Tonight's DG will take the form of two half-DGs, both tackling very interesting but also very different topics:

• Up first is Rajiv Shah, who will tell us all about the axiom of choice, one of the most interesting and possibly also most controversial axioms of set theory. In its most simple form, the axiom meerly states that if one has any given collection of sets, one can find a function from this collection which simply picks an element from each set. While this seems intuitively obvious and is clearly useful in many different settings, a few mathematicians have argued almost ever since its introduction in 1904 by Zermelo, that the axiom of choice is not a desirable axiom because of certain very unintuitive consequences, especially with regard to non-measurable sets (the most spectacular example of this is perhaps the theorem known as the Banach-Tarski paradox, which states that given a ball in , one can, using only isometries, split it into no more than 6 pieces and rearange those pieces into two balls identical to the first).

• Next up will be Sam Derbyshire with a short talk about the Gauss-Bonnet Theorem, a result in differential geometry which unexpectedly relates two very important properties of a surface: its curvature and its Euler characteristic. As one of the nicest and most important theorems in the field, it has attracted quite a bit of attention and motivated many modern developments of the subject. Despite this, it remains very accessible and can provide a great introduction to understanding the geometry and topology of surfaces.

As usual, the talks will start at 7:30 PM in MS.04 and will be followed by our usual Monday evening pub social.

In differential geometry of surfaces, the Gauss-Bonnet Theorem was one of the first and most important local-to-global Theorems: it provides a surprising relation between a local, geometric invariant (the (Gaussian) curvature) and a global, topological invariant (the Euler characteristic). As such, it was an important foundational theorem for the study of higher dimensional spaces (manifolds), and the development of a generalised Gauss-Bonnet Theorem led to many of the concepts of modern differential geometry (and algebraic topology), in particular, the theory of characteristic classes. As such, some of the first "proofs" of the generalised Gauss-Bonnet theorem depended on then not yet proved theorems such as the Nash embedding theorem (which states than any Riemannian manifold can be isometrically embedded in for some ), allowing us to consider manifolds as embedded manifolds in Euclidean space. This first proof also made use of a similarly striking local-to-global result, relating the Euler characteristic of a manifold with the zeroes of any (continuous, tangent) vector field on it.

In this talk, we will go through the classical version of the Gauss-Bonnet Theorem (for surfaces), and some of the history of the development of the generalised Gauss-Bonnet theorem (also called the Gauss-Bonnet-Chern Theorem for Shiing-Shen Chern that was the first to give a complete proof not depending on unproved results), including the first proof using the Nash embedding theorem and a sketch of the modern theory of characteristic classes and of Chern's proof. Then we relocalise to the pub.

Irrational numbers abound in all of mathematics, as is easy to see from the ubiquity of constants such as and or even Cantor's theorem that "almost all" numbers are in fact irrational. To deal with these numbers, very often, it is useful to be able to approximate them by the much more familiar rational numbers and countless methods to do this have been devised, ever since antiquity and Archimedes' famous approximation of , to do just that. However, in Diophantine Approximation (as opposed to, say, Numerical Analysis), we are not directly concerned with finding methods to approximate given irrationals but rather with studying the theory of how this can be done and, in a way, study the general "number theoreic" relationship between rational and irrational numbers. In this evening's DG, we'll take a look at a few interesting topics in this field and see a few of the striking results that this fascinating theory holds. Then pub.

P.S. Bear in mind that the Revision Group for Analysis III is taking place in MS.04 just before and might last until 8 PM, so we might have to use MS.03/05 or start slightly later instead, but there definetly will be a DG.