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In this talk I will summarise the recent progress in AdS/CFT due to the construction of the new infinite family of Sasaki-Einstein metrics Y^{p,q}, and their dual superconformal gauge theories. I will review some aspects of Sasaki-Einstein geometry and the main features of the Y^{p,q} metrics. I will then discuss the use of toric geometry to obtain a description of the corresponding Y^{p,q} Calabi-Yau singularities. I will explain how the AdS/CFT dual N=1 supersymmetric gauge theories were constructed using the combined information obtained from the metrics and the toric singularities. A crucial check on the consistency of the construction is provided by the field theory technique of a-maximisation. In the last part of the talk I will briefly discuss the recently formulated geometric dual of this, i.e. "Z-minimisation".

Edward Witten is one of the world’s preeminent string theorists. Among his many accomplishments, he is widely known for showing how five different variations of string theory all belong within a single framework. His awards range from a MacArthur \'genius grant\' to the Fields Medal - the highest honour in the world of mathematics. Professor Witten will examine key discoveries made by physicists in the 20th century such as the detection of antimatter. He will then describe how many of today’s leading scientists are working at the high energy frontier of elementary particle accelerators in their quest to uncover the quantum structures of space and time. supersymmetry, Edward Witten, particle physics, quantum mechanics, waves, Rutherford, antimatter, antiparticle, quantum uncertainty principle, accelerator, space-time

A key issue in the context of (compact) extra dimensions is the one of their stability. Any stabilization mechanism is effective only up to some given energy scale; if they can approach this energy, 4$d observers can excite the fluctuations of the internal space, and probe its existence. Stabilization mechanisms introduce fields in the internal space; perturbations of these fields are mixed with perturbations of the metric, so that their study requires a complete GR treatment. After presenting the general framework, I will then discuss some relevant applications. I will present the exact coupling of the radion to codimension one branes, extending the regime of validity of the results in the literature. I will then focus on de Sitter compactifications, showing that the cosmological expansion has typically the effect of destabilizing the internal space. The final part of the talk will be devoted to related work in progress in less conventional areas of brane models: the localization of gravity towards the IR brane (corresponding to a dual description of emerging gravity from the CFT), and the inclusion of ghost fields in the bulk.