# Workshop on geometric measure theory and analysis on metric spaces

8-10 August 2022, B3.02, Zeeman Building (directions from Radcliffe hotel).

This conference is funded by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 948021.

## Schedule

### Monday 8 August

- 09:00 - 09:30: Welcome
*Zeeman common room* - 09:30 - 10:20: Giovanni Alberti (Pisa): Sets and currents tangent to non-involutive distributions of planes
- 10:30 - 10:50: Coffee break
- 11:00 - 11:50: Enrico Pasqualetto (SNS): The Rank-One Theorem on RCD spaces
- 12:00 - 14:00: Lunch
- 14:00 - 14:50: Pertti Mattila (Helsinki): Parabolic rectifiability
- 15:00 - 15:20: Coffee break
- 15:30 - 16:20: Andrea Marchese (Trento): Characterization of rectifiable measures, structure of flat chains, and closability of differential operators
- 16:30 - 17:20: Andrea Merlo (Fribourg): On the converse of Pansu’s differentiability theorem

### Tuesday 9 August

- 09:00 - 09:50: Paolo Bonicatto (Warwick): Decomposition of integral metric currents
- 10:00 - 10:20: Coffee break
- 10:30 - 11:20: Urs Lang (ETH): Weak and flat convergence of integral currents revisited
- 11:30 - 12:20: Giacomo Del Nin (Warwick): Transport of currents and geometric Rademacher-type theorems
- 12:30 - 14:00: Lunch
- 14:00 - 14:50: Elefterios Soultanis (Radboud): Curvewise differentiable structure on metric spaces
- 15:00 - 15:20: Coffee break
- 15:30 - 16:20: Tapio Rajala (Jyväskylä): BV and Sobolev extension domains
- 16:30 - 17:20: Matthew Hyde (Warwick): Weak lower density and uniform rectifiability

19:00: Conference dinner in Scarman conference centre.

### Wednesday 10 August

- 09:00 - 09:50: Marianna Csörnyei (Chicago): On the dimension of Besicovitch sets for rectifiable curves
- 10:00 - 10:20: Coffee break
- 10:30 - 11:20: Katrin Fässler (Jyväskylä): Kakeya sets in the Heisenberg group
- 11:30 - 12:20: Ana Lucía García Pulido (Liverpool): Bi-Lipschitz embeddings of the space of persistence barcodes
- 12:30 - 14:00: Lunch
- 14:00 - 14:50: Olga Maleva (Birmingham): Typical behaviour of typical Lipschitz mappings
- 15:00 - 15:20: Coffee break
- 15:30 - 16:20: Jakub Takáč (Warwick): Typical behaviour of 1-Lipschitz images of rectifiable metric spaces
- 16:30 - 17:20: Tuomas Orponen (Jyväskylä): On the Hausdorff dimension of radial projections

## Abstracts

### Giovanni Alberti (Pisa): Sets and currents tangent to non-involutive distributions of planes

A classical result in geometry, Frobenius theorem, states that there exist no \(k\)-dimensional surface which is tangent to a non-involutive distribution of \(k\)-planes \(V\). In this talk I consider some extensions of this statement to weaker notions of surfaces, such as rectifiable sets and currents. Following the work of Z. Balogh, I first consider a contact set \(E\), namely a set of points where a surface \(S\) is tangent to \(V\), and ask whether \(E\) must be \(k\)-negligible. The answer depends on a combination of the regularity of \(S\) and of the boundary of \(E\): at one end of the spectrum, if \(S\) is of class \(C^{1,1}\) no regularity is required on \(E\); at the other end, if \(S\) is only of class \(C^1\), then \(E\) must be in a certain fractional Sobolev class (and counterexamples show that these results are somewhat sharp). Passing to currents, Frobenius theorem holds for integral currents (a notion which entails a certain regularity of the boundary), but interestingly enough, the answer turns out to be more involved for normal current. These results are part of an ongoing research project with Annalisa Massaccesi (University of Padova), Andrea Merlo (University of Fribourg) and Evgeni Stepanov (Steklov Institute, Saint Petersburg).

### Paolo Bonicatto (Warwick): Decomposition of integral metric currents

Currents are nowadays a widely used tool in geometric measure theory and calculus of variations, as they allow one to give a weak formulation of a variety of geometric problems. The theory of normal and integral currents (initiated mostly by Federer and Fleming in the ’60s) was developed in the context of Euclidean spaces. In 2000, Ambrosio and Kirchheim introduced metric currents, defined on complete metric spaces. The talk will be devoted to integral metric currents: we show that integral currents can be decomposed as a sum of indecomposable components and, in the special case of one-dimensional integral currents, we also characterise the indecomposable ones as those associated with injective Lipschitz curves or injective Lipschitz loops. This generalises to the metric setting a previous result by Federer. Joint work with Giacomo Del Nin (Warwick) and Enrico Pasqualetto (Scuola Normale Superiore).

### Marianna Csörnyei (Chicago): On the dimension of Besicovitch sets for rectifiable curves

It is well-known that if a planar set contains a line segment in each direction, then it must have dimension 2. In this talk our aim is to study what the dimension of a planar set can be that contains a rotated copy in each direction of a given rectifiable set.

The talk is based on an ongoing joint research with I. Altaf and K. Hera.

### Giacomo Del Nin (Warwick): Transport of currents and geometric Rademacher-type theorems

In the first part of the talk I will briefly introduce and motivate the transport equation for currents in Euclidean spaces, with a special attention to the space-time formulation. I will then shift the focus to some rectifiability questions and Rademacher-type results: given a Lipschitz path of integral currents, I will discuss the existence of a “geometric derivative”, namely a vector field advecting the currents. Based on joint work with Paolo Bonicatto and Filip Rindler (Warwick).

### Katrin Fässler (Jyväskylä): Kakeya sets in the Heisenberg group

This talk concerns Kakeya sets in the first Heisenberg group. These sets differ from usual Kakeya sets in \(\mathbb{R}^3\) in that they are only required to contain line segments in ‘horizontal’ directions determined by the structure of the Heisenberg group. J. Liu proved the sharp lower bound for the Hausdorff dimension of such sets with respect to a natural non-Euclidean metric. I will report on this and related results that illustrate connections between Heisenberg geometry and incidence geometry in the Euclidean plane. This is based on ongoing collaboration with A. Pinamonti and P. Wald.

### Ana Lucía García Pulido (Liverpool): Bi-Lipschitz embeddings of the space of persistence barcodes

The space of persistence barcodes, equipped with the bottleneck metric, is a fundamental object in topological data analysis. However, the understanding of the metric geometry of this space is limited.

In this talk I will introduce the space of persistence barcodes and present recent ongoing research concerning its bi-Lipschitz embeddability into Hilbert space. This is joint work with David Bate.

### Matthew Hyde (Warwick): Weak lower density and uniform rectifiability

A classical result of Besicovitch states that if \(E\subseteq \mathbb{R}^{2}\) is a set of finite 1-dimensional Hausdorff measure such that \(\Theta^{1}_{*}(E,x)>\frac{3}{4}\) for \(\mathcal{H}^{1}\)-a.e. \(x \in E\), then \(E\) is 1-rectifiable. A similar criterion for higher dimensional sets follows from a result of Preiss. In this talk I will introduce a lower density condition which implies a stronger version of rectifiability, namely, uniform rectifiability. This is joint work with Jonas Azzam.

### Urs Lang (ETH): Weak and flat convergence of integral currents revisited

A classical and fundamental theorem in geometric measure theory states that a bounded and weakly convergent sequence of integral currents converges with respect to the flat metric topology. In Euclidean space, this result can be derived from the deformation theorem. An analogue for Ambrosio-Kirchheim currents in metric spaces satisfying a weak local convexity condition was established by Wenger, using an elaborate induction and decomposition argument. In a recent joint paper with Tommaso Goldhirsch we have given a new perspective on this theorem. The proof proceeds along the same lines as Wenger’s, but is simplified by the use of an elegant variational argument due to Huang-Kleiner-Stadler and a more quantitative version of weak convergence.

### Olga Maleva (Birmingham): Typical behaviour of typical Lipschitz mappings

It is now a classical result that a Lipschitz function on a ‘good’ Banach space is differentiable on a dense set. We approach this from a Baire category point of view. Our earlier result for finite-dimensional spaces that a typical 1-Lipschitz function has points of differentiability inside any given set \(A\), unless \(A\) can be covered by a countably many closed purely 1-unrectifiable sets, motivates the following question: Is a typical 1-Lipschitz mapping differentiable at a typical point of a given set? We show that no matter how good the Banach space or its subset is, a typical 1-Lipschitz mapping is non-differentiable at a typical point in a very strong sense. This is a joint work with Michael Dymond.

### Andrea Marchese (Trento): Characterization of rectifiable measures, structure of flat chains, and closability of differential operators

The decomposability bundle of a Euclidean Radon measure is roughly speaking a map which captures at almost every point all the tangential directions to the Lipschitz curves along which the measure can be decomposed. I will discuss the role of such tool in the proof of the converse of Rademacher’s theorem and some more recent applications: a characterization of rectifiable measures in terms of Lusin type approximation of Lipschitz functions by functions of class \(C^1\), a description of the structure of Federer-Fleming flat chains with finite mass, and a classification of those differential operators which are closable with respect to a measure.

### Pertti Mattila (Helsinki): Parabolic rectifiability

I shall discuss in the \(n\)-space with parabolic norm, definitions and characterizations of rectifiability in terms of approximate tangent planes and tangent measures.

### Andrea Merlo (Fribourg): On the converse of Pansu’s differentiability theorem

In this talk I will present two new results concerning differentiability of Lipschitz maps between Carnot groups.

The former is a suitable adaptation of Pansu-Rademacher differentiability theorem to general Radon measures. More precisely we construct a suitable bundle associated to the measure along which Lipschitz maps are differentiable, very much in the spirit of the results of Alberti-Marchese.

The latter is the converse of Pansu’s theorem. Namely, let \(G\) be a Carnot group and \(\mu\) a Radon measure on \(G\). Suppose further that every Lipschitz map between \(G\) and \(H\), some other Carnot group, is Pansu differentiable \(\mu\)-almost everywhere. We show that \(\mu\) must be absolutely continuous with respect to the Haar measure of \(G\).

This is a joint work with Guido De Philippis, Andrea Marchese, Andrea Pinamonti and Filip Rindler.

### Tuomas Orponen (Jyväskylä): On the Hausdorff dimension of radial projections

The radial projection to a vantage point \(x \in \mathbb{R}^{d}\) is the map \[\pi_{x}(y) = \frac{(x - y)}{|x - y|}.\] What is the relationship between the Hausdorff dimension of a set, and the Hausdorff dimension of its radial projections? How many vantage points can there be such that the radial projection has dimension strictly lower than the dimension of the set? I will discuss recent progress on this question in \(\mathbb{R}^{2}\), joint with Pablo Shmerkin.

### Enrico Pasqualetto (SNS): The Rank-One Theorem on RCD spaces

RCD spaces are metric measure spaces whose Riemannian Ricci curvature is bounded from below, in a synthetic sense. Despite the lack of any smooth structure, a very refined calculus has been developed in this framework. In the talk I provide a quick overview on such calculus and I use it to prove the Rank-One Theorem for maps of bounded variation from finite-dimensional RCD spaces to Euclidean spaces. Based on a joint work with Gioacchino Antonelli and Camillo Brena.

### Tapio Rajala (Jyväskylä): BV and Sobolev extension domains

I will discuss different geometric properties of BV- and Sobolev \(W^{1,p}\)-extension domains. In the case of planar simply connected domains, there are geometric characterizations of extension domains that lead to sharp size estimates on the Hausdorff dimension of the boundary and its set of self-intersections. In higher dimensions and for general domains, so far similar conditions have been obtained that are only sufficient or necessary. I will present one such condition for the range \(1 < p < 2\). In the limiting case of \(W^{1,1}\)-extension domains, this condition becomes a characterization that works also on general metric measure spaces.

### Elefterios Soultanis (Radboud): Curvewise differentiable structure on metric spaces

In his seminal work, Cheeger introduced a weak differentiable structure on metric spaces supporting a Poincare inequality, with respect to which Lipschitz functions can be differentiated. These Lipschitz differentiability structures have since been intensively studied and connected to the existence of many curve fragments. In this talk I present an analogous construction using curves and leading to a \(p\)-weak differentiable structure which is compatible with Sobolev spaces. These structures exist more generally than Cheeger’s but only allow differentiation along curves. I will present the basic ideas behind the construction and, time permitting, the comparison with Lipschitz differentiability structures.

### Jakub Takáč (Warwick): Typical behaviour of 1-Lipschitz images of rectifiable metric spaces

Suppose \(X\) is a complete metric space and \(E\) is a \(\mathcal{H}^n\)-measurable subset. Bate recently gave a characterisation of \(n\)-rectifiability in the spirit of Besicovitch-Federer, via the (complete) metric space \((Lip_1(X,\mathbb{R}^n),\lVert\cdot\rVert_{\infty})\) of 1-Lipschitz functions \(f\colon X \to \mathbb{R}^n\). If \(E\) is \(n\)-rectifiable, then a typical \(f\) in \(\textnormal{Lip}_1(X,\mathbb{R}^n)\) satisfies \(\mathcal{H}^n(f(E))>0\), while if \(E\) is purely \(n\)-unrectifiable, a typical \(f\) satisfies \(\mathcal{H}^n(f(E))=0\). This motivates the question of whether, assuming \(E\) is \(n\)-rectifiable, there is some \(\Delta>0\) such that a typical \(f\in\textnormal{Lip}_1(X,\mathbb{R}^m)\) (for some fixed \(m>n\)) satisfies \(\mathcal{H}^n(f(E))>\Delta\). The answer turns out to be complicated.

If \(X\) is Euclidian then a typical \(f\) even satisfies \(\mathcal{H}^n(f(E))= \mathcal{H}^n(E)\), but no such \(\Delta>0\) exists for \(X=\mathbb{R}^n_\infty\). We provide sufficient and necessary conditions for existence of such a \(\Delta>0\) for \(n=2\) and \(X=E=[-1,1]^2\), equipped with some *norm* \(|\cdot|_a\).
These conditions agree for \(p\)-norms or norms who’s unit ball has \(C^2\) boundary. We will also briefly touch on the connection of these problems to a particular type of Lipschitz extension problems.

## Participants

- Giovanni Alberti
- David Bate
- Paolo Bonicatto
- Marianna Csörnyei
- Damian Dabrowski
- Giacomo Del Nin
- Katrin Fässler
- Luis Miguel García Martin
- Ana Lucía García Pulido
- Matthew Hyde
- Hefin Lambley
- Urs Lang
- Olga Maleva
- Andrea Marchese
- András Máthé
- Pertti Mattila
- Andrea Merlo
- Tuomas Orponen
- Enrico Pasqualetto
- Mark Pollicott
- Tapio Rajala
- Filip Rindler
- Elefterios Soultanis
- Tim Sullivan
- Jakub Takáč
- Michele Villa
- Polina Vytnova
- Pietro Wald
- Julian Weigt