Related papers: Algebraic topology from geometric viewpoint

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques…

The manuscript is an overview of the motivations and foundations lying behind Voevodsky's ideas of constructing categories similar to the ordinary topological homotopy categories. The objects of these categories are strictly related to…

We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.

We construct a topology on a given algebraically closed field with a distinguished subfield which is also algebraically closed. This topology is finer than Zariski topology and it captures the sets definable in the pair of algebraically…

The paper has a form of a talk on the given topic. It consists of three parts. The first part of the paper contains main notions, the second one is devoted to logical geometry, the third part describes types and isotypeness. The problems…

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very active area with many interdisciplinary…

We give a survey of algorithms for computing topological invariants of semi-algebraic sets with special emphasis on the more recent developments in designing algorithms for computing the Betti numbers of semi-algebraic sets. Aside from…

We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized…

We describe the topology of singular real algebraic curves in a smooth surface. We enumerate and bound in terms of the degree the number of topological types of singular algebraic curves in the real projective plane.

On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic…

An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.

This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. The main feature of our approach is…

In the paper the main attention is paid to conditions on algebras from a given variety which provide coincidence of their algebraic geometries. The main part here play the notions mentioned in the title of the paper.

The book covers basics of noncommutative geometry and its applications in topology, algebraic geometry and number theory. A brief survey of main parts of noncommutative geometry with historical remarks, bibliography and a list of exercises…

Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling…

This paper is a survey of computational issues in algebraic geometry, with particular attention to the theory of Grobner bases and the regularity of an algebraic variety. 1. A geometric introduction to Grobner bases. 2. An algebraic…

This expository paper explores the interaction of group ordering with topological questions, especially in dimensions 2 and 3. Among the topics considered are surfaces, braid groups, 3-manifolds and their structures such as foliations and…

In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.

This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics:…