BogomolovMultiplier · Bogomolov Multiplier

Bogomolov Multiplier

This project can roughly be situated inside mathematics in a branch called group theory. This theory is a mathematical foundation of the concept of symmetry. Its beginnings go back to the study of solutions of polynomial equations as envisioned by Galois in 1830s. More precisely, Galois showed how to associate a group to a polynomial, and then use this group to deduce some properties about the roots of the polynomial. This idea was later made more abstract and the concept of an abstract group is present all over modern mathematics. One of the fundamental unresolved problems in the theory of groups asks whether or not every finite abstract group arises from a polynomial whose coefficients are rational numbers. This question therefore asks for an inverse procedure to the construction of Galois. It is known to have a positive solution for polynomials with more complicated coefficients than rational numbers (for example, for rational functions with complex coefficients). One can try to imitate the positive solution for some other coefficients to get an answer to the inverse Galois problem over rational numbers. These ideas led Noether to establish a programme on how to construct polynomials with a given finite abstract group G associated to them. The idea is to associate a certain (regular) representation of the abstract group on a vector space and then quotient by the action of G. One gets an algebraic variety, given by a set of polynomial equations. Noether conjectured that there is a way of solving these polynomial equations in terms of simple rational functions (meaning that the quotient variety would be what is called rational in algebraic geometry), and this would be enough to imply that her construction produces many polynomials with Galois group G. It turned out much later that Noether's problem does not always have a positive solution. Obstructions were developed to show that the polynomial equations arising from her constructions can not be solved as she imagined, and concrete groups G were presented for which these obstructions are non-trivial. This project dealt with understanding possibly the most basic of these obstructions, nowadays called the Bogomolov multiplier and denoted by B_0(G). The aim was to understand various structural aspects of this obstruction in relation to the abstract group G, use this to produce more negative examples, and to explore some extensions of this obstruction. The relevance of this, apart for the original motivation regarding the inverse Galois problem, is that many diverse interpretations of the Bogomolov multiplier are known, making this object a meeting-point for several important areas of mathematics such as geometry, homology, K-theory, representation theory, mathematical physics.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз