Global and local properties of spaces and mappings (project VEGA)
The aim of this project is to contribute to improving the knowledge of global geometric and topological properties of smooth manifolds, fibrations, algebraic varieties and other spaces and mappings, in a direct relation to our previous achievements, published in world-class publications.
By: Pavel Chalmoviansky
Principal investigator: prof. RNDr. Július Korbaš, CSc.
Cooperating investigators: Mgr. Tibor Macko, PhD. (MÚ SAV Bratislava), doc. RNDr. Pavel Chalmovianský, PhD., RNDr. Martin Sleziak, PhD., Mgr. Tomáš Rusin, PhD.
Project funding: Scientific Grant Agency of the Ministry of Education, science, research and sport of the Slovak Republic and the Slovak Academy of Sciences, project VEGA-1/0101/17
Project duration: 2017-2019
For certain specific systems of smooth manifolds (e.g. Grassmann manifolds), appearing as important in various aspects, we hope to find new results on the maximum number of everywhere linearly independent vector fields, Z/2Z-cohomology cup-length, Lyusternik-Shnireľman category and also on the characteristic rank, keeping in mind interplays and applications of these invariants. We wish to study qualitative properties of singularities of algebraic varieties, mostly in dimensions 1 and 2, over an algebraically closed field. We shall explore the effect of specific quadratic transformations (blowup or similar) on the internal structure of singularities. There are several valuation methods in intersection theory, as an important branch of algebraic geometry. We focus on a geometric interpretation of algebraic intersection in the sense of W. Vogel. In addition, we shall deal with questions related to the classification of high-dimensional manifolds in topological surgery theory or in the algebraic theory of surgery (invented by Andrew Ranicki). We shall also be interested in the structure set of a space as one of the main objects of study in surgery theory.