Mathematical Sciences: Applications of Functional Analysis and Geometric Measure Theory

点击翻译按钮获取中文摘要

Almgren will continue his long term program of research in the geometric calculus of variations and its applications. The objective is to seek a better understanding of the geometric structure of surfaces minimizing the integrals of parametric integrands. A specific example would be certain area minimizing surfaces. He will also study the evolution of surfaces towards such minimization and the feasibility of constructing algorithms with which to compute the mathematical framework for these analyses and their applications in continuum mechanics. A considerable effort will be devoted towards making many significant results more accessible. Geometric measure theory has proved extremely powerful in the hands of such practitioners as Almgren, but the techniques employed still remain unknown to the majority of researchers. Amongst the new directions to be followed will be investigations of relationships between area minimizing currents and various energies of mappings from the ambient space to spheres. Also work will be carried out into flows in spaces of rectifiable currents and varifolds.