Tonight's discussion group will take place as usual but we have no predefined topic so we'll have a free for all Dg: if you've got an interesting theorem or concept you'd like to show others, you'll be able to do so in a mini-Dg. Following this we will have our usual Monday evening pub social.

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.

Group, ring, field... These words are omnipresent in modern mathematics and one would be hard pressed to find an area of mathematics which does not make at least some use of them. Such has been the impact of abstract algebra in mathematics ever since it has been formalised a few centuries ago. As a result, it has possibly been one of the most studied areas of mathematics and this has given rise to a great deal of fascinating results.

Among the most beautiful of these is Wedderburn's theorem. As you may know, a ring is a set with two binary operations, usually denoted and , which are roughly similar to the usual multiplication in the integers for example. In particular, the addition is always commutative (it does not matter in which order we add things) but the multiplication need not be. What Wedderburn's theorem states is that, if we assume that the ring is finite and that every element except has a multiplicative inverse (another element such that their product is ), the ring must be commutative. This provides an unexpected and fascinating link between two seemingly unrelated concepts relating to the ring: it's multiplication and the number of elements in the ring itself. As such, it has become one of the most celebrated theorems in algebra and has been proved in many different ways (Maclagan Wedderburn himself, who, in 1905, was the first to prove it, provided no less than 3 different proofs of it).

So, if you are intrigued by this and want to know more, come to MS.04 tonight (Thursday) at 7:30 PM as we guide you through what is possibly the most elegant proof of this famous theorem, assuming no background in algebra whatsoever, after which we head on to the pub.

Continuing the theme from last Thursday, we take a look at some even more interesting theorems and problems, such as Euler's pentagonal number theorem and Fermat's two squares. As always, we won't assume any prior knowledge about the subject, from Thursday or otherwise. This starts at 7:30 PM in MS.04, after which we take a trip to the pub.

The sum is arguably the least mysterious operation in all of mathematics. If you look back to your first experiences in maths, you would probably find that they involved counting and summing. At the same time, studying simple sums of natural numbers can lead to some of the most difficult unsolved problems we have today: Goldbach's conjecture (whether every even number can be written as a sum of two primes) is a famous example of that.

One particularly interesting area which deals with sums of integers is the study of integer partitions. The most general example, unrestricted partitions, i.e. counting the number of ways we can express a given number as a sum of integers (without any sort of restriction on the sum), looks deceptively simple yet gives rise to a surprisingly deep and complex theory. It has been studied by some of the greatest mathematicians, from Euler to Hardy and Ramanujan, using methods ranging from elementary algebra to complex analysis and yields a huge number of fascinating identities and theorems, even when studied with simple combinatorial arguments.

The more classical number theoretic approach deals with proving if certain numbers can be represented as sums of integers from some given set and is no less full of elegant results. An example of this would be Fermat's two squares theorem, which tells us which positive integers can be repesented as sums of two squares and is widely regarded as one of the most beautiful theorems of number theory.

So, if any of this sounds interesting, don't hesitate to come this Thursday at 7:30 PM to MS.04 for a gentle introduction to the awesome world of additive number theory. Then pub.

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.

The factorial is one of the simplest operations in mathematics and crops up everywhere from combinatorics to analyisis. As a function, it is normally defined over the integers but, as often in mathematics, it is useful to look for ways in which to extend where certain functions or sequences make sense. For the factorial, if we are to be able to define it over the real (and later complex) numbers, this means looking for a more general formula which would remain valid for these while having some useful properties such as differentiability or convexity. In other words, we need to find a meaningful way to "join the dots".

As Euler, about 20 years old at the time, was appointed at the academy of St Petersburg, where he held his first important post as a professor, some of his colleagues there were working on exactly these types of problems, which they called "interpolation of sequences". As an example, Jakob Bernoulli had found an interesting formula which expressed the sequence in terms of which would still work if we replaced the integer by a real number; for example, if , we get the well known . Even so, two sequences, in particular, still bothered them. One was the so called harmonic series (which Euler managed to show was essentially equivalent to the natural logarithm) and the other was none other than the factorial . All this soon piqued Euler's interest and, starting from the clever observation that managed to create the function, effectively solved the problem (and much more!). It turns out, in fact, that the Gamma function is not only the sole sensible way of extending the factorial but gives rise to an incredible amount of fascinating mathematics.

So, if you've ever wondered what would be equal to, join us this Thursday at 7:30 PM in MS.04 as Dave McCormick takes us on a wild ride through analysis to show us what the Gamma function really is and some of its cool properties, after which we interpolate our way to the pub.

Polynomials are some of the most general objects in mathematics and polynomial rings form some of the most useful structures in algebra. Often, when faced with one, being able to reduce it to its most basic components, its irreducible factors, makes things much easier and tells us a lot about the polynomial itself and the concepts it relates to, in cases as varied as the characteristic polynomial of a matrix, a polynomial ring quotient and so on.

Unfortunately, our usual method of factoring polynomials tends to be quite slow (especially with regard to running it on a computer) and, as such, it's important to find better and quicker ways of doing it. A particularly interesting example is that of polynomials with coefficients in . To simplify them, we can multiply through the coefficients to obtain an equivalent polynomial with coefficients in and reducing the coefficients modulo a prime gives a polynomial over the finite field . Factoring polynomials over finite fields is many times more efficient than doing it over or and what's interesting is that the original factors over can be recovered from the factorisation over a finite field, making this a very attractive approach. Join us this Thursday at 7 pm in MS.04 as Richard Howell-Peak takes us through some ways of doing this and explains how the method can be used to factor polynomials into irreducibles (as always, no prior knowledge of any of these concepts is assumed), after which we head on to our regular pub social.

Discussion Groups, the WMS's regular informal talks about awesome maths, are kicking off this Monday with a look at one of the most baffling and most important unsolved problems in the whole of mathematics: the Riemann Hypothesis.

The story starts about 200 years ago when Gauss, then a bored 15 year old, started counting the prime numbers in regular intervals and noticed that despite the fact that they seemed to pop up randomly among the other integers, they grew at a more or less regular rate. His conjecture came to be known as the "Prime Number Theorem" and attracted many mathematicians, to no avail, until Gauss' own student, Bernhardt Riemann, published, in 1859, a paper that provided a brilliant new way of looking at the problem and, in doing so, changed mathematics forever. One phrase of that paper, in particular, mentioned a certain conjecture involving the location of the roots of a certain function. Riemann deemed this to be of some interest but gave up looking for a rigorous proof after some "fleeting futile attempts". This was the Riemann Hypothesis. Fast forward 150 years later, although Riemann's ideas have eventually led to a proof of the Prime Number Theorem and many other important results, we are only marginally closer to solving what has arguably become the holy grail of modern mathematics.

The talk will, as usual, be entirely informal, with emphasis on the interesting ideas rather than the technicalities and hence suitable for anyone from first-years to seasoned number theorists. So, if you've ever wondered what the Riemann Hypothesis was all about, if the primes really are that random, how to win a million dollars or just wanted to learn something interesting, come to MS.04 this Monday at 7 PM, followed by the traditional post-Dg trip to the pub.

Few theorems in the history of mathematics have acquired the almost legendary status that quadratic reciprocity holds in number theory. First conjectured by Euler, it provides a striking relationship between the primes in terms of the solvability of certain quadratic congruences and has become a cornerstone of elementary number theory. The great Carl Friedrich Gauss was the first one to provide a valid proof and liked it so much that he called it his "aureum theorema" (golden theorem) and provided no less than 8 different proofs during his lifetime.