Covariance identities may serve as powerful tools for derivation of Sobolev-type inequalities and the study of measure concentration phenomena. After some historical remarks about the Gaussian case, new covariance representations will be discussed for the uniform distribution on Euclidean spheres. Based on a joint work with Devraj Duggal.

In a striking contrast with the situation for classical linear spaces, Sobolev maps with values into a compact manifold \( \mathcal{N} \) \emph{need not} be approximable with smooth \( \mathcal{N} \)-valued maps.

While situations where the strong approximation property are completely characterized since the seminal work of F. Bethuel (1991) and its subsequent generalizations, the weak approximation problem remains widely open. In this talk, I will present a recent result obtained with Jean Van Schaftingen (IRMP, UCLouvain), featuring a new construction to produce analytical obstructions to the weak approximation property of Sobolev mappings.

We develop the spectral analysis of the Jacobi operator on the interfaces of soap-bubble clusters. By Plateau's laws, these always meet in threes at $120^{\circ}$-angles, and thus naturally interact via 3 linearly independent ``conformal" boundary conditions (a mixture of Dirichlet and Robin). This gives rise to a self-adjoint operator, whose spectral properties determine the \emph{stability} of the soap-bubbles -- whether an infinitesimal regular perturbation preserving volume to first order yields a non-negative second variation of area modulo the volume constraint. In essence, stability amounts to verifying a Poincar\'e-type inequality on soap-bubble clusters.

We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters, conjectured to be minimizing total perimeter in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$, are indeed stable. In fact, stability holds for all M\"obius-flat \emph{partitions}, in which several cells are allowed to have infinite volume. In the Gaussian setting, any partition in $\mathbb{G}^n$ ($n\geq 2$) obeying Plateau's laws and whose interfaces are all \emph{flat}, is stable. Our proof relies on a new conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary, and the construction of a good conformally flattening boundary potential. 

Joint work with Botong Xu.

I will discuss a rearrangement Sobolev inequality that originates from the classical Sobolev embedding. The stories in the title refer to my involvement, with different collaborators, over many years, discovering/rediscovering ... formulating and reformulating ... and connecting it with geometric measure theory, interpolation, good-lambda inequalities, Burkholder-Gundy extrapolation ... The talk will avoid technicalities and emphasize how the ideas came about. Although no new results will be presented I still hope that there could be something of interest for experts and newbies alike.

The celebrated Courant nodal domain theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, various attempts have been made to find an appropriate extension of this result to linear combinations of eigenfunctions. It turns out that such a generalisation of Courant's theorem, along with some others, can be achieved by a coarse nodal count, which disregards small oscillations. The proof uses multiscale polynomial approximation in Sobolev spaces and the theory of persistence barcodes originating in topological data analysis. The talk is based on a joint work with L. Buhovsky, J. Payette,  L. Polterovich, E. Shelukhin and V. Stojisavljevic.

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, and has close connections to the known regularity theory for stable minimal surfaces.

I will give an overview of the history of the problem, and discuss joint work in progress with Luca Spolaor and Salvatore Stuvard in which we establish a precise local description of two-dimensional mod(q) area-minimizing hypersurfaces close to branch point singularities, which are the most difficult singularities to tackle in general, due to multiple sheets of the surface collapsing together there. Together with existing recent structural results in this setting, this allows us to obtain a complete local understanding of the behaviour of such surfaces close to most singular points. I will emphasize the difficulties in extending this to higher dimension and codimension.

Given compact Riemannian manifolds M and N and p ∈ (1, ∞), the ques-tion of traces for Sobolev mappings consists in characterising the mappings from∂M to N that can arises of maps in the first-order Sobolev space W ̇ 1,p(M, N ).A direct application of Gagliardo’s characterisation of traces for the linear spacesW ̇ 1,p(M, R) shows that traces of maps in W ̇ 1,p(M, N ) should belong to the frac-tional Sobolev-Slobodecki ̆ı space W ̇ 1−1/p,p(∂M, N ). There is however no reasonfor Gagliardo’s linear extension to satisfy the nonlinear constraint imposed byN on the target.

In the case p > dimM, Sobolev mappings are continuous and thus traces ofSobolev maps are the mappings of W ̇ 1−1/p,p(∂M, N ) that are also restrictionsof continuous functions [2]. The critical case p = dimM can be treated similarlythanks to their vanishing mean oscillation property [2, 3, 6].

The case 1 < p < dimM is more delicate. It was first proved that whenthe first homotopy π1(N ), . . . , π⌊p−1⌋(N ) are trivial, then the trace operatorfrom W ̇ 1,p(M, N ) to W ̇ 1−1/p,p(∂M, N ) is surjective [4]. On the other hand,several conditions for the surjectivity have been known: topological obstructionsrequire π⌊p−1⌋(N ) to be trivial [2, 4] whereas analytical obstructions arise unlessthe groups π1(N ), . . . , π⌊p−1⌋(N ) are finite [1] and, when p ≥ 2 is an integer,πp−1(N ) is trivial [5].

In a recent work, I have completed the characterisation of the cases wherethe trace is surjective, proving that the known necessary conditions turn outto be sufficient [7]. I extend the traces thanks to a new construction whichworks on the domain rather than in the image. When p ≥ dimM the sameconstruction also provides a Sobolev extension with linear estimates for mapsthat have a continuous extension, provided that there are no known analyticalobstructions to such a control.

References
[1] F. Bethuel, A new obstruction to the extension problem for Sobolev maps between manifolds, J. Fixed Point Theory Appl. 15 (2014), no. 1, 155–183.

[2] F. Bethuel, F. Demengel, Extensions for Sobolev mappings between manifolds, Calc. Var. Partial Differential Equations 3 (1995), no. 4, 475–491.

[3] H. Brezis, L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.) 1 (1995), no. 2, 197–263.

[4] R. Hardt, Lin F., Mappings minimizing the Lp norm of the gradient, Comm. Pure Appl. Math. 40 (1987), no. 5, 555–588.

[5] P. Mironescu, J. Van Schaftingen, Trace theory for Sobolev mappings into a manifold, Ann. Fac. Sci. Toulouse Math. (6) 30 (2021), no. 2, 281–299.

[6] R. Schoen, K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982), no. 2, 307–335.

[7] J. Van Schaftingen, The extension of traces for Sobolev mappings between manifolds, arXiv:2403.18738.

We will report on some recent progress on sharp integral inequalities for the Laplace and Legendre transforms, related to classical inequalities in convex geometry such as the Blaschke-Santalo inequality. These inequalities will be established by proving monotonicity along heat or Fokker-Plank flow.  In particular, we will explain how duality (in the form of the Legendre transform) interacts with heat flow.

With their 2004 and 2007 papers, Jean Bourgain and Haim Brezis opened for all of us a new world of previously unrealized Sobolev inequalities in L1.  A decade later, after many interesting contributions by a number of authors, Jean Van Schaftingen characterized those vector-valued differential operators which support Lebesgue scale L1 inequalities, the so-called canceling operators.  As it is known that the gradient supports a Lorentz scale inequality, a trace inequality, and estimates for spaces around L1 which are better than those known for the Riesz potential in the same regime, it is natural to wonder whether the cancelation condition is sufficient to obtain similar inequalities for general operators.  In this talk we discuss the state of the art on these several inequalities for canceling operators and report on recent progress in the area, including contributions from Jean Van Schaftingen, Felipe Hernandez, Bogdan Raita, Dmitriy Stolyarov, Dominic Breit, Andrea Cianchi, and the speaker.

Kari Astala proved Gehring's conjecture by showing that the W^{1,2} solution of Beltrami equation self improves up to W^{1, 1+1/k -epsilon}. Here $k$ is the norm of Beltrami coefficient. Such maps are called quasi-regular.

But there are weakly quasi-regular maps, these are solutions in W^{1,1+k}. There was a problem of Astala-Iwaniec-Saksman whether weakly quasiregular maps are actually quasi-regular.This was proved by Stefanie Petermichl and myself, and I will talk about this result and its connection with harmonic analysis.

The solution of wave equations in spatial dimension n > 1 can lose regularity in L^p if p is not 2. In fact, if we only know that the initial data for the wave equation in R^{n+1} is in L^p with n > 1 and p not equal to 2, then at time t = 1, the solution can concentrate in space so much, that it fails to be in L^p(R^n). This phenomenon can be captured by a variant of the usual Sobolev space, where one takes into account the frequencies of the function in different directions. I will explain some ideas behind this. Joint work with Andrew Hassell, Pierre Portal and Jan Rozendaal.

It is well-known that even under favorable conditions that ensure both existence and (partial) regularity of minimizers their uniqueness is not guaranteed. In this talk we show how exactly uniqueness is connected to convexity of the variational integral. We also give some results showing how uniqueness (and regularity) of minimizers can be ensured using smallness conditions on the data. It is important to emphasize that these smallness conditions are too weak to allow for a direct application of any known Implicit Function Theorem.

The talk is based on joint work with Judith Campos Cordero (Mexico City), Bernd Kirchheim (Leipzig) and Jan Kolar (Prague).

Let (M, g) be a complete Riemannian manifold, ∆ the Laplace Beltrami operator and ∇ the Riemannian gradient. We discuss the validity of inequalities ∥|∇f|∥p ≲ ∆ 1 2 f p and  ∆ 1 2 f p ≲ ∥|∇f|∥p  , 1 < p < ∞, under various geometric assumptions. The role of Poincar ́e  inequalities will be highlighted. These are joint works with Baptiste Devyver.

I will survey an interesting analogy between martingales and functions. Though such similarities are widely known in harmonic analysis, the particular discrete martingale setting found first by S. Janson in late 70s and then rediscovered by Ayoush, Wojciechowski, and me around 2018, seems to shed light on problems for vectorial functions in L_1 norms and vectorial measures. Though the idea has lead to certain progress on so-called Bourgain—Brezis inequalities, there is no rigorous justification why martingales and functions on Euclidean space behave in a similar manner. In recent years, together with Ayoush, Spector, and Wojceichowski, we found new instances where the analogy works (for example, for rank-one theorems in the spirit of G. Alberti). This naturally settles many interesting questions.

The characterization of constant maps lacking classical derivatives (Bourgain et al.) sparked significant interest in the connection between derivatives and limits of BMO semi-norms. Ambrosio et al. found a description of perimeter as limits of BMO-type seminorms on ε-size cubes (as ε →0), and this was later extended to describe the total variation of SBV functions (BV functions without Cantor derivative). However, the uniformity of the ε-scale constraint limits the description of general BV functions, which poses the following question: can we relax this to achieve a BMO-type characterization for all BV functions?

Since the characterizing of constant maps lacking a classical derivative (Bourgain et al.), there has been a growing interest on the link between derivatives and limits of BMO seminorms. Ambrosio et al. described perimeter using mean oscillations, and this was later extended to describe the total variation of SBV functions as limits of BMO-type seminorms on ε-size cubes (as ε →0). However, this uniform ε-size constraint hinders a precise description for general BV functions. Is it possible to relax this model to find a BMO-type description of BV functions? 

In this talk, I will address this question by introducing the concept of local Poincaré constant (LPC), as a tool to understand the relation between the mean oscillation and the total variation at small scales. Along with this new quantity, I will introduce a geometric relaxation of the functionals by Ambrosio et al., by considering cubes of side-length smaller than or equal to ε. I will then explain how our functionals converge (as ε→0) to a functional defined on BV, represented by integration in terms of the LPC and the total variation. 

Furthermore, I will discuss a cell-formula representation for the LPC, defined as the maximum mean oscillation among BV blow-ups. This reveals that our new functional extends the original one from SBV to the broader class of BV functions. Finally, I will share a rigidity result for one-dimensional functions with additive oscillation.

This presentation includes joint work with Paolo Bonicatto (Trento), Sergio Conti (Bonn) and Giacomo del Nin (MPI Leipzig).