The variational bi-complex formulation of Maxwell’s equations

In this manuscript, we present the variational derivation of Maxwell’s equations by means of the variational bi-complex. We use this exercise to introduce the reader to the formulation of geometric variational problems and their implementation in a Computer Algebra System.

Discretization and Approximation of Surfaces Using Varifolds

We present some recent results on the possibility of extending the theory of varifolds to the realm of discrete surfaces of any dimension and codimension, for which robust notions of approximate curvatures, also allowing for singularities, can be defined. This framework has applications to discrete and computational geometry, as well as to geometric variational problems in discrete settings. We finally show some numerical tests on point clouds that support and confirm our theoretical findings.

Dimensional estimates for singular sets in geometric variational problems with free boundaries

We show that singular sets of free boundaries arising in codimension one anisotropic geometric variational problems are

On the Equivariant Implicit Function Theorem with Low Regularity and Applications to Geometric Variational Problems

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite-dimensional Lie group. Motivated by some geometric variational problems, we consider group actions that are not necessarily differentiable everywhere, but only on some dense subset. Applications are discussed in the context of harmonic maps, closed (pseudo-) Riemannian geodesics and constant mean curvature hypersurfaces.

GAUGE THEORY OF FADDEEV–SKYRME FUNCTIONALS

We study geometric variational problems for a class of nonlinear σ-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces, we obtain a weaker result on existence of minimizers in each 2-homotopy class.Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G → G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.