We analyze the all-loop Bethe ansatz for the sl(2) twist operator sector of the N=4 gauge theory in the limit of large spacetime spin at large but finite twist, and find a novel all-loop scaling function. This function obeys the Kotikov-Lipatov transcendentality principle and does not depend on the twist. We discuss possible phase factors for the S-matrix, leading to modifications at four-loop order and beyond. While these result in a four-loop breakdown of perturbative BMN-scaling, transcendentality may be preserved in the universal scaling function. One particularly natural dressing phase, unique up to one constant, modifies the overall contribution of all terms in the scaling function that contain odd zeta functions. Excitingly, we present evidence that this choice is non-perturbatively related to a recently conjectured crossing-symmetric phase factor for perturbative string theory on AdS(5)xS(5) once the constant is fixed to a particular value. Our proposal, if true, might therefore resolve the long-standing AdS/CFT discrepancies between gauge and string theory.

Standard methods like the Bethe ansatz will typically fail when the target space of an integrable model is non-compact like, for example, in the case of the nonlinear sigma models associated to AdS-spaces. New methods are needed. We will discuss some of them in the simplest possible example, the Sinh-Gordon model.

This is the first of a three lecture series aiming to provide a basic introduction to topological strings. For more details, please visit our Graduate Program webpage at: http://www.strings.ph.qmw.ac.uk/

I discuss some aspects of the conformal field theories obtained by placing D3 branes at conical singularities focusing on the rich interplay between the properties of the gauge theories and the geometry of the internal manifold. I discuss in particular the role of baryons, a character for counting all BPS baryonic operators and its geometrical interpretation.