A large number of problems in modern equilibrium statistical mechanics and theoretical physics, including important random matrix models describing transport phenomena in disordered media, can be studied through integrals on a manifold, possibly involving anti-commuting variables. A major challenge is to prove or disprove existence of critical values where a phase transition occurs and to understand the behavior at the critical point. The goal of this project is to develop tools to analyze situations where the measure is, at least partially, complex-valued and hence classical methods fail.