Welcome address

Some known and less known remarks about the Jacobi formula for the Jacobian

• Jean Mawhin

Petru Mironescu, in a series of works alone or in collaboration, especially with the late Haïm Brezis, has shown the important role played by a formula of Jacobi in the definition and study of the distributional Jacobian. In this lecture, we recall some aspects of the tortuous history of this formula described in [1], and comment on some new historical facts and some new links to classical problems not included in [1]. [1] H. Brezis, J. Mawhin and P. Mironescu, A brief history of the Jacobian, Comm. Contemp. Math. 26 (2) (2024), 233001.

The weak approximation problem for manifold-valued Sobolev mappings

• Antoine Detaille

In a striking contrast with the classical situation of real-valued Sobolev functions, a Sobolev mapping taking its values into a manifold N need not be a limit of smooth N-valued maps with respect to the strong convergence. A natural idea to try restoring the approximation property by smooth maps is to work with a weaker notion of convergence. Unlike the strong approximation problem, which is by now considered as well-understood, the picture of the weak approximation problem remains yet widely open. In this talk, I will present the history of this problem, some well-known results, as well as some recent progress.

Extrapolation methods for Γ-convergence and applications to image processing

• Oscar Dominguez

In this talk, we introduce, via the Gagliardo completion, an extrapolation framework within De Giorgi’s Γ-convergence theory and develop its applications to variational problems arising in image processing, in particular, the Rudin-Osher-Fatemi (ROF) model. Special emphasis is given to new tools connecting extrapolation theory and variational analysis, specifically the extrapolation of compactness and the construction of variational K-functionals. These tools allow for the derivation of quantitative rates of convergence of minimizers. As an application, we resolve the inverse regularity problem in ROF: Regularity of the observed image that guarantees a prescribed rate of convergence to the solution of ROF is governed by the decay of variational K-functionals, which is explicitly captured by the Brezis-Van Schaftingen-Yung spaces. Moreover, this extrapolation framework provides a unified perspective on several limiting results, including the celebrated Bourgain-Brezis-Mironescu formula and its Γ-convergence counterparts due to Ponce.

Minimizing N-harmonic maps with partially constrained boundary conditions of prescribed degree

• Adriano Pisante

In this talk we will focus for $N\geq 2$ on fractional $W^{1-1/N,N}$ maps from an $(N−1)$-dimensional sphere into itself as traces of maps with finite energies modelled on the $N$-Dirichlet integral. We consider energy minimization among maps with trace of prescribed topological degree. Our goal is to prove existence and quantitative stability for minimizers in homotopy classes. Due to the lack of compact Sobolev embeddings and conformal invariance, minimizing sequences may fail to converge strongly, leading to a concentration-compactness alternative and bubbling phenomena. When they occur, these concentration effects are investigated using PDEs tools like blow-up analysis and measure-theoretic tools such as Cartesian currents. Finally, we discuss genericity results under perturbations of the energy functional and we present some open problems. This is a joint ongoing work with Xavier Lamy (Toulouse).

Point singularities of manifold-valued Sobolev maps and approximability by smooth maps

• Giacomo Canevari

Manifold-valued Sobolev maps naturally arise in variational problems and models of partially ordered media, where the topology of the target can enforce the formation of singularities. These singularities may act as obstructions to familiar constructions, such as approximation by smooth maps. In this talk, given a positive integer $p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $N$. The target manifold is required to satisfy suitable topological conditions; however, in contrast with previous works in this area, we do not assume that $N$ is $(p-1)$-connected. Using tools from geometric measure theory --- namely, flat chains with coefficients in an appropriate homotopy group of $N$ --- we associate to each map $u$ in the weak sequential closure of smooth maps an object that captures its point singularities. The vanishing of this object characterizes local strong approximability by smooth maps. This talk is based on joint work with G. Orlandi (Verona).

On CMC 1-immersions of a closed surface into Hyperbolic 3-manifolds

• Gabriella Tarantello

Constant Mean Curvature (CMC) c-immersions of a closed orientable surface S (with genus g ≥ 2) into hyperbolic 3-manifolds emerged by the work of Uhlenbeck in connection with irreducible representations of the fundamental group of S into the Mobious group. In view of Bryant surfaces, the value c =1 of the mean curvature enters as a “critical” (yet significant) parameter in this context. It is responsible for natural “blow-up” phenomena, and for this reason the moduli space of such (CMC) 1-immersions remained elusive for long time. In recent work, we showed how to encompass the blow-up situation in terms of sharp orthogonality conditions, involving the image Z of the Kodaira map, for genus g = 2, and the (g −1)-secant variety of Z, for larger genus. In this way, under a generic condition, we can ensure existence and uniqueness of (CMC) 1-immersions of S into (germs) of hyperbolic 3-manifolds, and obtain a parametrization of the corresponding moduli space in terms of the tangent bundle of the Teichmueller space of S.

Rigidity and flexibility for infinite double bubbles

• Mike Novack

The classical double bubble theorem characterizes in every dimension surface-area minimizing partitions of R^n into three chambers, two of which have prescribed finite volume. In this talk we will discuss a variant of the double bubble problem in which two of the chambers have infinite volume. Unlike the standard double bubble problem, the characterization of minimizers depends on the dimension. Based on joint works with L. Bronsard, R. Neumayer, and A. Skorobogatova.

mod(q) area-minimizing surfaces: structure and singularities

• Anna Skorobogatova

One possible framework in which to study the Plateau problem is by using currents with multiplicities modulo q, for a fixed integer q. This setting allows for minimizing surfaces to exhibit codimension 1 singularities like triple junctions, which are seen in physical soap films, and has close connections to the known regularity theory for stable minimal submanifolds, while at the same time providing the framework of a minimization problem. I will give an overview of the history of the problem and discuss recent structural results, including joint work in progress with Luca Spolaor and Salvatore Stuvard for 2-dimensional mod(q) minimizers of arbitrary codimenson, where we are able to obtain a fairly complete local structural picture of the surface and its singular set, in the spirit of the works of Almgren-Chang and De Lellis-Spadaro-Spolaor for two-dimensional area-minimizing integral currents.

On Minimizing Harmonic Maps with Tangential Boundary Anchoring

• Dominik Stantejsky

Motivated by experiments with nematic liquid crystal droplets, we study harmonic maps on the three-dimensional unit ball that arise as minimizers of the one-constant approximation of the Oseen-Frank energy subject to strong anchoring planar boundary condition. Through a reflection method, we are able to study regularity of minimizers close to the boundary. We also obtain results on the type and location of defects that can occur, such as boundary ``boojums'' and interior vortices. The talk is based on joint work with Lia Bronsard and Andrew Colinet.

3D local minimizers to vector Allen-Cahn converging down to asymmetric tetrahedral cones

• Peter Sternberg

We consider the vector Allen-Cahn (Modica-Mortola) energy with a 4-well potential. The goal is to produce a stable, diffuse version of a stable, tetrahedral cone in the absence of any symmetry assumptions. This is accomplished via standard Gamma-convergence theory once we can exhibit a bounded 3D domain which is partitioned by such a cone into a locally minimizing 4 chamber configuration with respect to weighted surface area in a sufficiently weak topology. This is joint work with Abhishek Adimurthi.

Minimisers of nonlocal energies: the effect of anisotropy

• Lucia Scardia

Nonlocal energies are continuum models for large systems of particles with long-range interactions. Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers. But what can be said in the case of anisotropic kernels? I will present some results and partial answers in this direction obtained in collaboration with Maria Giovanna Mora, and with José Antonio Carrillo, Rupert Frank, Luca Rondi, Joan Mateu, Joan Verdera and Riccardo Cristoferi.

Geometry and Topology of Mappings with Derivatives of Small Rank

• Piotr Hajłasz

The main theme of this talk is the study of mappings—primarily continuously differentiable and Lipschitz—that are critical everywhere, in the sense that the rank of their derivative is small at every point. Such mappings arise naturally in a variety of contexts across analysis, geometry, and topology. I will show how ideas from different areas combine to address fundamental questions about these mappings, with emphasis on problems related to approximation, homotopy, contact structures, Heisenberg groups, and analysis on metric spaces.

A symmetry result related to the baby Skyrme model

• Etienne Sandier

We show a symmetry result in an exterior domain in dimension 2 for minimizing harmonic maps to the sphere perturbed by a jacobian term. We hope this could be useful in the study of various Skyrmion models.

Integrability of the full gradient for Griffith minimizers in dimension 2

• Antoine Lemenant

The Griffith functional, arising in variational models of crack propagation and linearized elasticity, shares many features with the classical Mumford–Shah functional. A major difficulty in this vectorial setting is that the energy only controls the symmetric part of the gradient, rather than the full gradient itself. In this talk, I will present a strategy to obtain the L^2-integrability of the full gradient of two-dimensional Griffith minimizers through local Korn-type inequalities. This is part of a recent joint work with Camille Labourie and Lorenzo Lamberti (Université de Lorraine, Nancy). Along the way, and because this is how the story behind this work began, we will briefly discuss a conjecture formulated by E. De Giorgi in 1991 on the behavior of Mumford–Shah minimizers near their singular set. In dimension two, this question can be addressed using tools developed by G. David, A. Bonnet, and J.-C. Léger, and I will explain how these ideas naturally lead to the Griffith setting.

Another look at Boojums in a Liquid Crystal Model

• Itai Shafrir

Several authors (Alama, Bronsard, Golovaty, Mironescu) have recently studied the minimizers $\{u_\varepsilon\}$ of the energy $ E_\varepsilon(u) = E^{g,\alpha}_\varepsilon(u)$, given by \begin{equation} %\label{eq:1} E_\varepsilon(u) = \frac{1}{2} \int_\Omega \left( |\nabla u|^2 + \frac{1}{2\varepsilon^2}(1 - |u|^2)^2 \right) \, dx + \frac{1}{2\varepsilon^s} \int_{\partial\Omega} {\widetilde W}(u,g) \, ds \qquad (1) \end{equation} over $H^1(\Omega, \mathbb{C})$, where \begin{equation} %\label{eq:2} {\widetilde W}(u,g) = \frac{1}{2}(|u|^2 - 1)^2 + [(u, g) - \cos\alpha]^2 \hspace{5em} (2) \end{equation} with $\alpha \in (0, \pi/2)$ and $s \in (0,1)$, and where $g : \partial\Omega \to S^1$ is a smooth function of degree $D \ge 1$. The motivation comes from a thin-film limit of the Landau-de Gennes energy for a nematic liquid crystal. For $s < \frac{1}{2\big((\alpha/\pi)^2+(1-\alpha/\pi)^2\big)}$, it was shown that in the limit $\varepsilon \to 0$, boundary defects (called ``boojums'') appear. However, the first term in (2) is not part of the physical model, but rather added artificially to the energy for "technical reasons". There are two main goals of our work: * To show that the results obtained for the energy (1) remain valid for the energy derived from the physical model, namely with $W(u,g) = [(u, g) - \cos\alpha]^2$, rather than $ {\widetilde W}(u,g)$ as defined in (2). * To obtain more precise information on the minimizers $\{u_\varepsilon\}$ as $\varepsilon \to 0$ in the ``boojums regime"; e.g., to show that $|u_\varepsilon| \to 1$ uniformly on $\bar\Omega$. This is a joint work with Dmitry Golovaty.

Conference dinner at [Aux Pieds Sous La Table][1] [1]: https://www.google.com/maps/place/Aux+Pieds+sous+la+Table/@43.6096419,1.4367035,17z/data=!3m1!4b1!4m6!3m5!1s0x12aebb5e1c781ccb:0xca7f21f0d48e84d7!8m2!3d43.6096419!4d1.4392784!16s%2Fg%2F11b6dp0y28?entry=ttu&g_ep=EgoyMDI2MDUyNy4wIKXMDSoASAFQAw%3D%3D

Traces of Sobolev Spaces

• Giovanni Leoni

This talk explores the trace theory of Sobolev spaces in the critical $p=1$ case. After reviewing the first-order case in which the trace of $W^{1,1}(\Omega)$ is $L^1(\partial\Omega)$, we focus on higher-order Sobolev spaces whose traces are Besov spaces. We will present new and classical results on the trace and lifting problem for these spaces.

Phase-field model for polycrystals

• Adriana Garroni

I will present a recent result obtained in collaboration with S. Conti, V. Crismale and A. Malusa. We consider a phase-field model \`a la Ambrosio Tortorelli in order to approximate sharp interface energies for grain boundaries accounting for the Read and Shockley law for small angle grain boundaries. The independent variable takes values in the orthogonal group O(d) modulo a lattice point group G, reflecting the crystallographic symmetries of the underlying lattice. We also consider a discrete version of this approximation which can be applicable to the reconstruction of grain boundaries from imaging data.

Quantitative concentration and Wehrl-type entropy inequalities for homogeneous polynomials

• Maria Angeles Garcia Ferrero

Let us consider two notions of concentration for homogeneous polynomials in d complex variables on the unit sphere: a local notion measuring the fraction of the L2-norm supported on a measurable subset and a global notion given by the generalized Wehrl entropy. Lieb and Solovej proved that the extremizers in both cases are monomials up to a unitary rotation. Their result generalizes the one by Lieb in 1978 on the Wehrl entropy conjecture for coherent states in representations of the Heisenberg group to symmetric representations of the groups SU(d). In this talk, we will focus on the stability of the previous inequalities. Namely, if the concentration is close to the optimal one, we will quantify how close the polynomial is to the extremizers. This is obtained in full generality in the case d=2, while in the case of higher dimensions restrictions on the size of the subset or on the degree of the polynomials arise. We will finally recover analogous stability results in the Bargmann–Fock space. This is a joint work with Joaquim Ortega-Cerdà (UB-CRM).

Stochastic homogenization of the double-porosity model

• Elise Bonhomme

In this talk, I will present joint work with Mitia Duerinckx and Antoine Gloria, in which we establish the homogenization of the so-called "double-porosity" model in a random setting where the resonant inclusions are neither uniformly bounded nor uniformly separated. This mesoscopic model (used to describe flows in fractured porous media) arises as the limit of a diffusion process in a highly heterogeneous material composed of two pure phases: a connected "healthy" phase (with conductivity of order 1), randomly perforated by a dense network of small inclusions belonging to a second, nearly "soft" phase whose conductivity scales like the square of their size and tends to zero. In this specific regime, so-called resonance phenomena occur, in the sense that the homogenized model retains memory of nontrivial interactions between the micro and macroscopic scales of the material.

A topological toolbox for Sobolev maps

• Augusto Ponce

Classical works by F. Bethuel and by F. Hang and F.-H. Lin identified the local and global topological obstructions preventing smooth maps from being dense in the Sobolev space $W^{1,p}(M^{m};N^{n})$ between two Riemannian manifolds when $p<m$. These obstructions are related to the extension of continuous maps from certain subsets of $M^{m}$ into $N^{n}$. Inspired by the notion of modulus introduced by B. Fuglede, one can capture in a robust way generic properties of Sobolev functions. Combining degree-theoretic ideas for VMO maps developed by H. Brezis and L. Nirenberg, it becomes possible to determine whether a given Sobolev map $u \colon M^{m}\to N^{n}$ does not carry topological obstructions to smooth approximation, even when such obstructions exist at the level of the manifolds $M^{m}$ and $N^{n}$. This talk is based on recent joint work with P. Bousquet (Toulouse) and J. Van Schaftingen (UCLouvain).emphasized text

On s-Stability of W^{s,n/s}-minimizing maps between spheres in homotopy classes

• Katarzyna Mazowiecka

We consider maps between spheres $\mathbb S^n$ to $\mathbb S^\ell$ that minimize the Sobolev-space energy $W^{s,n/s}$ for some $s \in (0,1)$ in a given homotopy class. The basic question is: in which homotopy class does a minimizer exist? This is a nontrivial question since the energy under consideration is conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of homotopy classes that generates the whole homotopy group $\pi_{n}(\mathbb S^\ell)$. Explicit examples are known if $n/s = 2$ or $s=1$. In this talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of homotopy classes in which minimizers exist can be chosen stable. We also discuss that the minimum $W^{s,n/s}$-energy in homotopy classes is continuously depending on $s$. Joint work with A. Schikorra

Existence of critical points of the n-Ginzburg-Landau energy with prescribed degree 1

• Rémy Rodiac

We will discuss the existence of critical points of the $n$-Ginzburg-Landau energy in the unit ball of $\mathbb R^n$ with prescribed degree one on the boundary. We will first prove that there does not exist any minimizer of this energy among maps with prescribed degree $d\neq 0$. Then we will show that we can devise a min-max scheme which allows us to prove the existence of critical points with prescribed degree one 1 if $\epsilon$, the inverse of the Ginzburg-Landau parameter, is large enough. A bubbling analysis is necessary because of the lack of compactness of the problem. This is a joint work with Dorian Martino and Katarzyna Mazowiecka.

Quantitative topology in Hoelder- and fractional Sobolev-spaces

• Armin Schikorra

For maps between -spheres we can estimate their degree in terms of their Hoelder- and fractional Sobolev-spaces norm in a sharp way. We will discuss some approaches about a similar estimate for maps from $\mathbb S^3$ to $\mathbb S^2$ and the Hopf degree. Based on joint works with A. Grzela, K. Mazowiecka, J. Van Schaftingen

A high-splay limit of the Oseen-Frank model with electrostatics

• Dmitry Golovaty

I will present a recent result demonstrating that the Oseen-Frank model which explicitly incorporates electrostatics converges to its highly anisotropic counterpart in which the splay contribution dominates. This is a joint work with Peter Sternberg.

Minimality of the vortex solution for Ginzburg-Landau systems

• Radu Ignat

We consider the standard Ginzburg-Landau system for N-dimensional maps defined in the unit ball for some parameter $\epsilon>0$. For a boundary data corresponding to a vortex of topological degree one, the aim is to prove the (radial) symmetry of the ground state of the system. We show this conjecture in any dimension N≥7 and for every $\epsilon>0$, and then, we also prove it in dimension N=4,5,6 provided that the admissible maps are curl-free.