Abstract. We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher Auslander-Reiten theory on them. Auslander-Reiten theory, especially the concept of almost split sequences and their existence theorem, is fundamental to study categories which appear in representation theory, for example, modules over artin algebras [ARS][GR][Ri], their functorially finite subcategories [AS][S], their derived categories [H], Cohen-Macaulay modules over Cohen-Macaulay rings [Y], lattices over orders [A2,3][RS], and coherent sheaves on projective curves [AR2][GL]. In these Auslander-Reiten theory, the number ‘2 ’ is quite symbolic. For one thing, almost split sequences give minimal projective resolutions of simple objects of projective dimension ‘2 ’ in functor categories. For another, Cohen-Macaulay rings and orders of Krull-dimension ‘2 ’ have fundamental sequences and provide us one of the most beautiful situation in representation theory [A4][E][RV][Y], which is closely related to McKay’s observation on simple singularities [Ma]. In this sense, usual Auslander-Reiten theory should be ‘2-dimensional ’ theory, and it would have natural importance to search a domain of higher Auslander-Reiten theory from the viewpoint of representation

Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related

Abstract. This is an introduction to some aspects of Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents

this paper was partially supported by the National Science Foundation.

this paper we prove the other direction. Specifically, we prove the following theorem.

Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

Abstract. For a finite abelian group G ⊂ GL(n, k), let Yθ be the coherent component of the moduli space of θ-stable representations of the McKay quiver. We calculate the G-equivariant k[x1,...,xn]-module parameterized by each point of Yθ via Gröbner bases. In the case Mθ ∼ = G-Hilb, we show that G-Hilb may be reducible and its coherent component Yθ ∼ = Hilb G may be nonnormal, giving examples for G in GL(3, k) and GL(6, k) respectively. The latter answers a question of Nakamura. 1.

Abstract. We prove (the excellent case of) Schreyer’s conjecture that a local ring with countable CM type has at most a one-dimensional singular locus. Furthermore we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type. Let (R,m) be a (commutative Noetherian) local ring of dimension d. Recall that a nonzero R-module M is called maximal Cohen–Macaulay (MCM) provided it is finitely generated and there exists an M-regular sequence {x1,...,xd} in the maximal ideal m. We say that R itself is Cohen– Macaulay (CM) if it is MCM as a module over itself. The CM local rings of finite CM-representation type (meaning that they have only finitely many nonisomorphic indecomposable MCM modules) have been carefully studied over the last twenty years. The complete equicharacteristic hypersurfaces of finite CM type have been completely classified ([6], [4], [9]), as have the complete equicharacteristic 2-dimensional normal domains ([2]). More generally, it is known that a CM local ring of finite CM type has at most an isolated singularity (proved by Auslander [1] in the complete case, Leuschke-Wiegand [10] in the excellent case, and Huneke-Leuschke [8] in general). Yoshino’s monograph [15] is a comprehensive source for information about rings of finite CM type. The related property of countable CM type has received much less attention. Buchweitz, Greuel, and Schreyer [4] classified the complete hypersurface singularities of countable CM type, but very little more has been learned since then. The open questions and conjectures in Schreyer’s 1987 survey article [12] have inspired work on both finite and countable CM-representation type. For example, Conjecture 7.3(a) states that a CM local ring R has finite CM type if and only if the m-adic completion has finite CM type; this was recently proved in case R is excellent in [10]. This paper is concerned with another of Schreyer’s conjectures:

We present a method to compute the full non–linear deformations of matrix factorizations for ADE minimal models. This method is based on the calculation of higher products in the cohomology, called Massey products. The algorithm yields a polynomial ring whose vanishing relations encode the obstructions of the deformations of the D–branes characterized by these matrix factorizations. This coincides with the critical locus of the effective superpotential which can be computed by integrating these relations. Our results for the effective superpotential are in agreement with those obtained from solving the A–infinity relations. We point out a relation to the superpotentials of Kazama–Suzuki models. We will illustrate

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal Cohen-Macaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen-Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands. Keywords: Krull-Schmidt, maximal Cohen-Macaulay, finite Cohen-Macaulay type 1