For centuries, imaginary numbers were viewed with suspicion. But once mathematicians stopped philosophising and asking "what are complex numbers?" and started asking "what can we do with complex numbers?", the field of complex analysis swelled from absolutely nothing to a huge and rich subject in just thirty years. This Monday, join us at 7:30pm in MS.04 as Dave McCormick takes us on a tour of geometry of the complex plane as you've never known it before:

• When we generalise differentiation and integration to functions of the complex plane, we discover the remarkable theorem that a function which is once differentiable on the complex plane is automatically infinitely differentiable, in complete contrast to the real line.

• There are natural connections between complex-differentiability and planar geometry; in particular, a map from the complex plane to itself is conformal -- meaning it preserves local angles -- if and only if it is complex-differentiable and its derivative never vanishes.

• Unfortunately, when we try and generalise conformality to higher-dimensional Euclidean space, it becomes an over-determined system of equations and degenerates to become rather boring and useless. We thus introduce the natural generalisation of quasiconformal maps, and explore some of their properties.

Don't forget, Monday at 7:30pm for an exciting tour of the geometry of conformal and quasiconformal mappings. After which we go to the pub.

Elliptic Curves have motivated a large part of the mathematics of the last century: in particular, they have allowed for the famous proof of Fermat's Last Theorem. They still are an area of active research, in pure mathematics (for example, motivated by the Birch and Swinnerton-Dyer conjecture, one of the seven Clay Math problems) but also in applied mathematics in relation to many applications in cryptography.

Over the complex numbers, elliptic curves have a particularly simply description: an elliptic curve is just a complex torus. But this conceals a lot of their inner structure! Trying to classify them up to isomorphism leads into many interesting problems, such as the idea of moduli spaces, the mysterious j-invariant and doubly periodic functions on the complex plane.

These elliptic curves come with a group structure (either coming from their description as a torus or by considering a chord-tangent group law, the two being related by the Weierstrass function), and we can thus consider their endomorphisms (the group homomorphisms from the elliptic curve to itself). Every elliptic curve over the complex numbers comes with a multiplication by endomorphism for every integer , but for most of them, that's the end of the story. Special elliptic curves, with more endomorphisms, are said to have complex multiplication.

The study of elliptic curves with complex multiplication relates perhaps surprisingly with class field theory and the study of quadratic imaginary fields (fields of the form for positive integers ); there is a direct relation between isomorphism classes of elliptic curves with given endomorphisms and corresponding quadratic imaginary fields.

Join us in MS.04 at 7:30 to learn about all this and more! Don't miss it - this is probably the last Discussion Group for a while due to exams starting for many of us. We will of course translate ourselves to the pub afterwards.