Interactions between inclusions in fluid two-dimensional membranes

Biological membranes are fluid surfaces with an elastic Hamiltonian function of the local curvature, mean and Gaussian. Proteins, that are included in such surfaces, deform them, and interact via the deformation they create. Since membranes exhibit large thermal fluctuations, these interactions have a 'Casimir' character. Proteins can be out-of-equilibrium objects actively changing conformation, while the membrane in which they live exhibit a complex dynamics ruled by the bilayer structure of the membrane. Proteins thus experience nontrivial dynamical interactions. I will introduce and discuss these topics.