The Plateau problem for wet films

• Maggi, Francesco (University of Texas at Austin)

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 capillarity theory. Finally, we introduce a diffused interface version of this wet soap films model, a new class of free boundary problems arising as the Euler-Lagrange equations of these diffused models, and discuss various open problems. This course is based on a series of joint works with Camillo De Lellis, Francesco Ghiraldin, Antonello Scardicchio, Salvatore Stuvard, Darren King, Michael Novack, and Daniel Restrepo.

On the de Rham complex in Carnot groups

• Tripaldi, Francesca (University of Leeds)

Room: Room PS1,ground floor, building E, 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 complex. This subcomplex reflects key features of the underlying Lie algebra and proves to be more intrinsic to the subRiemannian context than the classical de Rham complex. In this mini-course, we will introduce Carnot groups, explore their fundamental properties, and examine how the de Rham complex behaves in this setting. We will then construct the Rumin complex and carry out explicit computations in key examples, illustrating its relevance and effectiveness in subRiemannian geometry.

Quantitative differentiability and rectifiability

• Young, Robert (New York University)

Room: Room PS1,- ground floor, building E 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 these ideas let us answer questions like "How well can a function or a set be approximated by affine functions or planes at local scales?" or "How often can a function or set fail to be approximated by an affine function or plane?", and we will apply these ideas to geometry and analysis in Euclidean space and the Heisenberg group.

A user’s guide to distributional fractional spaces

• Stefani, Giorgio (Università di Padova)

Room: Room PS1 - ground floor, building E, 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 fractional analog of De Giorgi’s Blow-up Theorem.

On the de Rham complex in Carnot groups

• Tripaldi, Francesca (University of Leeds)

Room: Aula Dini 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 complex. This subcomplex reflects key features of the underlying Lie algebra and proves to be more intrinsic to the subRiemannian context than the classical de Rham complex. In this mini-course, we will introduce Carnot groups, explore their fundamental properties, and examine how the de Rham complex behaves in this setting. We will then construct the Rumin complex and carry out explicit computations in key examples, illustrating its relevance and effectiveness in subRiemannian geometry.

Quantitative differentiability and rectifiability

• Young, Robert (New York University)

Room: Aula Dini 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 these ideas let us answer questions like "How well can a function or a set be approximated by affine functions or planes at local scales?" or "How often can a function or set fail to be approximated by an affine function or plane?", and we will apply these ideas to geometry and analysis in Euclidean space and the Heisenberg group.

The Plateau problem for wet films

• Maggi, Francesco (University of Texas at Austin)

Room: Aula Dini 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 capillarity theory. Finally, we introduce a diffused interface version of this wet soap films model, a new class of free boundary problems arising as the Euler-Lagrange equations of these diffused models, and discuss various open problems. This course is based on a series of joint works with Camillo De Lellis, Francesco Ghiraldin, Antonello Scardicchio, Salvatore Stuvard, Darren King, Michael Novack, and Daniel Restrepo.

Besicovitch's 1/2 problem

• Massaccesi, Annalisa (Università di Padova)

Room: Aula Dini 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 collaboration with C. De Lellis, F. Glaudo and D. Vittone, we devise a strategy to improve the bound by means of a hierarchy of variational problems and we reach a proof that . In this seminar, I will try to explain the fairly intuitive geometric idea behind this strategy and I will try to summarize both the computational obstacles and the intrinsic obstacles that are still in the way.

Quantitative differentiability and rectifiability

• Young, Robert (New York University)

Room: Aula Dini 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 these ideas let us answer questions like "How well can a function or a set be approximated by affine functions or planes at local scales?" or "How often can a function or set fail to be approximated by an affine function or plane?", and we will apply these ideas to geometry and analysis in Euclidean space and the Heisenberg group.

The Plateau problem for wet films

• Maggi, Francesco (University of Texas at Austin)

Room: Aula Dini 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 capillarity theory. Finally, we introduce a diffused interface version of this wet soap films model, a new class of free boundary problems arising as the Euler-Lagrange equations of these diffused models, and discuss various open problems. This course is based on a series of joint works with Camillo De Lellis, Francesco Ghiraldin, Antonello Scardicchio, Salvatore Stuvard, Darren King, Michael Novack, and Daniel Restrepo.

On the de Rham complex in Carnot groups

• Tripaldi, Francesca (University of Leeds)

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 complex. This subcomplex reflects key features of the underlying Lie algebra and proves to be more intrinsic to the subRiemannian context than the classical de Rham complex. In this mini-course, we will introduce Carnot groups, explore their fundamental properties, and examine how the de Rham complex behaves in this setting. We will then construct the Rumin complex and carry out explicit computations in key examples, illustrating its relevance and effectiveness in subRiemannian geometry.

Regularity for area minimizing integral currents

• Resende, Reinaldo (Carnegie Mellon University)

Room: Aula Dini 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 highly singular behaviors. We will then present some of the latest results on boundary regularity and highlight open problems that remain unsolved. Following this, we will outline the key ideas behind the proof of a Hausdorff dimension estimate for the boundary singular set, particularly in the linearized setting for multi-valued Dirichlet minimizers. This talk is based on joint work with Ian Fleschler and, separately, with Stefano Nardulli.

The Plateau problem for wet films

• Maggi, Francesco (University of Texas at Austin)

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 capillarity theory. Finally, we introduce a diffused interface version of this wet soap films model, a new class of free boundary problems arising as the Euler-Lagrange equations of these diffused models, and discuss various open problems. This course is based on a series of joint works with Camillo De Lellis, Francesco Ghiraldin, Antonello Scardicchio, Salvatore Stuvard, Darren King, Michael Novack, and Daniel Restrepo.

On the de Rham complex in Carnot groups

• Tripaldi, Francesca (University of Leeds)

Room: Aula Dini 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 complex. This subcomplex reflects key features of the underlying Lie algebra and proves to be more intrinsic to the subRiemannian context than the classical de Rham complex. In this mini-course, we will introduce Carnot groups, explore their fundamental properties, and examine how the de Rham complex behaves in this setting. We will then construct the Rumin complex and carry out explicit computations in key examples, illustrating its relevance and effectiveness in subRiemannian geometry.

Quantitative differentiability and rectifiability

• Young, Robert (New York University)

Room: Aula Dini 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 these ideas let us answer questions like "How well can a function or a set be approximated by affine functions or planes at local scales?" or "How often can a function or set fail to be approximated by an affine function or plane?", and we will apply these ideas to geometry and analysis in Euclidean space and the Heisenberg group.