We give a division algebra R,C,H,O description of D = 3 Yang-Mills with N = 1,2,4,8 and hence, by tensoring left and right multiplets, a magic square RR, CR, CC, HR, HC, HH, OR, OC, OH, OO description of D = 3 supergravity with N = 2, 3, 4, 5, 6, 8, 9, 10, 12, 16.

This talk will have three parts 1. Past story - classical projective planes 2. Unfinished story - topology of the projective plane 3. Fairy story - physics and the projective plane

By generalizing the recent proof of the a-theorem, I derive constraints on the possible UV and IR asymptotics of 4D Lorentz invariant unitary quantum field theory. Within perturbation theory the only possible RG flow asymptotic is given by conformal field theory. I also give a non-perturbative argument that excludes theories with scale but not conformal invariance. This argument holds for theories in which the stress-energy tensor is sufficiently nontrivial in a definite technical sense.

I will describe how this number comes from quantum fluctuations during the earliest moments of the Big Bang to become the large scale structure of cosmology that we see today in the skies.

We construct six-dimensional (1,0) superconformal models with non-abelian gauge couplings for multiple tensor and hypermultiplets. The non-abelian gauge invariance for the two-form potentials is achieved by the introduction of a hierarchy of forms up to degree four, where the three and four forms are non-dynamical. This circumvents certain no-go theorems and turns out to be essential also for the formulation of supersymmetry. The inclusion of the hypermultiplets completes the field content to that of superconformal (2,0) theories of multiple M5 branes, though only (1,0) supersymmetry is manifestly realized. The interacting tensor and hypermultiplets form together with the other (non-dynamical) forms a new on-shell representation of the (1,0) supersymmetry algebra. In general these models provide only equations of motions. For a subclass also a Lagrangian formulation exists which captures certain aspects of the first order dynamics.

This is the first Polygon meeting, which is meant to be complementary to Triangles, with local speakers.

I will present a series of new results on classical and quantum properties of maximal supergravity in 4 dimensions, emphasizing the role of the recently discovered "new SO(8) theories". I will explain how this changes our understanding of some general properties of supergravity theories and comment on the impact of these results on the gauge/gravity correspondence.

We discuss various recent computations that come to bear on the conjectured holographic duality between Vasiliev's theory of 3d higher spin gravity and families of 2d CFTs with W symmetry, at both zero and finite temperature. These rely on novel, purely algebraic methods for computing bulk-boundary propagators for scalar fields in Vasiliev theory. Among other topics to be discussed, our results further clarify the physical content of the term "higher spin black hole" and the nature of the classical limit required for the duality.

Localization is a non-perturbative method to exactly compute path integrals in supersymmetric theories. I will describe how localization can be thus used to test holographic duality in a non-conformal setting, namely for N=2* theory, which is a massive deformation of N=4 super Yang-Mills, and for which the dual supergravity background is known explicitly.

We show that for every asymptotically AdS solution compactified on a torus there is a corresponding Ricci-flat solution obtained by replacing the torus by a sphere, performing a Weyl rescaling of the metric and appropriately analytically continuing the dimension of the torus/sphere (as in generalized dimensional reduction). This correspondence should allow us to develop a holographic dictionary for Ricci-flat spacetimes. In particular, it maps Minkowski spacetime to AdS on a torus, the holographic stress energy tensor of AdS to the stress energy tensor due to a brane localized in the interior of spacetime and AdS black branes to (asymptotically flat) Schwarzschild black branes. Applying it to the known solutions describing the hydrodynamic regime in AdS/CFT, we compute the dispersion relation of the Gregory-Laflamme unstable modes through cubic order in the wavenumber, finding remarkable agreement with numerical data. We further obtain the fluid dual to Rindler spacetime and show that its transport coefficients through second order follow from the AdS ones.

I will summarise the Higgs search results and searches for supersymmetric particles from the LHC, commenting on the recent evidence found for rare decays of the Bs meson into mu+ mu-. I shall comment on what the Higgs means for the discovery prospects for supersymmetry.

I will summarise why SUSY may be a good idea phenomenologically, and what theoretical and phenomenological problems it can solve for us. I'll argue that there's more to life than the CMSSM, and discuss what we might learn from the continuing absence of BSM signals at the LHC.

I will discuss the ongoing effort to constrain the four-point function of conserved spin 1 and 2 currents tensors in a general conformal field theories in d>3 by applying the full set of the corresponding Ward identities.

In compactifications of M-Theory, certain singularities are expected to give rise to non-abelian gauge theories. In case the compactification manifold admits an torus fibration, such singularities become degenerations of the fibre. In this framework (which is called F-Theory), the behaviour of the fibre is well-understood for elliptic surfaces, but there remain puzzles both from the perspective of mathematics and physics in higher dimensions. After an introduction, I will report on some recent progress on the interplay of the structure of degenerate fibres and the effective field theory description.

We show how "generalised geometry", a class of extensions of conventional differential geometry first introduced by Hitchin, gives a natural way of formulating supergravity theories. The formulation unifies the bosonic fields and symmetries and has a natural action of O(d,d) or the exceptional groups E_d in d-dimensions. By introducing the analogue of the Levi-Civita connection we find that full set of bosonic equations of motion reduce to simply the vanishing of the generalised Ricci tensor. We show that the connection also encodes the supersymmetry variations and fermionic equations of motion. This formalism also gives natural extensions of complex, symplectic and other integrable structures, with implications for describing supersymmetric string theory backgrounds.