Compensated compactness of A-free measures on cones with quantified aperture
Adolfo Arroyo Rabasa (University of Pisa)
Abstract: The De Philippis–Rindler theorem establishes a profound geometric rigidity for PDE-constrained measures by demonstrating that A-free fields cannot concentrate mass along one-dimensional semi-lines outside the Tartar wave cone. Extending this rigidity to broader conical regions has remained a formidable problem. Because such cones possess a non-empty interior, the strict one-dimensional sign constraints underpinning the De Philippis–Rindler proof are lost, exposing the problem to unbounded singular integrals.
In this talk, I will discuss recent work establishing this phenomenon on conical regions. By constructing specialized A-free quasiconvex functions with targeted concavity properties, we derive a "spreading inequality" for A-free Young measure concentrations, showing that such microstructures cannot have arbitrarily small variance if their center of mass lies outside the wave cone. I will outline how this geometric penalty leads to an approximate compensated compactness result: L1-bounded A-free sequences are forced to be equi-integrable, provided their targets are asymptotically restricted to a cone around a subspace away from the wave cone. We explicitly quantify the allowed aperture of such cones, showing it is bounded by a power of the subspace's distance to the wave cone.
This is joint work with Averil Aussedat (Uni Pisa)
Alberti representations and rectifiability
David Bate (University of Warwick)
Abstract: An Alberti representation of a (finite, Borel) measure is a decomposition into 1-rectifiable measures. By Fubini's theorem, Lebesgue measure on [0,1]^n has n "independent" Alberti representations, each one consisting of curves parallel to a coordinate axis. This naturally leads to the existence of n independent Alberti representations of n-rectifiable subsets of Euclidean space, and can be extended further to rectifiable subsets of an arbitrary metric space.
This talk will consider the converse statement.
Prior work of myself and Sean Li shows that, if the n-dimensional Hausdorff measure of a metric space X has n independent Alberti
representations, then X is n-rectifiable provided X has positive lower n-density almost everywhere. This result led to important consequences, such as a non-linear counter part to the Besicovitch--Federer projection theorem and established the rectifiability of classes of metric currents, metric manifolds and differentiability spaces.
This talk will present recent work with Julian Weigt that shows that the existence of n independent Alberti representations alone is sufficient to ensure rectifiability.
Structure of metric 1-currents: approximation by normal and polyhedral currents and representation results via fragments
Emanuele Caputo (University of Trento)
Abstract: The goal of the talk is to present recent progress in Ambrosio-Kirchheim’s theory of metric currents. We first present a structure theorem for metric 1-currents as superpositions of oriented 1-rectifiable sets, called curve fragments, in complete and separable metric spaces, generalizing a result of Schioppa. This provides an instance in which pure 1-unrectifiability is characterized by the absence of 1-currents. Lack of metric currents in all weak tangents implies that the metric space is not only purely unrectifiable, but in fact biLipschitz to a snowflake. One crucial tool for the superposition result is a strict polyhedral approximation in the one-dimensional case. We provide examples showing that such an approximation fails for higher-dimensional currents in infinite-dimensional Hilbert spaces and certain Banach spaces. These are joint works with D. Bate, N. Cavallucci, J. Takáč, P. Valentine, and P. Wald.
A metric approach to the topology of 3-manifolds with nonnegative Ricci curvature
Alessandro Cucinotta (University of Oxford)
Abstract: A theorem of Gang Liu states that a complete noncompact 3-manifold with nonnegative Ricci curvature either has nonnegative sectional curvature or is homeomorphic to R^3. Liu’s proof is based on minimal-surface techniques developed by Schoen, Yau, and others.
I will present an alternative proof of this result, obtained in joint work with Mattia Magnabosco and Daniele Semola. The proof combines ideas from metric geometry and geometric measure theory, and leads to new topological restrictions for manifolds with nonnegative Ricci curvature in arbitrary dimension.
On divergence operators: Free space and vanishing charges
Thierry De Pauw (Westlake University)
Abstract: We use localized topologies to prove existence and optimal regularity results for the divergence equation $\mathrm{div}(v) = F$ in critical cases $v \in L_1(\Omega;\mathbb{R}^m)$ or $v \in C_0(\Omega;\mathbb{R}^m)$, i.e. we characterize those $F$ for which a solution $v$ exists whose norm is bounded by an appropriate norm of $F$. We assume $\Omega$ satisfies a Poincaré inequality or an extension property. We apply the general theory to give examples of admissible $F$ in each case.
A Simon-Smith min-max theory for anisotropic minimal surfaces with genus bound
Antonio De Rosa (Bocconi University)
Abstract: After briefly recalling our recent min-max constructions of anisotropic minimal surfaces via the Almgren-Pitts and Allen-Cahn approaches, I will present a Simon-Smith min-max theory for anisotropic minimal surfaces with controlled genus. One of the key ingredients is an anisotropic analogue of the theorem of Meeks-Simon-Yau: every minimizing sequence within a fixed isotopy class converges to a smooth stable anisotropic minimal surface, with lower semicontinuity of the genus. This result strengthens previous existence theorems for anisotropic Plateau problems and provides the compactness needed for a topological min-max construction. As an application, we prove the existence of closed anisotropic minimal surfaces with controlled genus in arbitrary closed three-manifolds. This is joint work with Aria Halavati and Ling Wang (Bocconi University)
Boundary rectifiability and compactness of integral currents via BV functions
Giacomo Del Nin (Max Planck Leipzig)
Abstract: We present a proof of the boundary rectifiability theorem for currents that is based on the theory of BV functions. We use a cylindrical projection argument to reduce to the case of top-dimensional currents (i.e., integer-valued BV functions), to which we can apply De Giorgi's structure theorem. As a consequence we also present a proof of the compactness theorem for integral currents that is ultimately based on the BV theory.
The product of Sierpiński carpets does not attain its conformal dimension
Sylvester Eriksson-Bique (University of Jyväskylä)
Abstract: The attainment problem asks when a quasisymmetric equivalence class of fractals contains a space minimizing Hausdorff dimension. This problem is well known and difficult, but for a long time even the simplest instances of it have been open problems, such as the Sierpiński carpet S_3. We show that S_3^k does not attain its conformal dimension for k\geq2. The proof involves a fundamentally new tool: the use of general p-energies for fractals. This is joint work with Riku Anttila.
Concepts in hyperbolic functional analysis
Nicola Gigli (SISSA)
Abstract: From recent progresses in the study of smooth and nonsmooth Lorentzian structures it emerges the need of a functional-analytic theory where, among other things, the relevant norms satisfy a reverse triangle inequality. Aim of the talk is to show that perhaps such a theory is possible.
How to prescribe the approximate derivative of a homeomorphism?
Zofia Grochulska (University of Jyväskylä)
Abstract: Study of the interplay between topological and analytical properties of homeomorphisms equipped with a derivative goes back to works of John Ball in nonlinear elasticity. We will see that, under mild assumptions, it is possible to find an almost everywhere (a.e.) approximately differentiable homeomorphism of the Euclidean unit cube with a given (prescribed) derivative a.e. I will discuss the connections of this result with some recent results about Sobolev homeomorphisms and show a few highlights of the proof. This is joint work with Paweł Goldstein (University of Warsaw) and Piotr Hajłasz (University of Pittsburgh).
Spectral synthesis, fractals and complexity
Alex Iosevich (Rutgers University)
Abstract: We are going to discuss variants of the classical spectral synthesis problem, introduced by Agmon and Hormander in the 70s, and developed systematically by Agranovsky-Narayanan, Senthil Raani and others. The basic question is, given a locally integrable function with the Fourier transform supported on a set of a given upper packing dimension, what is the threshold $p_0$ such that if the function is in $Lp$ for $p$ smaller than this threshold, then it is identically zero. Connections with partial differential equations and analogous manifestations on compact Riemannian manifolds will also be discussed.
A wetting theorem for soap films
Francesco Maggi (University of Texas at Austin)
Abstract: In this joint work with Michael Novack and Daniel Restrepo we analyze the convergence of a wet soap films model to its “dry” limit, and prove that wet regions converge in Hausdorff distance to the set of Plateau type singularities of the film.
Towards understanding the geometric structure of differentiability sets
Olga Maleva (University of Birmingham)
Abstract: Universal differentiability sets (UDS) contain a point of differentiability of every Lipschitz function. Of special interest are UDS which are null, of small dimension and so on. While the first examples were in Euclidean spaces of dimension 2 and above, more recently UDS have been considered and constructed in various types of normed and metric spaces. It is an open problem to describe UDS in geometric measure theory terms. I will talk about recent progress in related questions on differentiability which has potential to advance towards the description of closed UDS.
Singular integrals with special structure: dyadic methods that capture invariance
Henri Martikainen (Washington University in St. Louis)
Abstract: Singular integral operators (SIOs) have played a key role in many results in geometric measure theory in the last few decades. Indeed, they connect to PDEs via the link between elliptic operators and SIOs, and then to uniform rectifiability via David–Semmes style results. A deep understanding of the mapping properties of SIOs is central to these arguments, and this understanding usually comes via some T1/Tb style boundedness characterizations. In this talk I discuss some modern variants of this singular integral side of the theory. The proofs of Tb theorems are based on multiresolution analysis, and dyadic versions of these are especially important, since they are what is needed in the non-doubling settings that arise in many GMT problems. The fundamental question I study is the following. If we have singular integrals with very particular invariances, can we develop adapted dyadic multiresolution methods — refined T1 theorems, if you will — that capture the extra structure, namely, their particular invariance? That is, for certain special subclasses of SIOs, we aim to give very sharp dyadic representations of them so that their special structure is not lost in the process. Our goal is to reap the benefits of the special structure by being able to prove enhanced estimates for the SIOs that are not true for general SIOs, and this is best done on the dyadic side. But it can only be done if the invariance is maintained. In particular, I will discuss our recent work in two key situations — flag SIOs and Zygmund-invariant SIOs.
Currents with coefficients in a group and branched transport problems
Annalisa Massaccesi (University of Padua)
Abstract: In this seminar I will review the theory of flat G-chains, as they were introduced by H. W. Fleming in 1966, and currents with coefficients in groups with the aim of showing some recent applications to variants of the branched optimal transport. One development of the theory concerns its application to the Steiner tree problem and other minimal network problems which are related with a Eulerian formulation of the branched optimal transport. Starting from a 2016 paper by A. Marchese and myself, I will show how these problems and their variants are equivalent to a mass-minimization problem in the framework of currents with coefficients in a (suitably chosen) normed group. The variants I'm referring to include the multicommodity flow, the mailing problem and new models for robust and resilient traffic plans, as shown in a recent paper in collaboration with L. De Masi and A. Marchese.
Parabolic rectifiable and fractal sets
Pertti Mattila (University of Helsinki)
Abstract: Parabolic metric in $\mathbb{R}^n\times\mathbb{R}$ is induced by the parabolic 'norm' $\|(x,t)\|=(|x|^2+|t|)^{1/2}$. The related Hausdorff measures have been used for a long time in connection of the heath equation and other parabolic PDEs. During the last decades parabolic David-Semmes uniform (quantitative) rectifiability has been developed by a number of authors, with many deep connections to harmonic analysis, PDEs and other topics. In the talk I shall discuss more recent work on parabolic Besicovitch-Federer qualitative rectifiability, and on parabolic fractal geometry.
Analysis vs. Geometry for the Regularity Problem for elliptic operators in Rough Domains
Mihalis Mourgoglou (EHU Bilbao)
Abstract: A central theme in modern harmonic analysis and elliptic PDE is the interplay between the analytic properties of boundary value problems and the geometry of the underlying domain. In this talk, I will discuss this principle in the context of the Regularity problem for divergence-form elliptic operators in rough domains.
On the one hand, quantitative geometric assumptions on the boundary, such as uniform rectifiability, imply solvability of the Regularity problem with Sobolev data. On the other hand, solvability itself carries geometric information. I will describe recent results showing that the solvability of the Regularity problem characterizes uniform rectifiability of the boundary, thus establishing an equivalence between analytic and geometric regularity.
I will also discuss a qualitative counterpart of this picture. In this setting, quantitative estimates are replaced by qualitative assumptions, and solvability leads naturally to rectifiability of the boundary. This perspective connects boundary value problems, harmonic measure, and geometric measure theory, and suggests new directions toward free boundary problems where geometry is recovered from analytic information.
A topological characterisation of indecomposability and related results
Enrico Pasqualetto (University of Jyväskylä)
Abstract: I will discuss indecomposability of sets of finite perimeter in the setting of PI spaces (i.e. doubling metric measure spaces supporting a 1-Poincaré inequality).
The focus of the presentation will be on a recent result, obtained in collaboration with Paolo Bonicatto and Panu Lahti, which provides a topological characterisation of indecomposability: a set of finite perimeter is indecomposable if and only if it has a representative that is open and connected in the 1-fine topology. We proved this statement (which is new even for Euclidean spaces) in a class of PI spaces containing all Riemannian manifolds, Carnot groups and finite-dimensional RCD spaces. I will also discuss two related decomposition theorems, stating that every set of finite perimeter can be decomposed into maximal indecomposable components and that every BV function can be written as a series of monotone BV functions.
On metric currents and their connection to differentiability and PDEs
Jakub Takáč (University of Trento)
Abstract: A metric $k$-current is an appropriate generalisation of an oriented $k$ dimensional manifold, which makes sense in general metric spaces. The way this is done is by axiomatizing properties of the \emph{action} of the $k$-current $T$ on $k+1$ tuples of Lipschitz functions $(f, \pi_1, \dots, \pi_k)$ which intuitively correspond to differential $k$-forms:
$$
(f, \pi_1, \dots, \pi_k) = f \textnormal{d} \pi_1\wedge \dots \wedge \textnormal{d}\pi_k.
$$
The remarkable property of the axiomatisation is that it leads to very fruitful theory, allowing one to use currents as suitable tools for a number of geometric problems, yet the structure of general metric $k$-currents is not well understood even in $\mathbb{R}^d$.
The problem lies in the fact that one of the axioms, \emph{joint continuity}, is very difficult to understand. Since the definition is via duality with the forms of the type $f\textnormal{d}\pi_1\wedge\dots\wedge\textnormal{d}\pi_k$, one natural way to understand the structure of metric currents is to understand the differential forms which arise in this way. This naturally leads to questions concerning differentiability of Lipschitz maps and regularity of PDEs (at least when the ambient space is Euclidean). I will discuss the recent progress in this direction.
Dimension drop for harmonic measure
Xavier Tolsa (UAB Barcelona)
Abstract: In this talk I will survey the results obtained in recent years regarding the Hausdorff dimension of harmonic measure. In particular, I will talk about the so called dimension drop for harmonic measure. This is the phenomenon that occurs when the dimension of harmonic measure is smaller than the dimension of the boundary of the domain. I will mainly focus in the particular case of Ahlfors-regular sets of codimension larger than one, where there have been some recent advances in connection with a question posed by A. Volberg.
Dual attractors of iterated function systems and the dimension drop conjecture
Sascha Troscheit (Uppsala University)
Abstract: The dimension drop conjecture (or overlaps conjecture) is one of the central open questions in fractal geometry. It asserts that the only way for a self-similar set to have Hausdorff dimension strictly less than its "naturally predicted" dimension is for there to be an exact overlap in its construction. Much progress has been made on the topic over the last ten years, but most of it relies on a technical condition known as the exponential separation condition. Recently, in joint work with B. Bárány and I. Kolossváry, we showed that this condition can be expressed naturally through a "dual iterated function system" and in terms of the invariant dual attractor that exists in the space of analytic maps. In this talk, I will explain how these dual attractors can give new insight into the dimension drop conjecture and what we can learn about the original set through the study of the geometry of this infinite dimensional attractor.
Characterising 1-rectifiable metric spaces via connected tangent spaces
Phoebe Valentine (University of Warwick)
Abstract: Characterising rectifiability using the existence of tangents is a very classical technique in GMT and positive results can be found even when using very weak notions of a tangent. This work goes further to only require that tangents are supported on a connected set. Despite this weak requirement, we can still prove that 1-rectifiability in complete metric spaces is implied by the existence of connected tangents almost everywhere. This, combined with prior results of Bate, establishes a characterisation of 1-rectifiability in complete metric spaces, which is in particular also new for Euclidean space. Inspired by geometric ideas from Besicovitch in the plane, we achieve our result by exploiting the inherent ’gappiness’ of purely 1-unrectifiable sets. To this end, we will give rigorous meaning to this 'gappiness' and show how it may be quantified.
Lipschitz differentiation without Poincaré inequalities
Pietro Wald (University of Warwick)
Abstract: Cheeger’s seminal 1999 paper initiated the study of metric measure spaces that admit a generalised differentiable structure. In such spaces, Lipschitz functions-real-valued and, in some cases, Banach-valued-are differentiable almost everywhere. Since then, much work has gone into determining the precise geometric and analytic conditions under which such structures exist.
The work of Bate-Li and Eriksson-Bique constitutes a major step in this direction. They prove that differentiation into every Banach space with the Radon-Nikodym property is possible only if the space supports suitable Poincar\'e inequalities. However, such a condition is not necessary for differentiation into some specific targets, such as Hilbert space, as shown by an ingenious construction of Schioppa.
In this talk, I will give a brief overview of the theory and present new results from joint work with David Bate. We develop a general procedure to construct new examples with properties akin to Schioppa's, apply it to Laakso spaces, and investigate the Banach targets for which differentiation is possible. These results answer a question of Schioppa.