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.
