Local and nonlocal methods in geometric analysis

Contribution List

5. Gamma-convergence of the p-Dirichlet energy for manifold-valued maps

Giacomo Canevari (Università degli Studi di Verona)

05/11/2025, 14:00

In this talk, we will consider the p-Dirichlet energy of maps with values in a closed manifold. We will discuss a Gamma-convergence result in the limit as the exponent p approaches a certain critical value k, which is defined in terms of the topology of the target manifold. This particular limit is associated with the emergence of topological singularities of codimension k, which can be...

4. About some approximation problems for Sobolev maps to manifolds.

Antoine Detaille

05/11/2025, 15:00

In a striking contrast with the classical case of real-valued Sobolev function, a Sobolev map with values into a given compact manifold N need not be approximable with smooth N-valued maps.
This observation, initially due to Schoen and Uhlenbeck (1983), gave rise to a whole area of research concerned with questions related to approximability properties of Sobolev mappings with values into a...

3. Diffuse decay of excess in codimension two

Aria Halavati

05/11/2025, 16:30

Entire critical points of the Abelian-Higgs functional are known to blow down to (generalized) minimal submanifolds of codimension 2. We focus on the multiplicity-one regime, where the blow-down consists of a single sheet. In this talk, I will present results from three papers that together provide the analytical tools to prove that entire critical points in this regime exhibit an improvement...

1. (1st lesson) Regularity for area-minimizing and almost-minimizing currents

Anna Skorobogatova (ETH Zurich)

06/11/2025, 09:00

The Plateau problem concerns the surfaces of least m-dimensional area spanning a given (m-1)-dimensional boundary. To guarantee existence of minimizers and desirable compactness properties for sequences of surfaces, one must consider a weak notion of surface, which allows area-minimizing "surfaces” to have singularities. Two particularly natural frameworks for this problem in arbitrary...

7. (Non)local minimal surfaces and phase coexistence models.( 1st lesson)

Enrico Valdinoci (University of Western Australia)

06/11/2025, 11:00

We present some aspects of the theory of nonlocal minimal surfaces, with special attention to regularity, sheeting, maximum principle, and stickiness. We also link this theory to the study of long-range phase transition and to the analysis of its symmetry properties. Similarities and differences with the classical cases will be outlined.

8. A strict maximum principle for nonlocal minimal surfaces

Serena Dipierro (University of Western Australia)

06/11/2025, 14:00

Suppose that two nonlocal minimal surfaces are included one into the other and touch at a point. Then, they must coincide. But this is perhaps less obvious than what it seems at first glance.

2. Ambrosio-Tortorelli approach to topological singularities and connections with jump minimizing liftings

Roberta Marziani (Università degli studi di Siena)

06/11/2025, 15:00

We study the Gamma-convergence of Ambrosio-Tortorelli-type functionals, for maps u defined on an open bounded set Ω ⊂ R^n and taking values in the unit circle S^1 ⊂ R^2. Depending on the domain of the functional, two different Gamma-limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map u whose measure of the jump set is...

6. Regularity for stable phase transitions

Enric Florit-Simon (ETH Zürich)

06/11/2025, 16:30

We will discuss recent progress in the regularity for stable solutions to the Allen-Cahn equation, with a focus on the connections with minimal surface theory and two influential conjectures of De Giorgi and Yau. Part of this talk is based on recent joint work with Joaquim Serra which classifies all stable solutions to Allen-Cahn with bounded energy density in four dimensions.

10. (Non)local minimal surfaces and phase coexistence models.( 2nd lesson)

Enrico Valdinoci (University of Western Australia)

07/11/2025, 09:00

9. (2nd lesson) Regularity for area-minimizing and almost-minimizing currents

Anna Skorobogatova (ETH Zurich)

07/11/2025, 11:00