Polynomials are one of the most fundamental algebraic objects in mathematics, yet they often appear to have many surprising properties - an example is a polynomial in many variables whose set of positive values is precisely the set of primes. Galois theory restricts its attention to polynomials of one variable, which are much better behaved.

As such, Galois theory first arose through the study of polynomial equations of small degree - after the familiar quadratic formula (known to the Babylonians), Tartaglia and Cardano were the first to find the cubic formula for the roots of a cubic polynomial in 1535, and Cardano's student, Ferrari, found the quartic equation in 1545. But a quintic equation remained elusive despite many mathematicians' attempts at trying to find it - indeed, in 1799 Ruffini sketched a proof of the non-existence of such a formula, which was fleshed out in 1824 by Abel, and such a discovery was one of the main motivations for Galois theory.
To prove this theorem, Ruffini extensively studied permutations of roots of polynomials, which initiated the study of Galois theory a few decades later when Galois considered groups as a generalisation of permutations, hence allowing statements about polynomials and fields to be translated into algebraic statements.

Join us at 7:30 in MS.04 as Sam Derbyshire explains why the quintic isn't soluble by radicals, why the 65 537-gon is constructible by ruler and compass and more. After which we construct our way to the pub.

At the meeting point between Analysis and Algebra lies the study of Hilbert Spaces, possibly one of the most fruitful theories of the past few centuries. Its applications to Physics and the theory of PDEs (for example) as well as its intrinsic mathematical interest have made it a central area in analysis.

The basic idea is a simple one: instead of studying functions on their own as with classical analysis, we instead direct our attention to spaces of functions. The easiest example of these are vector spaces of functions, but this concept proves too limiting to do any interesting sort of analysis with: it's hard to get anywhere in analysis without the concept of a distance or norm. As a result, we study vector spaces with what's known as an inner product (something very much like the dot product of ) which in turn grants us a useful way to measure "sizes of vectors" in our space by defining a norm. One last thing which we would like to have is that the space is complete with regard to this norm, which roughly means that all sequences which should converge in the space (Cauchy sequences, to be exact) do indeed converge. This is how we arrive at the definition of a Hilbert space: a complete inner product space.

With this in mind, we need to find interesting spaces of functions to study. The most natural way to proceed is to see how we can turn the vector space of continuous functions on some interval into a Hilbert space and it turns out that, using integration, we can define the most important of these, the space. We can then come to the crucial consequence of this study: an extremely useful basis of this space which leads us to the theory of Fourier series, central in all of analysis.

So, if you're interested to see how all this works and more, come to MS.04 tonight (Monday 17th of November) at 7:30 as Dave McCormick sheds some light on the inner workings of Hilbert spaces, after which we head on to the pub.