Ouchi (Tokyo): Automorphisms of positive entropy on some hyperKahler manifolds via derived automorphisms of K3 surfaces. 
*** 18:00 Conference dinner at Kyoto Univ. Coop restaurant ***
17th June 9:3011:00
Stefan Schröer (Heinrich Heine Universität) 
Wild quotient singularities 
Abstract  I give a general introduction to wild quotient singularities, with an emphasis on the actions of Z/pZ on power series ring of characteristic p. I will start with an historical overview, and also touch modular representation theory, and finally describe some joint work with Dino Lorenzini. 
References  [1] M. Artin: Wildly ramified $Z/2$ actions in dimension two. Proc.
Amer. Math. Soc. 52 (1975), 6064. [2] H. Ito, S. Schr\"oer: Wildly Ramified Actions and Surfaces of General Type Arising from ArtinSchreier Curves. In: C. Faber, G. Farkas and R. de Jong (eds.), Geometry and arithmetic, pp. 213241. Eur. Math. Soc., Z\"urich, 2012. [3] H. Ito, S. Schr\"oer: Wild quotient surface singularities whose dual graphs are not starshaped. Asian J. Math. 19 (2015), 951986. [4] D. Lorenzini: Wild quotient singularities of surfaces. Math. Z. 275 (2013), 211–232. [5] D. Lorenzini, S. Schr\"oer: Moderately ramified actions and their rings of invariants in positive characteristics. Work in progress. [6] B. Peskin: Quotientsingularities and wild $p$cyclic actions. J. Algebra 81 (1983), 7299. [7] M. Romagny: Effective models of group schemes. J. Algebraic Geom. 21 (2012), 643682. 
12:3013:30
Shuji Saito (Tokyo I. Tech.)  McKay principle and Hasse princible 
Abstract  Quite a few examples have been observed which show that an arithmetic method can play a significant role for a geometric question. In this talk we present such an example. On the geometric side the McKay principle is formulated for the homotopy type of the dual complex of the exceptional divisors of a resolution of a quotient singularity. To prove it, we use the cohomological Hasse principle, which is a higher dimensional generalization of the globallocal principle for division algebras over number fields, a classical theorem in number theory. This is a joint work with M. Kerz. 
References 
13:4514:45
Takehiko Yasuda (Osaka)  Computational aspects of the wild McKay correspondence 
Abstract  I will talk about interesting computations appearing in the wild
McKay correspondence, a certain generalization of the McKay correspondence
to arbitrary characteristic. This version of the McKay correspondence roughly
says that a stringy invariant of a quotient singularity is equal to a weighted
count of Galois extensions of a local field (power series fields or padic
fields). It is a fun to observe how phenomena for local fields and ones
for singularities correspond one another through the McKay correspondence.
I will present concrete computations realizing such coincidences as:
1. Bhargava's mass formula vs. EllingsrudStromme's generating function of the Hilbert scheme of points
2. Convergence/divergence of weighted counts of Galois extensions vs. the property of singularities' (not) being log terminal
3. Dualities in mass formulas vs. the Poincare duality and equisingularities

References  [1] T. Yasuda, The wild McKay correspondence via motivic integration (Japanese), to appear in Sugaku [2] M.M. Wood and T. Yasuda, Mass formulas for local Galois representations and quotient singularities II: dualities and resolution of singularities, arXiv:1505.07577. [3] T. Yasuda, The wild McKay correspondence and padic measures, arXiv:1412.5260, to appear in Journal of the European Mathematical Society. [4] T. Yasuda, Wilder McKay correspondences, Nagoya Mathematical Journal, Volume 221, Issue 01, pp 111  164 [5] M.M. Wood and T. Yasuda, Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions, Int Math Res Notices, Volume 2015, Issue 23, pp 1259012619. [6] T. Yasuda, Toward motivic integration over wild DeligneMumford stacks, arXiv:1302.2982, to appear in the proceedings of the conference "Higher Dimensional Algebraic Geometry  in honour of Professor Yujiro Kawamata's sixtieth birthday" [7] T. Yasuda, The pcyclic McKay correspondence via motivic integration, Compositio Mathematica, 150, 11251168. 