Michael Green (Plenary)
String theory and its dualities
This talk will begin with an introductory overview of string theory. This will lead to a discussion of the role of the dualities that provide connections between very different perturbative and nonperturbative aspects of the theory and provide fascinating interconnections between quantum field theory and quantum gravity.
Michael Atiyah (Plenary)
A Panoramic View of Mathematics
Climbing a mountain is strenuous and hazardous, but the view from the top can be spectacular and makes it all worth while. There is a clear analogy with mathematics. In both cases one has to train properly, collect the right tools and gear, practice on the lower slopes, examine maps and do some background reading. After that it is a matter of hard work, patience and skill.
I will look back on my 60 years of mathematics, describing what views I have seen from the heights and what challenges lie ahead for the next generation. There are many more mountain ranges to explore.
Eliezer Rabinovici (Phenomenology and High Energy Physics)
SESAME - A Personal Perspective
I am researcher in theoretical high energy physics. This is not a talk about that subject but rather a talk describing a parallel universe called SESAME. In it a rather unique effort is made to nurture both high quality research and scientific collaboration in the Middle East. SESAME is a light source being constructed in Jordan involving people from the region and beyond it. I will present a personal perspective on how the project came about, where it is now and where it may be heading for.
Netta Cohen (Mathematical Biology)
Understanding the neural control of behaviour: lessons from the nematode C. elegans
Marc Lackenby (Differential Geometry)
The geometry of knots
Thurston's geometrisation conjecture has been a longstanding goal of 3-manifold theory, and Perelman's recent solution to it is a major breakthrough. However, in the case of knot complements, the conjecture was solved by Thurston himself in the late 1970s. As a consequence, the complement of a "generic" knot admits a unique hyperbolic structure. For these knots, one can therefore consider geometric invariants, such as their hyperbolic volume, or the length and location of their shortest geodesics. In my talk, I'll discuss how one can compute this geometric structure in practice, what it is good for, and how it relates to the topology of the knot.
Tim Gowers (Combinatorics and Analysis)
Szemerédi's theorem and additive combinatorics
Szemerédi's theorem asserts that for every positive constant c and every positive integer k there exists N such that every subset of {1,2,...,N} of cardinality c contains an arithmetic progression of length k. This theorem lies right at the centre of the relatively new field of additive combinatorics, and has led to developments in extremal graph theory, ergodic theory, Fourier analysis, the theory of hypergraphs, and analytic number theory. I shall discuss how it can be that the theorem has so many facets, and in the process will give an introduction to additive combinatorics as a whole.
Ruth Gregory (Cosmology)
Braneworld Black Holes
Braneworlds are a fascinating way of hiding extra dimensions by confining ourselves to live on a brane. The Randall-Sundrum in particular has links with string theory via living in anti de Sitter space. I'll describe research into black holes on braneworlds, discussing both observational aspects, as well as the theory. In particular, we will examine a claim that a braneworld black hole would tell us how Hawking radiation back reacts on spacetime, thus solving one of the outstanding problems of quantum gravity - the ultimate fate of an evaporating black hole.
Philip Dawid (Statistics and Probability)
Checking out your probabilities
You are a TV weather forecaster who has to announce, each evening, the probability that it will rain the next day. How should you go about this task, and how can your forecasting performance be judged (and how can it be compared to that of the forecaster on the other channel) in the light of the actual weather? Motivated by such meteorological concerns – but with applications far beyond – statisticians have developed a variety of methods and instruments, such as proper scoring rules and calibration diagrams, to help address them. I will describe some of these techniques, and their mathematical basis and implications. You might fnd these methods useful as you go about your every day business (with or without your umbrella).
John Hinch (Fluids)
A Load of Balls in Newton's Cradle
How many balls fly off the end, and at what velocity? How many balls rebound, and at what velocity? While the governing equations for Newton's cradle are in `A'-level mechanics, the asymptotic analysis for `a load of' is not. In order to gain insights into the flow of granular materials (a pile of sand), it might be helpful to understand inertial multi-particle interactions between grains.
Dan Segal (Algebra)
Profinite groups: algebra and topology
Profinite groups arise in nature as Galois groups of infinite algebraic extensions. But they have an interesting theory in their own right, and this talk is about one aspect of it.
A profinite group is a compact topological group that is built out of finite groups. The properties of the topological group reflect group-theoretic properties of all the finite groups; if we forget the topology we wouldn't expect this to remain true, and it doesn't in general. However, it does in an important special case (where the profinite group is topologically finitely generated). This surprising result (proposed by Serre in the 1970s, proved in 2007) is related to algebraic properties of finite groups, specifically the behaviour of word-values in these groups.
An example of the kind of property that is relevant is the following recent theorem: Given natural numbers $d$ and $q$, there exists $f$ such that in any $d$-generator finite group, every product of $q$th powers is equal to a product of $f$ $q$th powers. The proofs are hard and depend ultimately on the classification of finite simple groups. Interesting questions include: to what extent do results like this hold for other group words? This is equivalent to asking which verbal subgroups are necessarily closed in every finitely generated profinite group.
Samir Siksek (Number Theory)
The Generalized Fermat conjecture
The generalized Fermat conjecture (due to Beukers and Zagier, and independently Darmon and Granville) concerns the equation x^p+y^q=z^r in coprime integers x,y,z. This conjecture has been called the "new holy grail of number theory". We shall survey what is known and explore some recent directions.
Jose Figueroa-O'Farrill (String theory and High Energy Physics)
Supersymmetric M2-branes and ADE
Whatever M-theory ends up being, M2-branes are supposed to play a fundamental rôle. Motivated by the recent progress in the understanding of the three-dimensional conformal field theories which describe their low energy dynamics, I will present some results on the classification of supersymmetric M2-brane geometries. For sufficiently supersymmetric configurations, there is an ADE-like classification.
Yoav Git (Finance)
Who moved my mortgage?
We present a short case study on when Maths is useful and when it is not within finance. We track the evolution of the humble mortgage through the last twenty years to its recent zenith of notoriety. We observe external factors that affected its product development and the contributions Mathematics (and quants yielding it)made in making the product better... or worse. Along the way we demonstrate how to fit a mortgage prepayment model using 23 parameters and a fool-proof method of calling the top of the housing market
Richard Thomas (Algebraic Geometry)
Counting curves in algebraic geometry
This talk will assume no prior knowledge of geometry (just holomorphic functions). One can try to study "complex manifolds" or "algebraic varieties" via invariants that "count the holomorphic curves in them". This talk will be about explaining the notions in inverted commas; there are at least 4 different ways to define the the last one.
Richard Kaye (Foundations)
Nonstandard Mathematics
One of the more welcome by-products of the work on the logical foundations of mathematics in the 1930s, 1940s and 1950s was a number of new mathematical methods for constructions of new number systems and other objects, and reasoning about these objects. In the 1960s Abraham Robinson developed and applied some of these techniques to analysis, introducing a rigorous theory of analysis based on infinitesimals in which the calculation of derivatives, integrals etc. can be carried out rigorously in the same way as Leibniz and Newton used - and the same way as some of us may have been taught prior to attending university and learning about the concept of "limit".
Robinson coined the term "nonstandard analysis" for his methods, and proponents of the theory tend to use the phrase without hyphen to emphasise the technical meaning of the word. (There is nothing non-standard or improper about Robinson's mathematics.) At its simplest level, infinitesimals can be seen as mathematical objects encoding and hiding the additional complexity of the "for all epsilon there is a delta" quantifiers normally (and tediously) employed in analysis when done via limits. Where nonstandard methods really shine is in constructing solutions and examples. Nor does the method only apply in the realm of analysis: nonstandard methods can be applied to construct new infinite objects whenever we have a suitable family of finite objects, and the new nonstandard object (usually infinite) can be understood and described by applying the familiar theory of these finite objects.
In this talk I will illustrate these ideas by describing a nonstandard symmetric group, a group S_n acting on a nonstandard (infinite) set of n letters but behaving in many respects like the finite symmetric groups. There are some surprises, and it turns out that S_n has an infinite family of normal subgroups, in particular a unique maximal one, and by factoring out by this maximal normal subgroup we obtain a topological group with very elegant features that might have been difficult to manufacture by any other process.
Site designed using Blueprint.