Introduction: This article will present just one example of a general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution. It was the motivating example from two lectures on the BGG resolution given at the 19th Czech Winter School on Geometry and Physics held in Srní in January 1999. This article may be seen as a technical example to go with a more elementary introduction which will appear elsewhere [M. Eastwood, Notices Am. Math. Soc. 46, No. 11, 1368-1376 (1999)]. In fact, there...

An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\{,\cdots ,\}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$, i.e., $$\sum _{\sigma \in S_{2n-1}}(sign\sigma )\{\{f_{{\sigma }_1},\cdots ,f_{{\sigma }_n}\},f_{{\sigma }_{n+1}},\cdots ,f_{{\sigma }_{2n-1}}\}=0,$$$S_{2n...}$

Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra ${𝒟}_K^k$ for any positive integer $k$. This is spanned over $K$ by $d_0,...,d_k$, and has comultiplication $\Delta$ and counit $\epsilon$ defined by $\Delta \left(d_i\right)={\sum }_{j=0}^i{d}_j\otimes d_{i-j}$ and $\epsilon \left(d_i\right)={\delta }_{0,i}$ (Kronecker’s delta) for any $i$. This note presents a representation of the coalgebra ${𝒟}_K^k$ by using smooth spaces and a procedure of microlocalization. The author gives an interpretation of this result following the principles of the quantum theory of geometric spaces.

Summary: In this paper the isometries of the dual space were investigated. The dual structural equations of a Killing tensor of order two were found. The general results are applied to the case of the flat space.

This paper gives an exposition of algebraic K-theory, which studies functors $K_n:\text{Rings}\to \text{Abelian}\text{Groups}$, $n$ an integer. Classically $n=0,1$ introduced by Bass in the mid 60’s (based on ideas of Grothendieck and others) and $n=2$ introduced by Milnor [Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton University Press, 1971: Zbl 0237.18005]. These functors are defined and applications to topological K-theory (Swan), number theory, topology and geometry (the Wall finiteness obstruction to a CW-complex being finite,...

A certain family of homogeneous spaces is investigated. Basic invariant operators for each of these structures are presented and some analogies to Levi-Civita connections of Riemannian geometry are pointed out.

Summary: In order to get a mathematical understanding of the BRS-transformation and the Slavnov-Taylor identities, we treat them in a finite dimensional setting. We show that in this setting the BRS-transformation is a vector field on a certain supermanifold. The connection to the BRS-complex will be established. Finally we treat the generating functional and the Slavnov-Taylor identity in this setting.

This is an exposition of a general machinery developed by M. G. Eastwood, T. N. Bailey, C. R. Graham which analyses some real integral transforms using complex methods. The machinery deals with double fibrations $M\subset \Omega \stackrel{\eta }{\to }\leftarrow \tilde{\Omega }@>\tau >>X$$(\Omega$ complex manifold; $M$ totally real, real-analytic submanifold;...

List of ParticipantsOrganizing committee: Vasil Tsanov – Sofia University (Chairman), Harry Aleksiev – High School for Management and Laguages in Zlatograd (Local Organizer), Leon Farhy – Sofia University (Scientific Secretary), Emil Horozov – Sofia University, Ivailo Mladenov – Bulgarian Academy of Sciences, Angel Zhivkov – Sofia University.

A non-holonomic 3-web is defined by two operators $P$ and $B$ such that $P$ is a projector, $B$ is involutory, and they are connected via the relation $PB+BP=B$. The so-called parallelizing connection with respect to which the 3-web distributions are parallel is defined. Some simple properties of such connections are found.

An idea for quantization by means of geometric observables is explained, which is a kind of the sheaf theoretical methods. First the formulation of differential geometry by using the structure sheaf is explained. The point of view to get interesting noncommutative observable algebras of geometric fields is introduced. The idea is to deform the algebra $C^{\infty }(M,ℝ)$ by suitable interaction structures. As an example of such structures the Poisson-structure is mentioned and this leads naturally to deformation...

The paper deals with locally connected continua $X$ in the Euclidean plane. Theorem 1 asserts that there exists a simple closed curve in $X$ that separates two given points $x$, $y$ of $X$ if there is a subset $L$ of $X$ (a point or an arc) with this property. In Theorem 2 the two points $x$, $y$ are replaced by two closed and connected disjoint subsets $A$, $B$. Again – under some additional preconditions – the existence of a simple closed curve disconnecting $A$ and $B$ is stated.