The 5th edition of the Mathematica Summer School on Theoretical Physics was held in Italy, in ICTP-Trieste from the 11th to the 16th of March of 2013. Application form available here. The school was be devoted to Conformal Field Theories in higher dimensions and the Conformal Bootstrap program. For a longer description see below

Lecturers:

**Davide Gaiotto**, *Perimeter Institute, Canada*

**Nikolay Gromov**, *Kings College London, UK*

**Gregory Korchemsky**, *CEA Saclay, France*

**Joao Penedones**, *Porto University, Portugal*

**David Simmons-Duffin,** *Institute for Advanced Studies, USA*

**Pedro Vieira**, *Perimeter Institute, Canada*

The event is organized partly as an activity within the GATIS network

One physical motivation for studying the four-dimensional conformal theories is to learn more about Gauge theories such as QCD. Four-dimensional CFTs also often pop up in various scenarios of physics beyond the Standard Model, meant to solve puzzles related to flavor physics, the hierarchy problem, etc. Three-dimensional CFTs describe the critical behavior of condensed matter systems and holographically define quantum gravity in four dimensions.

In recent years, great progress has been achieved towards the exact solution of some three- and four-dimensional CFTs, thanks to the AdS/CFT correspondence and to the application of integrability techniques. The best known examples are N=4 SYM in d=4 and the ABJM theory in d=3, for which the exact spectrum is largely understood. A complete solution of these theories will be a major breakthrough in theoretical physics. Given the spectrum, it is very natural to ask what are the constraints of crossing symmetry on higher-point functions. Are there hidden symmetries such as the recently uncovered permutation symmetry, awaiting to be unveiled? Can Integrability help computing higher point functions?

The question can also be asked holographically, and we intend to discuss higher-point correlation functions in AdS/CFT. In particular, representations via Mellin-type integrals recently led to efficient recursion relation techniques for computing Witten diagrams.

An independent, but clearly related, line of research that has had notable recent success is the study of general bounds in CFTs that follow from crossing symmetry and unitarity. This method can also be applied to important models which are not known to be integrable, such as the 3D Ising model. Finally, we would like to see if recent insights about RG flows connecting UV and IR conformal fixed points, like the proof of the a-theorem in 4-dimensions and the F-conjecture in 3-dimensions, can be put to use in the context of the bootstrap. A possible connection with the ongoing studies of the entanglement entropy is also interesting to explore.