Non compact euclidean cone 3-manifolds with cone angles less than 2���

We study of noncompact Euclidean cone manifolds with cone angles less than c&2�� and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corol lary we classify those with cone angles & 2��/3 and those with cone angles = 2��/3.

Parametric analysis of different compact tension specimens for fracture toughness characterisation in woven composite materials

Estudi elaborat a partir d���una estada a l��� Imperial College London, entre juliol i novembre de 2006. En aquest treball s���ha investigat la geometria m��s apropiada per a la caracteritzaci�� de la tenacitat a fractura intralaminar de materials comp��sits laminats amb teixit. L���objectiu ��s assegurar la propagaci�� de l���esquerda sense que la proveta falli abans per cap altre mecanisme de dany per tal de permetre la caracteritzaci�� experimental de la tenacitat a fractura intralaminar de materials comp��sits laminats amb teixit. Amb aquesta fi, s���ha dut a terme l���an��lisi param��trica de diferents tipus de provetes mitjan��ant el m��tode dels elements finits (FE) combinat amb la virtual crack closure technique (VCCT). Les geometries de les provetes analitzades corresponen a la proveta de l���assaig compact tension (CT) i diferents variacions com la extended compact tension (ECT), la proveta widened compact tension (WCT), tapered compact tension (TCT) i doubly-tapered compact tension (2TCT). Com a resultat d���aquestes an��lisis s���han derivat diferents conclusions per obtenir la geometria de proveta m��s apropiada per a la caracteritzaci�� de la tenacitat a fractura intralaminar de materials comp��sits laminats amb teixit. A m��s, tamb�� s���han dut a terme una s��rie d���assaigs experimentals per tal de validar els resultats de les an��lisis param��triques. La concordan��a trobada entre els resultats num��rics i experimentals ��s bona tot i la pres��ncia d���efectes no previstos durant els assaigs experimentals.

A curious example of two model categories and some associated differential graded algebras

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Homotopy limits for 2-categories

We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.

Model structures on the category of small double categories

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofi brant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves and categorifications.

Derived categories and Grothendieck duality

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Cluster algebras, quiver representations and triangulated categories

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.

On the homotopy theory of spectral categories

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A Thomason model structure on the category of small n-fold categories

We construct a cofibrantly generated Thomason model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.

Persistent bundles over a two dimensional compact set

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Finiteness obstructions and Euler characteristics of categories

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Monads in double categories

We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and dene what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.

Euler characteristics of categories and homotopy colimits

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Spectrum of some trianguled categories

In this paper, we compute the triangular spectrum (as de fined by P. Balmer) of two classes of tensor triangulated categories which are quite common in algebraic geometry. One of them is the derived category of G-equivariant sheaves on a smooth scheme X, for a fi nite group G. The other class is the derived category of split superschemes.