Mathematical aspects of phase separation under long-range interactions

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Many physical systems exhibit complex patterns. For example, a system may display spatial regions of  two different phases with the interface between them resembling a complex surface. Often, the observed structure can be explained  or simulated by a variational principle in which a free energyof any admissible structure can be computed, with the observed structure being the one with the lowest energy. The resulting variational problems are challenging to say the least. If one wishes to analysis them without first adopting assumptions which severley limit possible geometric complexities, one needs advanced mathematical analysis.This proposal pertains to both mathematical and physical paradigms for energy-driven pattern  formation associated with  competing short and long-range interactions. The physical paradigm is provided by diblock copolymers which have a remarkable ability for self-assembly into many fascinating structures; the interfaces of which resemble triply-periodic surfaces whose mean curvature is constant.The goal of this proposal is two-fold: First, using the variational problem (the mathematical paradigm), we will address some central questions in the calculus of variations concerning scale and geometry. Second, within the general framework of variational problems, we will address the underlying physical application; phase separation in diblock copolymers melts.The techniques expoited range from rigorous analysis and geometry to statistical physics and numerical simulations.