ENDRISP · Energy Driven Pattern Formation: Continuous Symmetry Breaking and Positive Temperature

Spontaneous pattern formation is an ubiquitous and fascinating phenomenon in nature, of enormous importance for the present and future of applications. A crucial aspect, both at zero and positive temperature, is that in dimension d>1 the physical states retain less symmetries than the interactions involved in the models. This phenomenon, known as symmetry breaking, coupled with nonlocality makes any attempt of giving a mathematical explanation of pattern formation extremely challenging and still largely not accessible with the current tools.ENDRISP aims at gaining deep insight into the mathematical mechanisms behind the phenomenon of energy-driven pattern formation, focusing on the following foremost questions in the field: Explain continuous symmetry breaking mechanisms for isotropic functionals; Show pattern formation and symmetry breaking in the positive temperature setting.The focus of ENDRISP will be on the formation of periodic one-dimensional structures in general dimension (i.e. striped/lamellar structures), as they are generally the first type of patterns to emerge from uniform phases when symmetry breaking occurs, in regimes in which the attractive and repulsive forces are of the same order. We will use precise nonlocal quantities of geometric character coming from the free energy to control the curvature of the physical states and derive rigidity and stability estimates. The techniques we developed to show pattern formation for continuous functionals retaining discrete symmetry together with our experience on representation theorems in the positive temperature setting and the new ideas in ENDRISP will be the ideal basement to the project. Major advances in Calculus of Variations, Geometric Measure Theory and Statistical Mechanics will be required, for which the transfer of knowledge with the supervisors, empowered by the unique training offered by Courant Institute in the outgoing phase and by GSSI in the return phase, will be extremely beneficial.