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These notes are based on lectures given at the Third International School on Geometry and Physics at the Centre de Recerca Matem`atica in Barcelona, March 26–30, 2012. The aim ofthe School’s four lect...
We study the relaxation of crystal surfaces in 2+1 dimensions via the motion ofinteracting atomic steps. The goal is the rigorous derivation of the continuum limit. The starting point is a discrete sc...
We study the continuum limit in 2+1 dimensions of nanoscale anisotropic diffusion processes on crystal surfaces relaxing to become flat below roughening. Our main result is a continuum law for the sur...
We study the effect of an external electric field E on macroscopic relaxation laws for crystal surfaces in 2 + 1 dimensions. We derive a nonlinear, fourth-order partial differential equation (PDE) for...
We apply classical homogenization to derive macroscopic relaxation laws for crystal surfaces with distinct inhomogeneities at the microscale. The proposed method relies on a formal multiscale expansio...
Abstract: We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of t...
Abstract: We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of t...
Abstract: A family of algebraic surfaces with many nondegenerate real singularities is introduced with the help of a construction that has been used in previous works for the generation of substitutio...
Abstract: Concavity properties prevent the existence of significant landscapes in energy surfaces obtained by energy minimizations under constraints. The contradiction holds at finite temperatures T a...
We propose a method to decompose the total energy of a supercell containing defects into contributions of individual atoms, using the energy density formalism within density functional theory. The s...
A generalized Hamiltonian definition of ghost surfaces (surfaces defined by an action-gradient flow) is given and specialized to the usual Lagrangian definition. Numerical calculations show uncorrecte...

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