This monograph discusses specific examples of selfdual gauge field structures, including the Chern Simons model, the abelian Higgs model, and Yang Mills gauge field theory. The author builds a foundation for gauge theory and selfdual vortices by introducing the basic mathematical language of gauge theory and formulating examples of Chern Simons Higgs theories (in both abelian and non-abelian settings). Thereafter, the Electroweak theory and self-gravitating Electroweak strings are examined. The final chapters treat elliptic problems involving Chern Simmons models, concentration-compactness principles, and Maxwell Chern Simons vortices.

This volume contains the notes of the lectures delivered at the CIME course GeometricAnalysis andPDEsduringtheweekofJune11-162007inCetraro (Cosenza). The school consisted in six courses held by M. Gursky (PDEs in Conformal Geometry), E. Lanconelli (Heat kernels in sub-Riemannian s- tings), A. Malchiodi(Concentration of solutions for some singularly perturbed Neumann problems), G. Tarantello (On some elliptic problems in the study of selfdual Chern-Simons vortices), X. J. Wang (Thek-Hessian Equation)and P. Yang (Minimal Surfaces in CR Geometry). Geometric PDEs are a ?eld of research which is currently very active, as it makes it possible to treat classical problems in geometry and has had a dramatic impact on the comprehension of three- and four-dimensional ma- folds in the last several years. On one hand the geometric structure of these PDEs might cause general di?culties due to the presence of some invariance (translations, dilations, choice of gauge, etc. ), which results in a lack of c- pactness of the functional embeddings for the spaces of functions associated with the problems. On the other hand, a geometric intuition or result might contribute enormously to the search for natural quantities to keep track of, andtoproveregularityoraprioriestimatesonsolutions. Thistwo-foldaspect of the study makes it both challenging and complex, and requires the use of severalre?nedtechniquestoovercomethemajordi?cultiesencountered