Title: An approach to topological singularities through Mumford-Shah type functionals 

Abstract:  We introduce a notion of distributional Jacobian for BV maps.  We will use such a definition to prove a compactness and Gamma-convergence result for a new model describing the emergence of topological singularities in two dimensions, in the spirit of Ginzburg-Landau (GL) and core-radius (CR) approaches. Within our framework, the order parameter is an SBV map taking values in the unit sphere of the plane; the bulk energy is the squared L^2 norm of the approximate gradient whereas the penalization term is given by the length of the jump set, scaled by a small parameter.  We show that at any logarithmic scale our functional is “variationally equivalent” to the “standard” (CR) and (GL) model. Joint work with V. Crismale, R. Scala, and N. Van Goethem.

Title: EDP convergence for evolutionary systems with gradient flow structure

Abstract: We discuss the notion of EDP convergence for gradient systems. This convergence is based on De Giorgi's energy-dissipation principle (EDP) and allows us to study coarse-graining limits for families of gradient systems. While the energies simply converge in the sense of De Giorgi's Gamma convergence, the derivation of the effective dissipation potential is more involved. Several examples will be discussed, that show the flexibility of the theory, e.g., to allow situations where starting from quadratic dissipation potentials we arrive at effective dissipation potentials that are no longer quadratic.

Title: On the Minimization of Nonlocal Interaction Energies 

Abstract: Nonlocal interaction energies play a central role in describing the collective behavior of large particle systems in a wide range of applications. In this lecture we will focus on interactions that are short-range repulsive and long-range attractive. We will review the key results on the existence and uniqueness of minimizers, and present their explicit characterization in the classical case of isotropic kernels with Riesz-type repulsion and quadratic attraction. We will then show how a complete characterization can be given for a broad class of anisotropic variants of the repulsive kernel. If time permits, we will conclude with a discussion of open problems and future directions. This talk is based on joint work with several collaborators: R. Frank, J. Mateu, L. Rondi, L. Scardia, and J. Verdera.

Title: Skyrmions in ultrathin magnetic films: an overview

Abstract: I will present an overview of the current results on existence and asymptotic properties of magnetic skyrmions defined as topologically nontrivial maps of degree +1 from the plane to a sphere which minimize a micromagnetic energy containing the exchange, perpendicular magnetic anisotropy  and interfacial Dzyaloshinskii-Moriya interaction (DMI) terms. In ultrathin films, the stray field energy simply renormalizes the anisotropy constant at leading order, but in finite samples it also produces additional non-trivial contributions at the sample edges, promoting nontrivial spin textures. Starting with the whole space problem, I will first discuss the existence of single skyrmions as global energy minimizers at sufficiently small DMI strength. Then, using the quantitative rigidity of the harmonic maps I will present the asymptotic characterization of single skyrmion profiles both in infinite and finite samples. Lastly, I will touch upon the question of existence of multi-skyrmion solutions as minimizers with higher topological degree and present recent existence results obtained jointly with T. Simon and V. Slastikov.