University of Granada

Title: Variational problems in sub-Finsler Heisenberg groups

Abstract: We discuss recent work on variational problems in sub-Finsler Heisenberg groups, including the Pansu-Wulff shapes, variational formulas, regularity and area-minimizing cones.

University of Luxemburg

Title: Semialgebraic Geometry, a practical toolkit

Abstract: Semialgebraic geometry is the theory of sets and functions defined using polynomial inequalities. This restriction imposes a significant degree of regularity on all the objects involved, excluding many of the pathological counterexamples in the smooth category. For instance, every semialgebraic set can be expressed as a disjoint union of smooth manifolds that satisfy the axioms of a Whitney stratification and can be triangulated. Specifically, every semialgebraic subset of the real line is a finite union of intervals. Moreover, the triangulation can be tuned with any given semialgebraic map (Hardt's Semialgebraic Triviality).

Despite this rigidity, semialgebraic geometry is richer than the pure algebraic one, based only on polynomial equalities, in that the class of semialgebraic sets is stable under more operations (projections above all, by Tarski-Seidenberg), to the point that the semialgebraic is a useful compromise between the algebraic and the smooth, the rigidity and the flexibility. 

But why should one care? One reason is that semialgebraic sets naturally appear in various analytical contexts, and familiarity with their rich theory provides immediate access to its many powerful tools.

This mini-course will begin with foundational concepts, exploring the structure of semialgebraic sets, the Tarski-Seidenberg theorem, stratifications, Semialgebraic Triviality, and basics of o-minimal structures. The second part of the mini-course will focus on the interplay between semialgebraic geometry and other areas of mathematics, reviewing instances of contemporary research in analysis, singularity theory, and probability.

University of Milan

Title: An introduction to Brakke's mean curvature flow

Abstract: Arising as the gradient flow of the area functional, the mean curvature flow (MCF hereafter) is the fundamental parabolic problem in extrinsic differential geometry. The unknown is a time-parametrised family of (immersed) k-dimensional surfaces evolving so that their normal velocity equals the mean curvature vector of the immersion. The corresponding initial value problem is well-posed when the initial datum is a closed immersed submanifold, but it is easy to see that the flow develops singularities in finite time in dimension k ≥ 2. Upon the occurrence of singularities, the surfaces may undergo topology changes, and the PDE description becomes meaningless. At the same time, it is interesting for the applications (particularly to materials science) to gain understanding of a flow of mean curvature type for possibly non-smooth objects.

The Brakke flow is a weak formulation of MCF which leverages on the flexibility of Geometric Measure Theory to describe the evolution by mean curvature of generalised surfaces through singularities and topology changes. In this mini course I will aim at presenting the basics of the theory of Brakke flows, with an emphasis on compactness properties, monotonicity formulas, and partial regularity.

University of Valencia

Title: Ricci flow in a nutshell

Abstract: Geometric flows have been used to address successfully key questions in Differential Geometry like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, the differentiable sphere theorem or, more recently, the generalized Smale conjecture. During this mini-course we will give an intuitive introduction to Ricci flow, which is sort of a non-linear version of the heat equation for the Riemannian metric. The equation should be understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance, some kind of constant curvature. We will emphasize the features that convert evolution equations into a powerful tool in geometry.