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In this set of notes, we present some recent developments on the fractional \( {(-\Delta)^{s}} u = u - {{u}^{3}}\) Allen-Cahn equation with special attention to -convergence results, energy and density estimates, convergence of level sets, Hamiltonian estimates, rigidity and symmetry results.

Long-range Phase Coexistence Models: Recent Progress on the Fractional Allen-Cahn Equation