The equations for Abelian Higgs vortices (magnetic flux vortices) on a plane or a more general surface are generally not integrable, but for vortices on a hyperbolic plane of curvature -1/2 they are. This talk will present (almost explicit) vortex solutions on certain compact hyperbolic surfaces. Also to be discussed are two asymptotically solvable problems for vortices: the effective vortex motion on a large surface with small curvature, and the structure of vortex solutions on a small surface where the vortices are about to dissolve (and the equations linearize). These results (obtained with N. Rink and with N. Romao) bring vortex theory closer to classical results on the complex and metric geometry of Riemann surfaces.

Let us say that two conjugacy classes of a group commute if they contain representatives that commute. When G is a finite group with a normal subgroup N such that G/N is cyclic, one can use this definition, together with Hall's Marriage Theorem, to describe the distribution of the conjugacy classes of G across the cosets of N. I will give an overview of this result, and then talk about some more recent work on commuting conjugacy classes in symmetric and general linear groups. This talk is on joint work with John Britnell.

According to AdS/CFT a remarkable correspondence exists between strings in AdS5 x S5 and operators in N=4 SYM. A particularly important case is that of fast-spinning folded closed strings and the so called twist-operators in the gauge theory. This is a remarkable tool for uncovering and checking the detailed structure of the AdS/CFT correspondence and its integrability properties. In this talk I will show how to match the expression of the anomalous dimension of twist operators as computed from the quantum superstring with the result obtained from the Bethe ansatz of SYM. This agreement resolves a long-standing disagreement between gauge and string sides of the AdS/CFT duality and provides a highly nontrivial strong coupling test of SYM integrability.

Some aspects of the modular representation theory of a finite group can be described by a tree. Such trees have been determined for almost all finite simple groups, but some cases remain unknown. Starting from the example of the group SL2(q) I will explain how geometric methods can be used to solve this problem for finite reductive groups.

I will speak about a new research project called 'Evolution as an information dynamic system', which involves collaboration between four universities in the United Kingdom. This is a three year project started this year, 2010, and its aim is to develop new understanding of information dynamics in evolution and biology. In particular, we are going to derive new optimality conditions for some evolutionary operators, such as mutation and recombination. Evolutionary states will be represented by probability measures on the space of genetic sequences, and different operators produce different evolution of the states. We define the optimality conditions for evolution based on the maximisation of utility (or fitness) of information principle. The optimal evolution in this sense achieves the shortest 'information distance', and it can be different from an evolution optimal in another sense, such as the shortest convergence time. We argue that the former achieves a better adaptation of organisms living in a dynam ic environment. I will present several early results related to the optimisation of mutation rate parameter. I will review these results in the light of the classical theories of adaptation (e.g. Fisher's geometric model) and error threshold. Then I will outline some future theoretical and experimental work of the project.

Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this talk, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation and we formalise lumping and its direct link to graph automorphism. Lumping is a powerful technique that exploits graph symmetry and allows to keep the model exact while considerably reducing the number of equations. Finally, for pairwise model two different closures are presented, one well established and one that has been recently proposed. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models work well for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness. Keywords: network, epidemic, Markov chain, moment closure.

Somos sequences are generated by a rational recurrence, which is specified by a quadratic relation between adjacent iterates. Michael Somos noticed that, for some special choices of initial values, such recurrences could unexpectedly produce sequences of integers. Examples of Somos sequences were known somewhat earlier in number theory, from Morgan Ward's elliptic analogues of Fibonacci and Lucas sequences. In algebraic combinatorics, Somos recurrences provide a basic example of the Laurent phenomenon, which is a cornerstone of Fomin and Zelevinsky's theory of cluster algebras. This introductory talk reviews the history of Somos sequences and their connections with these and other areas of mathematics and theoretical physics, including solvable statistical mechanics (the hard hexagon model) and discrete integrable systems (QRT maps and the discrete Hirota equation).

We study the Renyi entropy of the one-dimensional XYZ spin-1/2 chain in the entirety of its phase diagram. The model has several quantum critical lines corresponding to rotated XXZ chains in their paramagnetic phase, and four tri-critical points where these phases join. Two of these points are described by a conformal field theory and close to them the entropy scales as the logarithm of its mass gap. The other two points are not conformal and the entropy has a peculiar singular behavior in their neighbors, characteristic of an essential singularity. At these non-conformal points the model undergoes a discontinuous transition, with a level crossing in the ground state and a quadratic excitation spectrum. We propose the entropy as an efficient tool to determine the discontinuous or continuous nature of a phase transition also in more complicated models.

Helically symmetry is an exact generalisation of two-dimensionality and axisymmetry. The flow down a helical pipe is investigated under the assumption of helical symmetry. The implications for blood flow in the body are discussed. It is shown that the observed torsion of arteries may have fluid dynamical benefits. The blood is then replaced by a liquid metal, and it is found that the same flow can give rise to the spontaneous generation of magnetic field, known as a dynamo. Animations, but no experiments will be shown, in the absence of a volunteer for this surgical procedure.

Given an r-dimensional family of degree d plane curves, it is a classical (Victorian) question how many there are with r nodes. I will attempt to explain what this means, what form Goettsche and others conjectured for the answer (for curves on arbitrary complex surfaces), and a short proof.

The ABJM model is a superconformal Chern-Simons theory with N = 6 supersymmetry which is believed to be integrable in the planar limit. However, there is a coupling dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling. We compute this function to four loops in perturbation theory by an explicit Feynman diagram calculation for both the ABJM model and the ABJ extension. We then compute the four-loop wrapping correction for a scalar operator in the 20 of SU(4) and find that it agrees with a recent prediction from the ABJM Y-system of Gromov, Kazakov and Vieira. We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.

Evolutionary dynamics models have been mainly studied on homogeneous infinite populations. However, real populations are neither homogeneously mixed nor infinite. We investigate the stochastic evolutionary game dynamics on structured populations represented by graphs. We consider three simple graphs of finite number of vertices: the star, the circle and the complete graph. We present exact formulae for the fixation probability of a single mutant individual introduced into the graph and the speed of the evolutionary process, namely the mean time to absorption (either mutant fixation or extinction) and the mean time to mutant fixation. Through numerical examples we show the significant impact of the structure of the population, the population size and the payoff matrix on the above quantities.

We present some new perspectives on N=1 gauge theories, especially SQCD, D-Brane Quiver Theories and the MSSM, from the stand-point of recent advances in computational and algorithmic algebraic geometry and commutative algebra. We introduce the plethystic program which systematically count gauge invariants and encodes certain hidden symmetries. Moreover, we discuss special structures of the vacuum moduli space, such as that of SQCD being Calabi-Yau.

The Renormalisation Group tells us that, quite generically, quantum and classical many-body systems exhibit interesting scaling behaviour near critical points. Quantum critical points form a subset of these, corresponding to zero-temperature phase transitions. In the last decade, the AdS/CFT correspondence between conformal field theories on one side and string or M-Theory on the other side has developed into a rich and exciting subject. Until recently the applications of this 'duality' were mostly focused on theories relevant to model quantum field theories that are considered interesting from the point of view of high-energy particle physics - the prime example being N=4 SYM in d=4. In this talk I will describe how similar methods can be fruitfully applied to the theory of quantum critical phenomena in 2+1 dimensions. These systems are of practical interest, as it has been proposed that a quantum critical point underlies the strange behaviour of high-TC superconductors falling into the copper-oxide group.

Category theory is used to study structures in various branches of mathematics, and higher-dimensional category theory is being developed to study higher-dimensional versions of those structures. Examples include higher homotopy theory, higher stacks and gerbes, extended topological quantum field theories, concurrency, type theory, and higher-dimensional representation theory. In this talk we will present two general methods for categorifying things, that is, for adding extra dimensions: enrichment and internalisation. We will show how these have been applied to the definition and study of 2-vector spaces, with 2-representation theory in mind. This talk will be introductory. In particular, it should not be necessary to be familiar with any category theory, although it will of course help

We review the recently uncovered connections between three classes of theories: A_r quiver matrix models, d=2 conformal A_r Toda field theories and d=4 N=2 conformal A_r quiver gauge theories

We investigate two examples of models of populations with structure. These are different in character, with the common theme that the structure has an important influence on population outcomes. In the first part we consider a model of kleptoparasitism, the stealing of food from one animal by another. We investigate a model where individuals are allowed to fight in groups of more than two, as often occurs in real populations, but which has not featured in previous theoretical models. We find the equilibrium distribution of the population amongst various behavioural states, conditional upon the strategies played and environmental parameters, and then find evolutionarily stable strategies (ESSs) for the challenging behaviour of the participants. We show that ESSs can only come from a restricted subset of the possible strategies and that there is always at least one ESS. We show that there can be multiple ESSs, and indeed that the number of ESSs is unbounded. Finally we discuss the biological circumstances when particular ESSs occur in terms of key parameters such as the availability of food and the cost of fighting. The second part of the talk concerns the study of evolutionary dynamics on populations with some non-homogeneous structure, a topic in which there is a rapidly growing interest. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. This process is a Markov chain and we prove several mathematical results. For example we prove that for all but a restricted set of graphs, (almost) all states are accessible from the possible initial states. We then consider graphs within this restricted set or with considerable symmetry. To find the fixation probability of a line graph we relate this to a two-dimensional random walk which is not spatially homogeneous. We investigate our solutions numerically and find that for mutants with fitness greater than the resident, the existence of population structure helps the spread of the mutants. Thus it may be that models assuming well-mixed populations consistently underestimate the rate of evolutionary change.

I will start by giving an introduction to polynomial knot invariants, as well as to Khovanov's more recent idea of knot homology. The goal is to find bi- and tri-graded vector spaces whose graded Euler characteristics are classical polynomial knot invariants (like the Jones or HOMFLYPT polynomial). I will then explain how HOMFLYPT homology can be given a transparent construction using some heavy machinery from geometric representation theory. This gives a bridge between link homology and techniques which have been developed for studying the characters of finite groups of Lie type.