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We study the cluster combinatorics of d-cluster tilting objects in d-cluster categories. Using mutations of maximal rigid objects in dcluster categories, which are defined in a similar way to mutations for d-cluster tilting objects, we prove the equivalences between dcluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of d + 1 triangles of d-cluster tilting objects in [O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (1) (2008) 117-168], we prove that any almost complete d-cluster tilting object has exactly d + 1 complements, compute the extension groups between these complements, and study the middle terms of these d + 1 triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established for d-cluster categories in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006) 572-618]. They are applied to the Fomin-Reading generalized cluster complexes of finite root systems defined and studied in [S. Fomin, N. Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. 44 (2005) 2709-2757; H. Thomas, Defining an m-cluster category, J. Algebra 318 (2007) 37-46; K. Baur, R. Marsh, A geometric description of m-cluster categories, Trans. Amer. Math. Soc. 360 (2008) 5789-5803; K. Baur, R. Marsh, A geometric description of the m-cluster categories of type D n , preprint, arXiv:math.RT/0610512; see also Int. Math. Res. Not. 2007 (2007), doi:10.1093/imrn/rnm011], and to that of infinite root systems [B. Zhu, Generalized cluster complexes via quiver representations, J. Algebraic Combin. 27 (2008) 25-54].

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We study the cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of $\operatorname{Ext}^1$ as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.Comment: Final version, to appear in Compositio Mathematic

Dedicated to Professor Idun Reiten on the occasion of her seventieth birthday. AbstractWe study extension spaces, cotorsion pairs and their mutations in the cluster category of a marked surface without punctures. Under the one-to-one correspondence between the curves, valued closed curves in the marked surface and the indecomposable objects in the associated cluster category, we prove that the dimension of extension space of two indecomposable objects in the cluster categories equals to the intersection number of the corresponding curves. By using this result, we prove that there are no non-trivial t−structures in the cluster categories when the surface is connected. Based on this result, we give a classification of cotorsion pairs in these categories. Moreover we define the notion of paintings of a marked surface without punctures and their rotations. They are a geometric model of cotorsion pairs and of their mutations respectively.

Because of the unique optical properties of gold nanomaterials, the preparation of gold nanomaterials with excellent chirality has received extensive attention. In order to develop a simple fabrication method for three-dimensional chiral Au nanostructures with a size of several hundred nanometers, chiral gold nanoparticles were developed to transfer chirality of a peptide to gold nanoparticles. In this study, the controlled synthesis of asymmetric gold nanopolyhedrons was achieved. The asymmetric gold nanopolyhedrons prepared via peptide-directed growth can exhibit strong circular dichroism (∼±50 mdeg) couplets in the visible range (500–600 nm). Also, the morphology of chiral Au nanododecahedrons-peptide particles showed distorted and asymmetric properties. In order to prove that the size and spatial structure of gold nanopolyhedrons have an influence on their chiral optical properties, Au nanotrioctahedron-peptide particles were prepared by using Au nanotrioctahedrons with different morphologies. Au nanotrioctahedron-peptide particles also exhibited circular dichromatic couplets in the visible region.

For a d‐Calabi–Yau triangulated category 𝒷 with a d‐cluster tilting subcategory 𝒯, the decomposition of 𝒷 is determined by the decomposition of 𝒯 satisfying ‘vanishing condition’ of negative extension groups, namely, 𝒷=⊕i∈I𝒷i, where 𝒷i, i∈I, are triangulated subcategories, if and only if 𝒯=⊕i∈I𝒯i, where 𝒯i, i∈I, are subcategories with Hom𝒷(𝒯i[t], 𝒯j)=0, ∀ 0⩽t⩽d−2 and i≠j. This induces that for any two cluster tilting objects T and T′ in a 2‐Calabi–Yau triangulated category 𝒷, the Gabriel quiver of the endomorphism algebra End𝒷T is connected if and only if so is that of End𝒷T′. As an application, we prove that the indecomposable 2‐Calabi–Yau triangulated categories with cluster tilting objects have no non‐trivial t‐structures and no non‐trivial co‐t‐structures. This allows us to give a classification of torsion pairs in those triangulated categories, and to determine further the hearts of torsion pairs in the sense of Nakaoka, which are equivalent to the module categories over the endomorphism algebras of the cores of the torsion pairs. We also discuss the relationship between mutations of torsion pairs and mutations of cluster tilting objects.

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