Geometric Measure Theory I
9:10-12:10 Every Mon., September 9, 2019 - January 3, 2020
M210, Department of Mathematics, NTNU
Speaker:
Ulrich Menne (National Taiwan Normal University)
Background and Purpose：
A very successful strategy for the study of geometric variational problems is to firstly prove existence in an enlarged class of competitors by means of compactness theorems and subsequently study the regularity of the solution therein. For instance, instead of considering only smooth submanifolds, one proves existence in the classes of boundaries of sets of finite perimeter, integral currents, or integral varifolds—all of which are based on the more basic concept of rectifiable set. In the ensuing regularity theory, the generality of varifolds allows to unify a substantial part of the treatment. The purpose of the course is to develop, after providing the necessary infrastructure, the concept of rectifiable set as well as key elements of the theory of varifolds.
Outline
We assume sound familiarity with the concepts of measure, measurable function, Lebesgue integration, and product measure. In the initial part of the one-year course, we focus on developing the relevant concepts from advanced measure theory (Riesz representation theorem, covering theorems, derivatives of measures), functional analysis (locally convex spaces, weak topology), multilinear algebra (exterior algebra, alternating forms) and basic submanifold geometry (second fundamental form, Grassmann manifold). Then, we develop Hausdorff measures and the area of Lipschitzian maps to treat rectifiable sets. Finally, the basic theory of varifolds is developed (first variation, monotonicity identity) to treat the isoperimetric inequality and compactness theorems.