Let $\mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional semigroups. A semigroup $X$ is called C$-$closed$ if closed in each $Y\in \mathcal containing as discrete subsemigroup; $projectively$ for congruence $\approx$ on the quotient $X/_\approx$ C$-closed. $chain$-$finite$ any infinite set $I\subseteq X$ there are elements $x,y\in I$ such that $xy\noti...

Zariski groups are @0-stable groups with an axiomatically given Zariski topology and thus abstract generalizations of algebraic groups. A large part of algebraic geometry can be developed for Zariski groups. As a main result, any simple smooth Zariski group interprets an algebraically closed eld, hence is almost an algebraic group over an algebraically closed eld.

Let G be a reductive algebraic group over an algebraically closed field k. An algebraic characteristic class of degree i for principal G-bundles on schemes is a function c assigning to each principal G-bundle E → X an element c(E) in the Chow group AX, natural with respect to pullbacks. These classes are analogous to topological characteristic classes (which take values in cohomology), and two ...

Triangulated categories were introduced independently in algebraic geometry by Verdier [7, 8], based on ideas of Grothendieck, and in algebraic topology by Puppe [6]. These constructions have since played a crucial role in representation theory, algebraic geometry, commutative algebra, algebraic topology and other areas of mathematics (and even theoretical physics). Recently, Geiss, Keller and ...

Algebra Situs is a branch of mathematics which has its roots in Jones’ construction of his polynomial invariant of links and Drinfeld’s work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and sta...

Let ~ C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category ~ C is called ~ C-closed if for each morphism Φ ⊂ X×Y in the category ~ C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are ~...

In this talk, an algorithm is proposed to determine the topology of an implicit real algebraic surface in R. The algorithm consists of four steps: surface projection, projection curve topology determination, surface patch composition, and combination of surface patches. The algorithm gives an intrinsic representation for the topology of the surface. Some examples are used to show that the algor...

Within Martin-Lőf type theory ([4]), G. Sambin initiated the intuitionistic formal topology which includes Scott algebraic domain theory as a special case (unary formal topology)([7]). In [6], he introduced the notions of (algebraic) information base and translation, and proved the equivalence between the category of (algebraic) information bases and the category of (algebraic) Scott domains. I...