Advanced Topics in Real Analysis

The aim of the course is to provide a modern and robust treatment of measure and integration theory, and of the theory of differentiability of functions, initiated in Mathematical Analysis 4 and Real Analysis. In particular, we will describe the generalization of the fundamental theorems of Calculus to weakly differentiable functions and non-smooth sets in Euclidean spaces, providing a systematic introduction to Geometric Measure Theory (GMT). In this framework, we will study the almost everywhere differentiability of Lipschitz maps, the existence of tangent spaces to rectifiable sets almost everywhere, and the theories of sets of finite perimeter and functions of bounded variation in arbitrary dimension. Moreover, we will discuss the application of GMT techniques to geometric variational problems, and we will provide an introduction to the regularity theory and the analysis of singularities of solutions, with an emphasis on the weak theory of minimal surfaces.

Knowledge of basic notions and techniques in Geometric Measure Theory in R^n: Lipschitz maps, rectifiable sets, area and coarea formulas and applications, theory of BV functions and sets of finite perimeter, densities and tangent cones, variation formulas for the perimeter and partial regularity of perimeter minimizing sets.