In this talk Owen Daniel will introduce the basic notions of continuum percolation, in particular constructing the Boolean (or Gilbert Disc) model. In lattice percolation we declare some edges to be 'open', and then ask what the probability that there is an infinite open 'cluster' is. In continuum percolation we lose the restrictive nature of the lattice, but have to be careful in how to introduce the notion of an open set (not the same as the topological sense). After introducing the central ideas, Owen will summarise the surprising findings of Roy and Tanemura (2002).

We are excited to announce this coming Wednesday we will have renowned mathematician Ronnie Brown coming to talk for the maths society on the
following:

"The importance of group theory in mathematics and science is well known, from quantum physics to crystallography, notions of Galois theory, and intuitions of symmetry. This talk will explain some of the intuitions behind the search for higher dimensional versions, how the ideas came unstuck for decades, but succeeded by a back to basics approach. The talk will include a video and demonstrations of the Dirac string trick, loops on knots, and how we work with some 2-dimensional formulae.

No background knowledge required!'"

The talk will take place in MS.01 starting at 7pm after which we head to the pub.

Do come along!

There are various notions of intersection theory in different areas of mathematics; for instance in algebraic topology it is a key factor in the understanding of Poincaré duality and products in cohomology. The situation there is rather easy to understand: say we want to intersect two curves in the plane; we want to obtain just a finite set of points. This can go wrong in many ways: the plane isn't compact, so we could end up with infinitely many intersections (i.e. the graph of and the -axis). More importantly, we have to deal with more complicated types of intersections: tangency, or curves sharing a common arc. The way to manage these issues in this situation is to move the curves slightly (with a homotopy), so that they intersect transversely (which means that the tangents of the two curves at a point of intersection are not parallel).

But the situation here is rather loose, as we can easily just draw smooth curves without intersection, and we have no way of really understanding what intersection numbers we end up getting. However, if we restrict to algebraic curves (given by polynomial equations, say a parabola or a circle ), then the situation is much more rigid. A basic insight comes from the Fundamental Theorem of Algebra: the intersection of the graph of a degree polynomial with the x-axis should give us exactly -intersection points. The issues we have here are:

• We need to work over the complex numbers, rather than over (because is not algebraically closed, so for instance has no solutions).
• We need to count the intersections with multiplicity (the parabola has a double intersection with the -axis at ).
• We need to take care of issues of parallelism: any two distinct lines intersect in exactly one point, except if they are parallel, in which case the intersection has escaped to infinity.

Once we deal with these issues, we are naturally lead to a vast generalisation of the Fundamental Theorem of Algebra: Bezout's Theorem. Join us on Thursday at 7:30pm as Sam Derbyshire starts exploring the realms of intersection theory in algebraic geometry.

We are excited to announce that we will next Thursday be hosting another wonderful speaker. David Corfield from Kent University and of n-Category Cafe fame will talk to us on the following:

"After four years of mathematics at undergraduate and Masters level, I turned to philosophy. In this talk I shall discuss what I came to understand about mathematics in the process of studying it as a philosopher: that mathematics cannot be pr...operly understood merely in terms of a collection of theorems, but should be seen as governed by a large number of interacting research programmes, each with its understanding of its own history, concepts and outlook. These aspects of mathematics are embodied in the blog that I jointly run - The n-Category Cafe, where we promote informal exposition of ideas, collaborative and critical conversation, driven by the demands of a research programme to take mathematics up the category theoretic ladder."

The talk will take place in L4 in the physics department.

Do come along!

Synthetic differential geometry is an attempt to reformulate differential geometry in a way that infinitesimals are allowed. There are several reasons to attempt such a reformulation; the most important is that even if we do not admit it, we think of constructions like the tangent vector or the vector field as infinitesimal objects. The main object of this talk is to give a naive, yet rigourous, introduction to the notion of microlinearity in synthetic differential geometry. SDG will be approached as a theory in itself (within the framework of intuisionistic logic), without any prerequisites in logic, category theory or classical differential geometry. If any of this intrigues you then come along to hear more from Stephanos Papanikolopoulos on Thursday evening. After which we will head to the pub.

Galois representations are ubiquitous in modern number theory and they provide a powerful tool. Quoting Barry Mazur, Galois representations offer a bridge between the two worlds of arithmetic geometry, such as that of elliptic curves and the analytic theory of that of modular forms. They also had an important part in the proof of Fermat's Last theorem. In this talk we will give an introduction to the beautiful world of Galois representations, starting with the basics of Galois theory, introducing the absolute Galois group of the rationals (the beast) and investigate some spaces over which it acts. No prerequisites apart from groups and rings. If any of this sounds exciting then join us in MS.05 as Lambros Mavrides will shed light upon some of this fascinating maths. After which we will head over to the pub.

We are pleased to announce that we will be hosting the second of our two special Fields medal talks. We will hear about the following:

David Loeffler will talk to us about the Langlands program and Ngô Bảo Châu's contribution in proving the Fundamental Lemma.

Florian Theil will talk to us about the importance of the work of Cédric Villani on the Boltzmann equation.

Arguably the first major result within topology is the classification of surfaces. This was well known in the mid 19th century, most likely a by-product of interest in the differential geometry of curves and surfaces. It was not until Poincaré that the result was placed within its correct context. In his seminal paper "Analysis Situs" (arguably the first on topology), he defined higher dimensional analogues of curves and surfaces; so called manifolds along with some methods with which to study them. The question that arose was again that of classification. Putting aside what one may mean by this (and therefore any discussion of "computable", "algorithm" and the like!) we wish to know what approaches have been developed to tackle this rather unwieldy problem.

One approach is that of studying the functions on our space which leads to ideas such as Morse Theory and the study of PDEs on our manifold. A second approach is that of finding suitable algebraic invariants which we can associate to our space which leads to homology and cohomology theory, homotopy groups, K-theory etc. Perhaps historically the first algebraic invariant is the fundamental group. This is something close to the set of loops up to continuous deformation. Poincaré realised the importance the fundamental group in the study of 3-manifolds. He asked seemingly one of the most basic questions in the realm of this program of generalising the above classification; if a "nice" 3-manifold has fundamental group zero, then is it in fact the 3-sphere (up to some suitable equivalence). This led naturally to generalisations to higher dimensions (despite no known proof at the time). Surprisingly this generalisation was proved for all dimensions greater than 4 by Smale in the 1960s (and for n=4 in the 1980s) as a result of some of the ideas in surgery theory.

We will try to discuss some of the background to the problem along with generalisations and ideas related to Smale's solution of this. If any of this sounds interesting come along to MS.05 at 7:30 after which we will head to the pub.

The Fields Medals 2010: Who Got Them, and What Were They For?

This Wednesday we are pleased to announce that we will be hosting the first of our two special Fields medal talks. We contacted the winners, but unfortunately they weren't available to join us (too busy spending their prize money). But have no fear, a few brave and charismatic members of our own department have stepped in to take the role. In particular we will be hos...ting:

Stefan Adams on Smirnov's Conformal Invariance Theorem for percolation, and
Mark Pollicott on Lindenstrauss's applications of ergodic theory to number theory.

Both speakers are experts in these fields, but promise that their talks will be accessible to non-specialists. In particular all undergraduate students are encouraged to come, as this will be an excellent opportunity to hear about some hot topics in modern mathematics.

Afterwards there will be food and drinks, and an opportunity to talk further about the Fields medals, as well as to talk about non maths related things: such as cars, football, and the amount of people who don't like bacon.

Many structures in mathematics are created by putting forward axioms: axioms for a group, for a field, for a vector space, etc. Then we then study objects that satisfy those axioms (also called models of those axioms). Those would be groups, fields, vector spaces, etc.

One can actually study the relation between axiomatic theories and their models, to get many results about what models are possible, what size they might have, etc.

A surprising application is to set theory. One might think that the axioms of set theory uniquely determine what a set is. But that is not so: there are many different, inequivalent models of set theory (assuming that set theory is consistent, i.e. that one cannot derive a contradiction from the axioms!).
This phenomenon was discovered with the solution of the Continuum Hypothesis, whose answer depends on which model of set theory is taken. Here, we say that the Continuum Hypothesis is undecidable: the axioms of set theory alone do not settle the question.

Come along at 7:30 in MS.05 as Sam Derbyshire introduces model theory and its applications to various other areas of mathematics. No prior knowledge is assumed, so students from all years should be able to understand.

As usual, after the talk we will head over to the Duck for some refreshments. Hope to see you there!