TITLE: Building Abelian functions with generalised Hirota operators. Matthew England (Glasgow) ABSTRACT: We consider symmetric generalisations of Hirota's bilinear operator and how such operators can be used to build Abelian functions. The Abelian functions associated to a curve are those functions periodic with respect the period lattice of the curve. They have been the subject of increased study over recent years and have found a range of applications. A key problem for working with such functions is the identification of bases for the relevant vector spaces. We address this problem by defining new infinite classes of Abelian functions using generalised Hirota operators acting on the Kleinian sigma function. These new functions have a prescribed pole structure and omit the Kleinian $\wp$-functions as a subclass. We present some explicit examples of vector space bases built using the new functions, revealing some previously unseen similarities between bases of functions associated to curves of the same genus. Geometric quanitzation, Dirac monopoles and coadjoint orbits Graham Kemp The phase space of a classical Dirac monopole is a magnetic cotangent bundle over S2 and can be identified with a coadjoint orbit of E(3), the Euclidean group of motions, as was shown by Novikov and Smeltzer. If G is a compact Lie group, then a magnetic cotangent bundle over a coadjoint orbit of G can be considered as the phase space of a generalized Dirac monopole. These can be identified with certain special coadjoint orbits of TG - the tangent bundle of G. I will explain how the geometric quantization of these phase spaces and induced representations of G are related. This continues earlier work with A.P. Veselov. The unexpected simplicity of Automorphic Lie Algebras Sara Lombardo Originally motivated by the problem of reduction of Lax pairs, Automorphic Lie Algebras are interesting objects in their own right. A first step towards their classification was presented in [Sara Lombardo and Jan A. Sanders, On the Classification of Automorphic Lie Algebras, 2010, Communication in Mathematical Physics, Volume 299, Number 3, 793-824], where sl_2 automorphic algebras associated to finite groups where considered. These groups are the five groups of Klein¹s classification, namely, the cyclic group, the dihedral group, the tetrahedral group, the octahedral group and the icosahedral group. In this talk I will report on the ongoing research programme (carried out in collaboration with the VU Amsterdam) which aims at a complete classification of Automorphic Lie Algebras associated to finite groups. The results for higher dimensional algebras are strikingly simple. Solitons on tori and soliton crystals Martin Speight There are many numerical studies in the topological solitons literature in which a period lattice \Omega is chosen and energy minimizers on the torus \R^k/\Omega are found. Such minimizers are interpreted as soliton crystals, that is infinite periodic arrays of static solitons in stable equilibrium. However, the important question of whether the field is stable, or even critical, with respect to variations of the period lattice \Omega, is generally ignored in these studies. Let us call a minimizer a "lattice" if its energy is critical with respect to variations of \Omega, and a "crystal" if it is, in addition, stable with respect to such variations. In this talk, I will describe how the question of whether a minimizer is a crystal can be formulated in terms of its stress tensor. The general framework will be applied to the nuclear Skyrme model and its planar analogue, the baby Skyrme model. It will be shown that every baby Skyrmion lattice is a crystal, and that the Skyrme "crystal" of Castillejo, Kugler at al is, likewise, a crystal in our stronger sense. Anne Taormina K3 elliptic genus: glimpse of a new Moonshine Abstract: I will review how ideas stemming from string theory recently led to a remarkable and still unexplained connection between the elliptic genus of a K3 surface and the sporadic group M24 (Mathieu 24). It has sparked the interest of Number Theorists, Algebraic Geometers, Group Theorists and Mathematical Physicists worldwide over the last year or so, and its explanation may require novel ideas. M24 has been linked to the classical geometry of K3 surfaces by Mukai, who classified the finite groups of symplectic automorphisms of all K3 surfaces in 1988 and showed that they are all subgroups of M23, each having at least 5 orbits on a set S of 24 elements. M23 is the stabiliser in M24 of an element in the set S, when viewing M24 as the group of permutations of 24 elements preserving the extended binary Golay code G24. Using lattice techniques, I will highlight the existence of an `overarching symmetry' in this context. Namely I will indicate how a group, itself a maximal subgroup of M23 containing all the polarization preserving symplectic automorphism groups of Kummer K3 surfaces with induced polarization, has an action on the Niemeier lattice of type 24A_1induced by the action of two individual polarization preserving symplectic automorphism groups on that same Niemeier lattice. The discovery of this overarching symmetry may provide a first step in understanding the connection between the K3 elliptic genus and M24. Bethe Ansatz and nonlinear wave equations Roberto Tateo (Torino) Abstract: We discuss the recently discovered off-critical variant of the correspondence between the spectral theory of ordinary differential equations and the Bethe Ansatz for relativistic integrable quantum field theories. Quantum Affine Algebras and Extended T-systems Charles Young Abstract: I will discuss some aspects of the representation theory of quantum affine algebras. In particular, I shall describe a system of short exact sequences of tensor products of representations, which contains the T-system as a special case. If time allows, I will also talk about self-extensions of representations, and quantum-affine analogs of Verma modules. This is based on joint work in http://arxiv.org/abs/1112.6376 and http://arxiv.org/abs/1104.3094. Universality in Lie algebras and Chern-Simons theory A.P. Veselov (Loughborough) Abstract. I will show that the eigenvalues of certain natural Casimir operators in the adjoint representation of simple Lie algebras can be expressed rationally in the universal Vogel's parameters and give explicit formulae for the corresponding generating functions. Similar universal formulae can be given for some quantities in Chern-Simons theory on a 3D sphere. The talk is based on joint results with Mkrtchyan and Sergeev. TITLE: Universal properties of two-dimensional percolation Jacopo Viti ABSTRACT: Percolation is the best known example of geometrical phase transition. In two dimensions a plethora of exact results were derived at criticality (critical exponents, crossing probabilities) using Conformal Field Theory. In this talk I will discuss a possible field theoretical description for the scaling region of off-critical percolation, exploiting the $q\rightarrow 1$ limit of the integrable q-color Potts field theory. I will apply the formalism to the computation of universal amplitude ratios and off-critical crossing probabilities. All the predictions can be checked by numerical simulations.