Volume 11 (2011), Number 4. Abstracts
S. Anan′in and C. Grossi. Coordinate-Free Classic Geometries [PDF]
This paper is devoted to a coordinate-free approach to several classic geometries such as hyperbolic (real, complex, quaternionic), elliptic (spherical, Fubini–Study), and lorentzian (de Sitter, anti de Sitter) ones. These geometries carry a certain simple structure that is in some sense stronger than the riemannian structure. Their basic geometrical objects have linear nature and provide natural compactifications of classic spaces. The usual riemannian concepts are easily derivable from the strong structure and thus gain their coordinate-free form. Many examples illustrate fruitful features of the approach. The framework introduced here has already been shown to be adequate for solving problems concerning particular classic spaces.
Keywords. Classic geometries, hyperbolic geometry, Fubini–Study metric, parallel transport.
2000 Mathematics Subject Classification. 53A20 (53A35, 51M10).
Y. Boubakri, G.-M. Greuel, and T. Markwig. Normal Forms of Hypersurface Singularities in Positive Characteristic [PDF]
The main purpose of this article is to lay the foundations for a classification of isolated hypersurface singularities in positive characteristic. Although our article is in the spirit of Arnol'd who classified real and complex hypersurfaces in the 1970's with respect to right equivalence, several new phenomena occur in positive characteristic. Already the notion of isolated singularity is different for right resp. contact equivalence over fields of characteristic other than zero. The heart of this paper consists of the study of different notions of non-degeneracy and the associated piecewise filtrations induced by the Newton diagram of a power series f. We introduce the conditions AC and AAC, which modify and generalise the conditions A and AA of Arnol'd resp. Wall and which allow the classification with respect to contact equivalence in any characteristic. Using this we deduce normal forms and rather sharp determinacy bounds for f with respect to right and contact equivalence. We apply this to classify hypersurface singularities of low modality in positive characteristic.
Keywords. Hypersurface singularities, finite determinacy, Milnor number, Tjurina number, normal forms, semi-quasihomogeneous, inner Newton non-degenerate.
2000 Mathematics Subject Classification. Primary: 58K50, 14B05, 32S10, 32S25, 58K40.
J. Brundan and C. Stroppel. Highest Weight Categories Arising from Khovanov's Diagram Algebra I: Cellularity [PDF]
This is the first of four articles studying some slight generalisations Hnm of Khovanov's diagram algebra, as well as quasi-hereditary covers Knm of these algebras in the sense of Rouquier, and certain infinite dimensional limiting versions K∞m, K±∞m and K∞∞. In this article we prove that Hnm is a cellular symmetric algebra and that Knm is a cellular quasi-hereditary algebra. In subsequent articles, we relate Hnm, Knm and K∞m to level two blocks of degenerate cyclotomic Hecke algebras, parabolic category O and the general linear supergroup, respectively.
Keywords. Highest weight category, cellular algebra, diagram algebra.
2000 Mathematics Subject Classification. 17B10, 16S37.
D. Kaledin. Derived Mackey Functors [PDF]
For a finite group G, the so-called G-Mackey functors form an abelian category M(G) that has many applications in the study of G-equivariant stable homotopy. One would expect that the derived category D(M(G)) would be similarly important as the “homological” counterpart of the G-equivariant stable homotopy category. It turns out that this is not so—D(M(G)) is pathological in many respects. We propose and study a replacement for D(M(G)), a certain triangulated category DM(G) of “derived Mackey functors” that contains M(G) but is different from D(M(G)). We show that standard features of the G-equivariant stable homotopy category such as the fixed points functors of two types have exact analogs for the category DM(G).
Keywords. Derived, Mackey functor.
2000 Mathematics Subject Classification. 18G99
S. Matveev and V. Turaev. A Semigroup of Theta-Curves in 3-Manifolds [PDF]
We establish an existence and uniqueness theorem for prime decompositions of theta-curves in 3-manifolds.
Keywords. 3-manifold, theta-graph, prime decomposition.
2000 Mathematics Subject Classification. Primary: 57M25; Secondary: 57M15.
R. Minlos. Remark on my Paper “On Point-like Interaction between n Fermions and Another Particle” [PDF]
This note contains a refinement of the estimates from the article mentioned in the title.
Keywords. Symmetric operator, selfadjoint extension, boundedness below, Ter-Martirosian–Skornyakov extension.
2000 Mathematics Subject Classification. 81Q10, 47S30.
Moscow Mathematical Journal