List of Talks
Giovanni Alberti (University of Pisa)
Frobenius theorem for non-smooth sets and currents.
Let $V$ be a distribution of $k$-planes in the Euclidean space and let $S$ be a $k$-dimensional surface tangent to $V$. Then Frobenius theorem states that $V$ must be involutive at every point of $S$.
In this talk I consider some generalization of this statement to weaker notions of surfaces, such as rectifiable sets and currents. I begin with the case where $S$ is a subset of a $k$-dimensional surface. In this case the validity of the statement depends on a combination of the regularity of the surface and of the boundary of $S$: at one extreme, when the surface is of class $C^{1,\alpha}$ with $\alpha$ sufficiently large, then no regularity is required on the boundary of $S$; at the other extreme, when the surface is only of class $C^1$, then the boundary of $S$ (viewed as a current) must have finite mass.
More generally, Frobenius theorem holds when $S$ is an integral current. What then if $S$ is a normal current but is not rectifiable? In this case the key is a certain geometric property of the boundary of $S$. These questions are strictly related to the problem of decomposing a normal current in terms of integral/rectifiable currents.
These results are part of an ongoing research project with Annalisa Massaccesi (University of Padova), Andrea Merlo (University Paris-Saclay) and Evgeni Stepanov (Steklov Institute, Saint Petersburg).
Luigi Ambrosio (SNS Pisa)
An invitation to the matching problem
In this lecture I will review a few old and new results on the matching problem, a question at the interface between Optimal Transport, Probability, Partial Differential Equations. The results I will present have been obtained in joint papers with Dario Trevisan, Federico Glaudo, Federico Stra and Michael Goldman.
Alessandro Carlotto (ETH Zurich)
Applying the Almgren-Pitts min-max theory to problems in geometry: a case study
In the last decade, the striking developments in the min-max theory for the area functional have led to the solution of a number of outstanding open questions in geometry, such as the Willmore conjecture, Freedman's conjecture on the energy of links, and, more recently, Yau's conjecture on the existence of infinitely many closed minimal hypersurfaces in all compact manifolds without boundary.
A well-known drawback of this method is the fact that, since one needs to go through weak notions of convergence (i.e. those typical of GMT), it is not topologically effective: it does not allow to control the topology of the critical points it produces. The aim of this lecture will be to present one of the very few exceptions to this rule:
I will describe some joint work with Giada Franz and Mario Schulz where we answer (in the affirmative) the well-known question whether there exist in B^3 (embedded) free boundary minimal surfaces of genus one and one boundary component. In fact, we prove a more general result: for any g there exists in B^3 an embedded free boundary minimal surface of genus g and connected boundary. This result provides a long-awaited analogue of the existence theorem obtained by Lawson in 1970 for closed minimal surfaces in round S^3.
Antonin Chambolle (CMAP, Ecole Polytechnique)
Effective surface tensions for periodic lattice systems
In this talk I will describe some properties of the effective surface tensions of discrete perimeters defined on a periodic lattice, as well as a process to numerically build periodic coefficients whose homogenised surface tension minimises some objective, such as a distance to the Euclidean norm. This is based on a joint work with L. Kreutz (Münster).
Guy David (University Paris-Sud)
A simple existence theorem for 2-dimensional minimal sets with some specific sliding Plateau conditions.
The lecture will present a recent result with Camille Labourie on the existence of minimal sets of dimension 2 in $n$-space, with some sliding boundary condition.
The result is a little anecdotic because the boundary $G$ is assumed to be composed of smooth curves that satisfy a condition close to asking that $G$ is contained in the boundary of a convex set (in ambient dimension 3). On the other hand this seems to be the first result for sliding boundaries and the condition is a natural way to avoid some blow-up limits from arising from a minimizing sequence.
The method is an illustration of recent theorems on limits of minimizing sequences and boundary regularity.
Giacomo Del Nin (University of Warwick)
Endpoint Fourier restriction and unrectifiability
In this talk we show that if a measure of dimension $s$ on $\mathbb{R}^d$ admits $(p,q)$ Fourier restriction for some endpoint exponents allowed by its dimension, namely $q=\frac{s}{d}p'$ for some $p>1$, then a dicotomy holds: the measure is either absolutely continuous or purely 1-unrectifiable. Based on a joint work with Andrea Merlo.
Kenneth Falconer (University of St. Andrews)
Intermediate dimensions
Jon Fraser, Tom Kempton and I recently introduced `intermediate dimensions’ which, for sets in Euclidean space which have differing Hausdorff and box counting dimensions, provide a continuum of dimensions with Hausdorff dimension at one end of the range and box-counting dimension at the other. We will discuss basic and more sophisticated properties of intermediate dimensions, illustrated by examples.
Adriana Garroni (University of Roma La Sapienza)
Rigidity estimates for incompatible fileds and applications
Nicola Gigli (SISSA Trieste)
Module-valued measures and applications
Consider a real valued BV function defined on a Riemannian manifold: in which space does it live its differential? The answer to this question is typically given by either using coordinates or via polar decomposition. When working on less regular structures, typically neither of these is a priori given, and questions like `what is it the differential of a BV function, or the Hessian of a convex function’ are often left unanswered.
In this talk I will present a framework to deal with this sort of problems. A key concept is that of module-valued measures defined over arbitrary metric spaces, a notion that seem to encapsulate several pre-existing ones in metric analysis, including in particular Ambrosio-Kirchheim’s metric currents.
Applications to BV calculus on RCD spaces will be discussed.
From a joint work with Camillo Brena.
Jonas Hirsch (University of Leipzig)
On bounded solutions of Linear elliptic operators with measurable coefficients - De Giorgi’s theorem revisited
Slawomir Kolasinski (University of Warsaw)
Geometric ellipticity
By a geometric variational problem I mean a problem of minimising a functional defined on k-dimensional geometric objects, like currents or varifolds, lying in n-dimensional ambient space. Most interesting (for me) are functionals defined as integrals, where the integrand depends on the point and the tangent k-plane at that point. One example is the k-dimensional Hausdorff measure generated by some non-Euclidean norm on $\mathbb{R}^n$
Ellipticity (AE), introduced by Almgren in the 1960s, is a condition on the functional ensuring existence and partial regularity of minimisers. The atomic condition (AC) was defined a few years ago by G. De Philippis, A. De Rosa, and F. Ghiraldin so to ensure rectifiability of critical points. Together with A. De Rosa we proved that (AC) implies (AE).
The problem with this theory is that there are virtually no specific non-trivial examples of functionals satisfying any of (AE) or (AC). A. De Rosa and R. Tione proposed recently the scalar atomic condition (SAC) which might be easier to verify.
In my talk I shall review definitions, properties, and relations between conditions (AE), (AC), and (SAC). I shall talk about my joint work with A. De Rosa (CPAM 2020) and also about ongoing work with my student Mariusz Janosz.
Gian Paolo Leonardi (University of Trento)
Some new results on the prescribed mean curvature problem
Finding graphs of prescribed mean curvature is one of the classical problems of Calculus of Variations. This problem is closely connected with non-parametric minimal surfaces, and it is strongly motivated by capillarity theory. After an introduction of the classical formulation together with some milestone results, we will describe some recent progress obtained in two different directions. First, the relaxation of the regularity assumptions on the domain, leading to the "weak-regularity" hypothesis. Second, the prescribed mean curvature measure problem.
Henri Martikainen (Washington University - St. Louis)
Singular integrals and genuinely multilinear weights in product spaces
We focus on a modern singular integral problem and prove genuinely multilinear weighted estimates for singular integrals in product spaces. We motivate the theory by looking at unweighted applications and the key role of extrapolation in these. Examples include a free access to fractional Leibniz rules of Muscalu, Pipher, Tao and Thiele and their recent improvements.
Keeping in mind the main theme of the conference, we attempt to use this particular problem to give an idea of the latest abstract methods underlying singular integral theory and their connections to GMT and PDEs.
Benoit Merlet
About the differential constraint $\partial_1 u=0$ or $\partial _2 u=0$
The talk is about joint works with Michael Goldman. We first consider the characteristic function $u$ of a measurable subset $E$ of the unit square. We show that if the cross derivative $\mu[u]=\partial_1\partial_2 u$ is a measure then $E$ is essentially a polyhedron with sides parallel to the axes and $\mu[u]$ is a finite sum of Dirac masses. We then generalize to higher dimensions and study the characteristic functions of sets in $\Omega_1\times\Omega_2$ assuming that $\mu[u]=\nabla_1\nabla_2 u$ is a measure. In this case, we show that $\mu[u]$ is rectifiable and has a tensor structure. Eventually, we consider the class of Lipschitz continuous functions $u$ defined on the unit square such that $\partial_1u \partial_2u = 0$ almost everywhere and $\mu[u]$ is a measure. Again, we show that $\mu[u]$ is a 1-rectifiable measure.
Andrea Merlo (University Paris-Saclay)
Marstrand-Mattila rectifiability criterion and rectifiable measures in Carnot groups
The Marstrand-Mattila rectifiability criterion in Euclidean spaces asserts that a measure with positive lower density, finite upper density and flat (possibly non-unique) tangent measures at almost every point is rectifiable. In this talk I will explain how to extend such criterion to 1-codimensional measures in general Carnot groups and I will use this as an occasion for introducing and discussing various definitions of rectifiability in Carnot groups.
Tuomas Orponen (University of Jyväskylä)
On the dimension of Furstenberg sets
A planar set “K" is called a Furstenberg (s,t)-set if there exists a t-dimensional family of lines such that every line in the family contains an s-dimensional piece of K. Furstenberg (1,1)-sets are essentially Kakeya sets, and it is known since the 70s that they have Hausdorff dimension 2. A problem of Wolff from the late 90s asks to determine the smallest possible Hausdorff dimension of Furstenberg (s,t)-sets, for general pairs (s,t). Wolff’s problem remains open for most pairs (s,t). I will report on recent progress, based on joint works with D. Dabrowski, P. Shmerkin, and M. Villa.
Giuseppe Savaré (Bocconi University, Milano)
Lipschitz approximation, Capacity and Capacitary Modulus in metric Newtonian-Sobolev spaces
We will present a new explicit construction of Lipschitz approximations of Newtonian-Sobolev functions on a complete metric-measure space and a novel concept of Capacitary Modulus of a family of rectifiable arcs, which provides a simple characterization of the Newtonian capacity.
As a byproduct, we will show that many useful properties of the Newtonian Capacity in spaces satisfying a doubling-Poincaré condition (as Choquet property, outer regularity, the existence of quasi-continuous representatives of Sobolev functions) hold in arbitrary complete metric-measure spaces.
Pablo Shmerkin (University of Torcuato di Tella)
Dimensions of distance sets
The Falconer distance set conjecture, relating the dimension of a subset of Euclidean space to that of the set of distances it spans, is one of the outstanding open problems in geometric measure theory. I will survey some recent progress on this problem, including the solution of the conjecture for sets of equal Hausdorff and packing dimension (including Ahlfors-David regular sets). Joint work with Hong Wang (UCLA).
Eugene Stepanov (Steklov Institute, St. Petersburg)
Reconstructing hidden manifolds from intrinsic distances: from multidimensional scaling to semidefinite programming
We will consider one of the important classes of manifold learning problems frequently arising in applications of statistical data analysis: reconstruct the unknown manifold or its embedding into a given (say, Euclidean) space (or at least some of its structural characteristics) knowing just the information on intrinsic distances between points in its "almost dense'' subset. One of the basic spectral methods to solve such problems is multidimensional scaling (MDS). We will study whether the manifold is reconstructed by MDS (and what is actually reconstructed), and provide some negative though notrivial answers to this question. As an alternative a variational reconstruction method based on semidefinite programming will be proposed.
Salvatore Stuvard (Universiry of Milan)
The regularity of mass minimizing currents modulo p
Integer rectifiable currents mod p are a class of generalized surfaces in which it is possible to define and solve Plateau's problem. The corresponding minimizers, mass minimizing currents mod p, are minimal surfaces exhibiting a far richer geometric complexity than the classical mass minimizing integral currents of Federer and Fleming. Such complexity is often observed in soap films.
In this talk, I will discuss the partial interior regularity theory for mass minimizing currents mod p. The focus will be on dimension bounds and fine structural properties (such as rectifiability and local finiteness of measure) of their singular set. The ultimate goal will be to reveal that "most" of the singularities of mass minimizing currents mod p present an interesting "regular free boundary" local structure. Based on joint works with Camillo De Lellis (IAS), Jonas Hirsch (U Leipzig), Andrea Marchese (U Trento), and Luca Spolaor (UCSD).
Xavier Tolsa (UAB Barcelona)
The measures with $L^2$-bounded Riesz transform and the Painlevé problem for Lipschitz harmonic functions
Sascha Troscheit (University of Vienna)
Quasi-self-similar sets with positive Hausdorff measure.
In this talk I will discuss the Hausdorff measure and content of a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. I will show that any Hausdorff measurable subset of such a set has comparable Hausdorff measure and Hausdorff content, which also proves that graph-directed and sub self-conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions.
When restricting to self-conformal subsets of the real line with Hausdorff dimension strictly less than one, we will additionally show that the weak separation condition is equivalent to Ahlfors regularity and its failure implies full Assouad dimension. In fact, this resolve a self-conformal extension of the dimension drop conjecture for self-conformal sets with positive Hausdorff measure by showing that its Hausdorff dimension falls below the expected value if and only if there are exact overlaps.
(Joint with Jasmina Angelevska and Antti Käenmäki.)
Bozhidar Velichkov (University of Pisa)
An epsilon-regularity theorem for the solutions of a free boundary system
Michele Villa (University of Oulu)
Characterising tangent points via square functions.
Geometric square functions appeared in the late eighties, when problems from harmonic and complex analysis pushed researchers to study rectifiable sets quantitatively. In this talk I will trace the origins of two of these functions, the Carleson's $\epsilon^2$ function and Jones' $\beta$ numbers. I will specifically focus on their connection to tangent points. This will give me the occasion to mention and sketch the proof of two recent results, which characterise tangent points of some classes of sets in terms of a Dini condition on $\epsilon$ and $\beta$.
Based on works with Matthew Hyde, Ben Jaye and Xavier Tolsa.