61 research outputs found
Tips and tricks for building a good paper: what editors want
No full textESSKA is constantly committed to promoting the improvement of scientific quality through the publication of books and the organization of dedicated conferences. In line with this commitment, this interview paper was crated with the aim of being useful for all the young scientists and orthopaedics keen in musculoskeletal and sport medicine research. Three Editors from the most important journals in our field were invited to participate: Jon Karlsson from Knee Surgery Sport Traumatology and Arthroscopy, Bruce Reider from The American Journal of Sport Medicine and Edward Wojtys from Sports Health. © 2020, The Author(s)
Infinitesimal invariants of variations of Hodge structures and geometry of surfaces of general type
No full textSi X est une variété complexe projective, on peut lui associer des données cohomologiques (comme par exemple ses groupes de cohomologie singulière, sa décomposition de Hodge, etc. . .). Si on arriveà tirer assez d’informations géométriques de ces données pour reconstruire X, on dit que l’on a résolu un problème de type-Torelli. Dans le problème de Torelli infinitésimal, on se donne comme données de départ les informations encodées par la différentielle de l’application des périodes. Ces données sont notées V ISH(X). L’objectif est de retrouver X à partir de V ISH(X). En 1983, R. Donagi montre que si X est une hypersurface lisse générique, V ISH(X) détermine X. Cette thèse montre un résultat semblable où cette fois la variété étudiée est une surface quintique singulière de P3 : la quintique de Togliatti, Σ5. Cette quintique possède le nombre maximal de nœuds (= 31) qu’une telle surface peut admettre (A.Beauville, 1979). On montre que si X est la résolution minimale de Σ5, V ISH(X) détermine les 31 nœuds de Σ5. Par ailleurs, ces nœuds sont dans une configuration spéciale que l’on peut lire dans V ISH(X). Cela détermine Σ5, c’est-à-dire que l’on montre un théorème de type Torelli pour la quintique de TogliattiLet X be a complex projective manifold. One can associate to X its cohomological data (for instance, its singular cohomological groups, its Hodge decomposition, etc...). When it is possible to extract enough geometric information from these data to recover X, one says that a Torelly-type theorem holds. For the infinitesimal Torelli-type problem, one is given the information encoded by the differential of the period map. These data are IV HS(X). The aim is to recover X from IV HS(X). In 1983, R. Donagi shows that if X is a generic smooth hypersurface then IV HS(X) determines X. This thesis shows a similar result for a singular quintic surface in P P 3 : the Togliatti quintic, Σ5. This quintic has 31 nodes. This is the maximal number of ordinary double-points a quintic surface in P3 canhave. We show that if X is the minimal resolution of Σ5, IV HS(X) determines the double points. These double-points are in a specialconfiguration which can be read in IV HS(X). This determines Σ5, i.e. we show a Torelli type theorem for the Togiatti quintic
Nonabelian Jacobian of projective surfaces: geometry and representation theory
No full textThe Jacobian of a smooth projective curve is undoubtedly one of the most remarkable and beautiful objects in algebraic geometry. This work is an attempt to develop an analogous theory for smooth projective surfaces - a theory of the nonabelian Jacobian of smooth projective surfaces. Just like its classical counterpart, our nonabelian Jacobian relates to vector bundles (of rank 2) on a surface as well as its Hilbert scheme of points. But it also comes equipped with the variation of Hodge-like structures, which produces a sheaf of reductive Lie algebras naturally attached to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra structure of the classical Jacobian. This feature naturally relates geometry of surfaces with the representation theory of reductive Lie algebras/groups. This work’s main focus is on providing an in-depth study of various aspects of this relation. It presents a substantial body of evidence that the sheaf of Lie algebras on the nonabelian Jacobian is an efficient tool for using the representation theory to systematically address various algebro-geometric problems. It also shows how to construct new invariants of representation theoretic origin on smooth projective surfaces
Configurations of points and strings
No full textLet X be a smooth projective variety of dimension n ≥ 2 . It is shown that a finite configuration of points on X subject to certain geometric conditions possesses rich inner structure. On the mathematical level this inner structure is a variation of Hodge-like structure. As a consequence one can attach to such point configurations:
Lie algebras and their representations;
a Fano toric variety whose hyperplane sections are Calabi–Yau varieties.
These features imply that the points cease to be zero-dimensional objects and acquire dynamics of linear operators “propagating” along the paths of a particular trivalent graph. Furthermore, following particular linear operators along the “shortest” paths of the graph, one creates, for every point of the configuration, a distinguished hyperplane section of the Fano variety in (ii), i.e. the points “open up” to become Calabi–Yau varieties. Thus one is led to a picture which is very suggestive of quantum gravity according to string theory
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