Geometric Measure Theory and related topics - SECOND WEEK

Contribution List

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

De Rosa, Antonio (Università Bocconi)

16/06/2025, 09:00

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...

3. 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.

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

16/06/2025, 14:30

13. Complexity methods in geometric measure theory

Prof. Csornyei, Marianna (University of Chicago)

16/06/2025, 16:30

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.

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

De Rosa, Antonio (Università Bocconi)

17/06/2025, 11:00

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...

6. 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.

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

17/06/2025, 14:30

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...

14. Complexity methods in geometric measure theory

Prof. Csornyei, Marianna (University of Chicago)

17/06/2025, 16:30

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.

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

De Rosa, Antonio (Università Bocconi)

18/06/2025, 11:00

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...

8. 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.

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

18/06/2025, 14:30

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...

15. Complexity methods in geometric measure theory

Prof. Csornyei, Marianna (University of Chicago)

18/06/2025, 16:30

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.

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

De Rosa, Antonio (Università Bocconi)

19/06/2025, 11:00

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...

10. 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.

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

19/06/2025, 14:30

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...

16. Complexity methods in geometric measure theory

Prof. Csornyei, Marianna (University of Chicago)

19/06/2025, 16:30

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.

2. Multi-bubble isoperimetric problems

Prof. Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure)....

5. Multi-bubble isoperimetric problems

Prof. Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure)....

7. Multi-bubble isoperimetric problems

Prof. Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure)....

12. Multi-bubble isoperimetric problems

Prof. Milman, Emanuel (Euskal Herriko Unibertsitatea/Universidad del País Vasco)

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems on more general spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ ($\mathbb{R}^n$ endowed with the standard Gaussian measure)....