Quantum field theories (QFTs) can be (relatively) easily studied in perturbation theory when they are

weakly coupled, but very little is known about them when they are strongly coupled.

Since some of the most interesting systems we have encountered in the last couple of decades are strongly coupled, new methods and approaches are sought after. Some of the available techniques include exact results about the Wilsonian Renormalization Group flow, dual descriptions in terms of a different-looking Quantum Field Theory (which allows to map strong coupling problems to weak coupling problems in a different theory), and dual descriptions in term of (Super)String Theory. Supersymmetry and Conformal Symmetry have played a pivotal role in these developments. How can we control strongly coupled many-body quantum systems? When is there a useful dual description (in terms of another QFT or a gravitational theory)?

Some of the most interesting Quantum Field Theories are conformal. What makes

Conformal Field Theories (CFTs) interesting is that they describe second order phase transitions, they are the end-points of renormalization group flows, and they also describe quantum gravity in Anti-de Sitter spaces. A better understanding of CFTs would thus shed light on many interesting branches of physics, ranging from statistical physics to quantum gravity. CFTs above two space-time dimensions remain elusive, and the full power of the conformal group is not yet fully appreciated. In which dimensions do non-trivial CFTs exist? What can be said about critical exponents at second-order phase transitions? What can we learn from CFTs about Quantum Gravity? Which CFTs can be connected by Renormalization Group Flows? Is the field theory which describes boiling water at the second order phase transition solvable?

Supersymmetry is a symmetry that maintains Bose-Fermi degeneracy, so the spectrum of bosons and fermions has to be identical. Supersymmetry also imposes severe constraints on the allowed interactions between the various particle species. These models are special, and lead to powerful tools that are applicable even for strongly coupled theories. Traditionally, the power of supersymmetry was mostly limited to studying the vacuum structure (more precisely, the chiral ring). Recently, it was realized that supersymmetry leads to remarkable simplifications also on curved spaces, and it allows to extract information that extends well beyond the chiral ring of the theory. These recent developments can also be used to evaluate the expectation values of non-local observables, such as the entanglement entropy of the vacuum. These observables are of exceeding interest in particle physics and also in condensed matter physics. Which additional quantities can be evaluated non-perturbatively in supersymmetric field theories? What can we learn about general aspects of quantum field theory such as dualities and anomalies? What do we learn about the vacuum of quantum field theory? What is the mathematical interpretation of supersymmetric theories on curved spaces?