"The goal of this project is to achieve breakthroughs in a few fundamental questions in 2D statistical physics, using techniques from complex analysis, probability, dynamical systems, geometric measure theory and theoretical physics.Over the last decade, we significantly expanded our understanding of 2D lattice models of statistical physics, their conformally invariant scaling limits and related random geometries. However, there seem to be serious obstacles, preventing further development and requiring novel ideas. We plan to attack those, in particular we intend to:(A) Describe new scaling limits by Schramm’s SLE curves and their generalizations,(B) Study discrete complex structures and use them to describe more 2D models,(C) Describe the scaling limits of random planar graphs by the Liouville Quantum Gravity,(D) Understand universality and lay framework for the Renormalization Group Formalism,(E) Go beyond the current setup of spin models and SLEs.These problems are known to be very difficult, but fundamental questions, which have the potential to lead to significant breakthroughs in our understanding of phase transitions, allowing for further progresses. In resolving them, we plan to exploit interactions of different subjects, and recent advances are encouraging."