In this course we introduce the formulation of the Plateau problem by Harrison and Pugh and discuss its connection with the theory of soap films and, in particular, with the derivation of Plateau laws. We then introduce the notions of “dry” and “wet” soap films, the notion of Plateau borders for wet soap films, and discuss the derivation of the resulting “wet Plateau laws” in the context of...
Carnot groups are a class of nilpotent Lie groups naturally equipped with a horizontal distribution. When endowed with a compatible metric, they form subRiemannian manifolds, which are central objects in non-holonomic systems and control theory.
To better capture the algebraic and geometric structure of such spaces, Rumin introduced a refinement of the de Rham complex, now known as the Rumin...
Differentiability and rectifiability describe whether a function or a set can be approximated by affine functions or planes, but both of these notions are infinitesimal, i.e., they deal with the properties of limits. Notions like uniform rectifiability let us quantify differentiability by considering approximations at scales that are small but positive. In this minicourse, we will explain how...
We provide an introduction to the distributional theory of fractional spaces. In the first part of the talk, we define the Riesz fractional gradient, explore its key properties, and introduce the distributional fractional Sobolev and $BV$ spaces along with their main features. In the second part, we discuss the properties of fractional variation and survey recent developments, including the...
Carnot groups are a class of nilpotent Lie groups naturally equipped with a horizontal distribution. When endowed with a compatible metric, they form subRiemannian manifolds, which are central objects in non-holonomic systems and control theory.
To better capture the algebraic and geometric structure of such spaces, Rumin introduced a refinement of the de Rham complex, now known as the Rumin...
Differentiability and rectifiability describe whether a function or a set can be approximated by affine functions or planes, but both of these notions are infinitesimal, i.e., they deal with the properties of limits. Notions like uniform rectifiability let us quantify differentiability by considering approximations at scales that are small but positive. In this minicourse, we will explain how...
In this course we introduce the formulation of the Plateau problem by Harrison and Pugh and discuss its connection with the theory of soap films and, in particular, with the derivation of Plateau laws. We then introduce the notions of “dry” and “wet” soap films, the notion of Plateau borders for wet soap films, and discuss the derivation of the resulting “wet Plateau laws” in the context of...
Besicovitch's problem
Besicovitch's problem investigates the smallest threshold guaranteeing rectifiability for a set with Hausdorff -dimensional finite measure when the lower density of the set is larger than almost everywhere. Besicovitch conjectured that (hence the name of the problem) and proved , then Preiss and Tišer improved the bound to . In a recent work in...
Differentiability and rectifiability describe whether a function or a set can be approximated by affine functions or planes, but both of these notions are infinitesimal, i.e., they deal with the properties of limits. Notions like uniform rectifiability let us quantify differentiability by considering approximations at scales that are small but positive. In this minicourse, we will explain how...
In this course we introduce the formulation of the Plateau problem by Harrison and Pugh and discuss its connection with the theory of soap films and, in particular, with the derivation of Plateau laws. We then introduce the notions of “dry” and “wet” soap films, the notion of Plateau borders for wet soap films, and discuss the derivation of the resulting “wet Plateau laws” in the context of...
Carnot groups are a class of nilpotent Lie groups naturally equipped with a horizontal distribution. When endowed with a compatible metric, they form subRiemannian manifolds, which are central objects in non-holonomic systems and control theory.
To better capture the algebraic and geometric structure of such spaces, Rumin introduced a refinement of the de Rham complex, now known as the Rumin...
We will explore the state of the art in interior and boundary regularity for solutions of the oriented Plateau problem, specifically in the framework of integral currents. After reviewing recent developments in interior regularity, we will shift our focus to the boundary setting. In this context, we will discuss the types of boundary points that naturally arise in the theory and introduce...
In this course we introduce the formulation of the Plateau problem by Harrison and Pugh and discuss its connection with the theory of soap films and, in particular, with the derivation of Plateau laws. We then introduce the notions of “dry” and “wet” soap films, the notion of Plateau borders for wet soap films, and discuss the derivation of the resulting “wet Plateau laws” in the context of...
Carnot groups are a class of nilpotent Lie groups naturally equipped with a horizontal distribution. When endowed with a compatible metric, they form subRiemannian manifolds, which are central objects in non-holonomic systems and control theory.
To better capture the algebraic and geometric structure of such spaces, Rumin introduced a refinement of the de Rham complex, now known as the Rumin...
Differentiability and rectifiability describe whether a function or a set can be approximated by affine functions or planes, but both of these notions are infinitesimal, i.e., they deal with the properties of limits. Notions like uniform rectifiability let us quantify differentiability by considering approximations at scales that are small but positive. In this minicourse, we will explain how...
In this course we introduce the formulation of the Plateau problem by Harrison and Pugh and discuss its connection with the theory of soap films and, in particular, with the derivation of Plateau laws. We then introduce the notions of “dry” and “wet” soap films, the notion of Plateau borders for wet soap films, and discuss the derivation of the resulting “wet Plateau laws” in the context of...
