Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

Room: Aula Dini The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

Room: Aula Dini

Complexity methods in geometric measure theory

• Csornyei, Marianna (University of Chicago)

Our aim is to introduce the computability-theoretic concept 'Kolgomorov complexity' and show how it can be used to obtain interesting results in Geometric Measure Theory.

Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

Room: Aula Bianchi The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

Room: Aula Bianchi This course presents a unified approach to extending boundary data from rough domains into the interior, with a focus on applications to boundary value problems for elliptic operators. We study recent advances in constructing \emph{smooth harmonic-type extensions} of BMO and \( L^p \) functions from the boundary \( \partial \Omega \) of a domain \( \Omega \subset \mathbb{R}^{n+1} \), where the geometry of \( \Omega \) may be highly irregular. The domains under consideration include: \begin{itemize} \item \textbf{Corkscrew domains} when \( \partial \Omega \) is \( n \)-dimensional and Ahlfors regular, \item and \textbf{complements of \( s \)-Ahlfors regular sets} when \( s < n \). \end{itemize} The core objectives of the course include: \begin{itemize} \item Constructing \emph{smooth interior extensions} of boundary functions with optimal control in terms of \emph{Carleson measures} and \emph{non-tangential maximal functions}, \item Establishing \emph{pointwise convergence} of these extensions back to the boundary data in a non-tangential sense, \item Showing how \emph{Lipschitz boundary data} yields Lipschitz continuous extensions up to the closure of the domain. \end{itemize} A significant portion of the course will be dedicated to \textbf{applications in elliptic boundary value problems}, particularly for \emph{divergence-form elliptic systems with rough (e.g., merely bounded, complex-valued) coefficients}. We will explore: \begin{itemize} \item The role of these extensions in solving \emph{Dirichlet problems with \( L^p \) and BMO boundary data}, \item Connections between \emph{interior regularity in Carleson or tent spaces} and the \emph{solvability of Poisson problems}, \item How these tools fit into the modern framework of harmonic analysis on non-smooth domains. \end{itemize} The course is aimed at graduate students and researchers interested in \emph{elliptic PDEs, harmonic analysis, and geometric measure theory}. It will balance theoretical development with motivation from concrete problems in analysis and PDE.

Complexity methods in geometric measure theory

• Csornyei, Marianna (University of Chicago)

Our aim is to introduce the computability-theoretic concept 'Kolgomorov complexity' and show how it can be used to obtain interesting results in Geometric Measure Theory.

Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

Room: Aula Dini The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

Room: Aula Dini This course presents a unified approach to extending boundary data from rough domains into the interior, with a focus on applications to boundary value problems for elliptic operators. We study recent advances in constructing \emph{smooth harmonic-type extensions} of BMO and \( L^p \) functions from the boundary \( \partial \Omega \) of a domain \( \Omega \subset \mathbb{R}^{n+1} \), where the geometry of \( \Omega \) may be highly irregular. The domains under consideration include: \begin{itemize} \item \textbf{Corkscrew domains} when \( \partial \Omega \) is \( n \)-dimensional and Ahlfors regular, \item and \textbf{complements of \( s \)-Ahlfors regular sets} when \( s < n \). \end{itemize} The core objectives of the course include: \begin{itemize} \item Constructing \emph{smooth interior extensions} of boundary functions with optimal control in terms of \emph{Carleson measures} and \emph{non-tangential maximal functions}, \item Establishing \emph{pointwise convergence} of these extensions back to the boundary data in a non-tangential sense, \item Showing how \emph{Lipschitz boundary data} yields Lipschitz continuous extensions up to the closure of the domain. \end{itemize} A significant portion of the course will be dedicated to \textbf{applications in elliptic boundary value problems}, particularly for \emph{divergence-form elliptic systems with rough (e.g., merely bounded, complex-valued) coefficients}. We will explore: \begin{itemize} \item The role of these extensions in solving \emph{Dirichlet problems with \( L^p \) and BMO boundary data}, \item Connections between \emph{interior regularity in Carleson or tent spaces} and the \emph{solvability of Poisson problems}, \item How these tools fit into the modern framework of harmonic analysis on non-smooth domains. \end{itemize} The course is aimed at graduate students and researchers interested in \emph{elliptic PDEs, harmonic analysis, and geometric measure theory}. It will balance theoretical development with motivation from concrete problems in analysis and PDE.

Complexity methods in geometric measure theory

• Csornyei, Marianna (University of Chicago)

Our aim is to introduce the computability-theoretic concept 'Kolgomorov complexity' and show how it can be used to obtain interesting results in Geometric Measure Theory.

Introduction to the theory of varifolds with applications to the min-max theory

• De Rosa, Antonio (Università Bocconi)

Room: Aula Dini The construction of critical points of the area, that are not necessarily area minimizers, typically requires to pass to the limit a sequence of almost critical submanifolds with area bounds, in order to find a limit that is critical for the area. To this aim, currents are not very effective, since the mass is only lower semicontinuous and we could end up with a trivial limit. This is not an issue for the Plateau problem, because of the assigned nontrivial boundary or homology class. To solve this problem, F. J. Almgren introduced varifolds in 1965, following an earlier notion of generalized surfaces by L. C. Young in 1951. The main difference compared to currents is that the mass of varifolds is continuous in compact sets. This allows to obtain one of the most important applications of the theory of varifolds: the min-max construction of closed minimal hypersurfaces in closed Riemannian manifolds. In this mini-course, we will give an introduction to the theory of varifolds, and we will discuss how we can use varifolds to construct (possibly anisotropic) minimal hypersurfaces via the min-max theory.

Varopoulos extensions of $BMO$ and $L^p$ functions in domains with Ahlfors-regular boundaries and applications to Boundary Value problems for elliptic PDEs with $L^\infty$ coefficients.

• Mourgoglou, Mihalis (Universidad del País Vasco/Euskal Herriko Unibertsitatea)

Room: Aula Dini This course presents a unified approach to extending boundary data from rough domains into the interior, with a focus on applications to boundary value problems for elliptic operators. We study recent advances in constructing \emph{smooth harmonic-type extensions} of BMO and \( L^p \) functions from the boundary \( \partial \Omega \) of a domain \( \Omega \subset \mathbb{R}^{n+1} \), where the geometry of \( \Omega \) may be highly irregular. The domains under consideration include: \begin{itemize} \item \textbf{Corkscrew domains} when \( \partial \Omega \) is \( n \)-dimensional and Ahlfors regular, \item and \textbf{complements of \( s \)-Ahlfors regular sets} when \( s < n \). \end{itemize} The core objectives of the course include: \begin{itemize} \item Constructing \emph{smooth interior extensions} of boundary functions with optimal control in terms of \emph{Carleson measures} and \emph{non-tangential maximal functions}, \item Establishing \emph{pointwise convergence} of these extensions back to the boundary data in a non-tangential sense, \item Showing how \emph{Lipschitz boundary data} yields Lipschitz continuous extensions up to the closure of the domain. \end{itemize} A significant portion of the course will be dedicated to \textbf{applications in elliptic boundary value problems}, particularly for \emph{divergence-form elliptic systems with rough (e.g., merely bounded, complex-valued) coefficients}. We will explore: \begin{itemize} \item The role of these extensions in solving \emph{Dirichlet problems with \( L^p \) and BMO boundary data}, \item Connections between \emph{interior regularity in Carleson or tent spaces} and the \emph{solvability of Poisson problems}, \item How these tools fit into the modern framework of harmonic analysis on non-smooth domains. \end{itemize} The course is aimed at graduate students and researchers interested in \emph{elliptic PDEs, harmonic analysis, and geometric measure theory}. It will balance theoretical development with motivation from concrete problems in analysis and PDE.

Complexity methods in geometric measure theory

• Csornyei, Marianna (University of Chicago)

Our aim is to introduce the computability-theoretic concept 'Kolgomorov complexity' and show how it can be used to obtain interesting results in Geometric Measure Theory.