For further information on events at the Mathematics Institute, see: go.warwick.ac.uk/mathseventsor contact: Mathematics Research Centre, Zeeman Building, University of Warwick, Coventry CV4 7AL, UKE-mail: mrc@maths.warwick.ac.uk Phone: +44(0)24 7652 2681

Geometric Measure Theory

Organisers: Giovanni Alberti (Pisa), David Preiss (Warwick)Local Organisers: Tomasz Kania (Warwick), Martin Rmoutil (Warwick),

Thomas Zürcher (Warwick)Venue: Room B3.03, Mathematics Institute, Zeeman Building,

University of Warwick, Coventry, UK

For more information please visit:http://www2.warwick.ac.uk/fac/sci/maths/research/events/2016-17/nonsymposium/gmt/

TimetableMonday

9:25-9:30 Announcements9:30-10:20 Marianna Csörnyei: The Kakeya needle prob-

lem and the existence of Besicovitch andNikodym sets for rectifiable setsCoffee break

11:00-11:50 Andrea Pinamonti & Gareth Speight: Maximaldirectional derivatives and universal differen-tiability sets in Carnot groups

12:00-12:25 Vasileios Chousionis: Nonnegative kernels andrectifiability in the Heisenberg group

12:30-12:55 Katrin Fässler: Quantitative rectifiability inthe Heisenberg groupLunch

15:00-15:50 Bruce Kleiner: Sobolev mappings betweenCarnot groups, the exterior derivative, andrigidityCoffee break

16:30-16:55 Guy C. David: Differentiability and rectifiabil-ity on metric planes

17:00-17:50 David Bate: Rectifiability in metric and Ba-nach spaces via arbitrarily small perturbationsDinner

2

Tuesday

9:25-9:30 Announcements9:30-10:20 Guido De Philippis & Filip Rindler: On the

converse of Rademacher theorem and the rigid-ity of measures in Lipschitz differentiabilityspacesCoffee break

11:00-11:50 Andrea Marchese: One-rectifiable representa-tions of Euclidean measures and applications

12:00-12:25 Olga Maleva: Where can real-valued Lipschitzfunctions on Rn be non-differentiable?

12:30-12:55 Valentino Magnani: Remarks on Hausdorffmeasure and differentiabilityLunch

15:00-15:25 Davide Vittone: Existence of tangent lines tosub-Riemannian geodesics

15:30-15:55 Dario Trevisan: On level sets in the HeisenberggroupCoffee break

16:30-16:55 Sean Li: Geometric characterizations of RNPdifferentiability

17:00-17:50 Andrea Schioppa: Calculus on Metric Spaces:Beyond the Poincaré InequalityDinner

3

Wednesday

9:25-9:30 Announcements9:30-10:20 Nageswari Shanmugalingam: In the setting

of metric measure spaces, what is Dirichletproblem for functions of least gradient?Coffee break

11:00-11:50 Jan Malý: Measuring families of curves byapproximation modulus

12:00-12:50 Dali Nimer: Geometry of uniform measuresLunch

15:00-15:25 Alan Chang: Small unions of affine subspacesand skeletons via Baire category

15:30-15:55 Kornélia Héra: Hausdorff dimension of unionof affine subspacesCoffee break

16:30-16:55 Tamás Keleti: Fubini type result for Hausdorffdimension

17:00-17:25 Michael Dymond: Mapping n grid pointsonto a square forces an arbitrarily large Lip-schitz constantConference dinner

4

Thursday

9:25-9:30 Announcements9:30-10:20 Jeremy Tyson: Tube formulas, uniform mea-

sures and heat content in the Heisenberg groupCoffee break

11:00-11:50 Ulrich Menne: Sobolev-type spaces on varifolds12:00-12:25 Thomas Zürcher: Sets where lip is infinite12:30-12:55 Annalisa Massaccesi: On the rank-one theorem

for BV functionsLunch

15:00-15:50 Tuomas Orponen: Sharpening Marstrand’sprojection theoremCoffee break

16:30-16:55 De-Jun Feng: Dimension of invariant measuresfor affine iterated function systems

17:00-17:50 Miklós Laczkovich: The story of squaring thecircleDinner

5

Friday

9:25-9:30 Announcements9:30-10:20 Séverine Rigot: Besicovitch covering property

on graded groups and applications to measuredifferentiationCoffee break

11:00-11:50 Elena Riss: A set of positive Gaussian measurewith uniformly zero density everywhere

12:00-12:25 András Máthé: A covering theorem for closedcovers of metric spaces

12:30-12:55 Matthew Badger: Geometry of Radon mea-sures via Hölder parameterizationsLunch

15:00-15:25 Sylvester Eriksson-Bique: Classification ofPoincaré inequalities and PI-rectifiability

15:30-15:55 Lukáš Malý: Poincaré inequalities that fail toconstitute an open-ended conditionCoffee break

16:30-16:55 Panu Lahti: Fine potential theory in metricspaces for p = 1

17:00-17:50 Tapio Rajala: On density of Sobolev functionson Euclidean domainsDinner

6

TalksMatthew Badger: Geometry of Radon measures via Hölder parameterizations . . . . . . . . . . . 9

David Bate: Rectifiability in metric and Banach spaces via arbitrarily small perturbations 9

Alan Chang: Small unions of affine subspaces and skeletons via Baire category . . . . . . . . . . 9

Vasileios Chousionis: Nonnegative kernels and rectifiability in the Heisenberg group . . . . . 9

Marianna Csörnyei: The Kakeya needle problem and the existence of Besicovitch andNikodym sets for rectifiable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Guy C. David: Differentiability and rectifiability on metric planes . . . . . . . . . . . . . . . . . . . . . . . 10

Guido De Philippis and Filip Rindler: On the converse of Rademacher theorem and therigidity of measures in Lipschitz differentiability spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Michael Dymond: Mapping n grid points onto a square forces an arbitrarily large Lip-schitz constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Sylvester Eriksson-Bique: Classification of Poincaré inequalities and PI-rectifiability . . . . . 10

Katrin Fässler: Quantitative rectifiability in the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . 11

De-Jun Feng: Dimension of invariant measures for affine iterated function systems . . . . . . . 11

Kornélia Héra: Hausdorff dimension of union of affine subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 11

Tamás Keleti: Fubini type result for Hausdorff dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Bruce Kleiner: Sobolev mappings between Carnot groups, the exterior derivative, andrigidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Miklós Laczkovich: The story of squaring the circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Panu Lahti: Fine potential theory in metric spaces for p = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Sean Li: Geometric characterizations of RNP differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Valentino Magnani: Remarks on Hausdorff measure and differentiability. . . . . . . . . . . . . . . . . 12

Olga Maleva: Where can real-valued Lipschitz functions on Rn be non-differentiable? . . . . 13

Jan Malý: Measuring families of curves by approximation modulus . . . . . . . . . . . . . . . . . . . . . . 13

Lukáš Malý: Poincaré inequalities that fail to constitute an open-ended condition. . . . . . . . 13

Andrea Marchese: One-rectifiable representations of Euclidean measures and applications 13

7

Annalisa Massaccesi: On the rank-one theorem for BV functions . . . . . . . . . . . . . . . . . . . . . . . . 14

András Máthé: A covering theorem for closed covers of metric spaces . . . . . . . . . . . . . . . . . . . 14

Ulrich Menne: Sobolev-type spaces on varifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Dali Nimer: Geometry of uniform measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Tuomas Orponen: Sharpening Marstrand’s projection theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Andrea Pinamonti and Gareth Speight: Maximal directional derivatives and universal dif-ferentiability sets in Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Tapio Rajala: On density of Sobolev functions on Euclidean domains . . . . . . . . . . . . . . . . . . . . 15

Séverine Rigot: Besicovitch covering property on graded groups and applications to mea-sure differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Filip Rindler and Guido De Philippis: On the converse of Rademacher theorem and therigidity of measures in Lipschitz differentiability spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Elena Riss: A set of positive Gaussian measure with uniformly zero density everywhere . 16

Andrea Schioppa: Calculus on Metric Spaces: Beyond the Poincaré Inequality . . . . . . . . . . . 16

Nageswari Shanmugalingam: In the setting of metric measure spaces, what is Dirichletproblem for functions of least gradient? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Gareth Speight and Andrea Pinamonti: Maximal directional derivatives and universal dif-ferentiability sets in Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Dario Trevisan: On level sets in the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Jeremy Tyson: Tube formulas, uniform measures and heat content in the Heisenberg group17

Davide Vittone: Existence of tangent lines to sub-Riemannian geodesics . . . . . . . . . . . . . . . . . 17

Thomas Zürcher: Sets where lip is infinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

8

Matthew Badger: Geometry of Radon measures via Hölder parameterizations

I’ll discuss joint work with Vyron Vellis, in which we investigate the influence that s-dimensional lower andupper spherical Hausdorff densities have on a Radon measure in n-dimensional Euclidean space when s is areal number between 0 and n. In particular, we give sufficient conditions on the densities which guaranteethat a measure is carried by or singular to (1/s)-Hölder curves. This extends part of the recent work ofBadger and Schul, which examined the case s = 1 (Lipschitz curves) in depth.

David Bate: Rectifiability in metric and Banach spaces via arbitrarily smallperturbations

This talk presents a new characterisation of compact, purely d-unrectifiable metric spaces X with finited-dimensional Hausdorff measure. It is shown that for any ε > 0 there exists a Lipschitz σ : X → σ(X)with Hd(σ(X)) < ε that perturbs the distances between points in X by at most ε. The key point of ourconstruction is that the Lipschitz constant of such a perturbation is independent of ε, which allows us toprove the converse statement.In fact, we construct such a perturbation σ : X ⊂ B → B for any Banach space B with an unconditionalbasis. The result for metric spaces is obtained using a suitable embedding into such a B (namely c0). Weobtain improved results if B is a Hilbert space or d = 1.This result is a significant generalisation of a result of H. Pugh who proved the characterisation for Ahlforsregular subsets of Euclidian space. His proof crucially relies on the Besicovitch–Federer projection theorem,which completely fails in the infinite dimensional setting, and our techniques are different. However, ourresults can be considered as an alternative to the projection theorem in this setting.

Alan Chang: Small unions of affine subspaces and skeletons via Baire category

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of thek-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we showthat a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular,this proves a conjecture of R. Thornton.Our results also show that Nikodym sets are typical among all sets which contain, for every x ∈ Rn, apunctured hyperplane H \ x through x. With similar methods we also construct a Borel subset of Rn ofLebesgue measure zero containing a hyperplane at every positive distance from every point.This is joint work with Marianna Csörnyei, Kornélia Héra, and Tamás Keleti.

Vasileios Chousionis: Nonnegative kernels and rectifiability in the Heisenberggroup

Following Calderón’s seminal work on the L2-boundedness of the Cauchy transform in 1977, a series offundamental contributions established the deep relations between singular integrals with odd kernels andrectifiability in Euclidean spaces. We find similar connections for 1-dimensional subsets of the Heisenberggroup. In stark contrast to the Euclidean setting our kernels are positive and even. Joint work with S. Li.

9

Marianna Csörnyei: The Kakeya needle problem and the existence of Besicovitchand Nikodym sets for rectifiable sets

We show that the classical results about rotating a line segment in arbitrarily small area, and the existenceof a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This isa joint work with Alan Chang.

Guy C. David: Differentiability and rectifiability on metric planes

In this talk, we discuss metric measure spaces that have both a differentiable structure in the sense of Cheegerand certain quantitative topological control; specifically, their blowups are topological planes. We showthat any differentiable structure on such a space is at most 2-dimensional, and furthermore that if it is2-dimensional the space is 2-rectifiable. This is partial progress on a question of Kleiner and Schioppa, and isjoint work with Bruce Kleiner.

Guido De Philippis and Filip Rindler: On the converse of Rademacher theoremand the rigidity of measures in Lipschitz differentiability spaces

Rademacher Theorem asserts that every Lipschitz function is differentiable almost everywhere with respectto the Lebesgue measure. Results in this spirit have been established by Pansu in Carnot groups and byCheeger in abstract metric measure spaces. A natural question is then the rigidity of those measures forwhich every Lipschitz function is differentiable almost everywhere.Aim of the talk is to discuss this issue in increasing generality. In particular we will present a proof of thefact that Rademacher Theorem can hold for a measure if and only if it is absolutely continuous with respectto the Lebesgue measure. The theorem is based on a new structural result for measures satisfying a PDEconstraint.We will also show some consequences of this fact concerning the structure of measures in Lipschitz differentia-bility spaces and discuss some ongoing work concerning the converse of Pansu Theorem.

Michael Dymond: Mapping n grid points onto a square forces an arbitrarilylarge Lipschitz constant

We prove that the regular n× n square grid of points in the integer lattice Z2 cannot be recovered from anarbitrary n2-element subset of Z2 via a mapping with prescribed Lipschitz constant (independent of n). Thisanswers negatively a question of Feige. Our resolution of Feige’s question takes place largely in a continuoussetting and is based on new results for Lipschitz mappings falling into two broad areas of interest, which westudy independently. Firstly we discuss Lipschitz regular mappings on Euclidean spaces, with emphasis ontheir bilipschitz decomposability in a sense comparable to that of the well known result of Jones. Secondly,we build on work of Burago and Kleiner and McMullen on non-realisable densities. We verify the existence,and further prevalence, of strongly non-realisable densities inside spaces of continuous functions. This is jointwork with Vojtěch Kaluža and Eva Kopecká.

10

Sylvester Eriksson-Bique: Classification of Poincaré inequalities and PI-rectifiability

I’ll discuss conditions equivalent to possessing Poincaré inequalities, and new constructions of PI-spaces byusing hyperbolic fillings.

Katrin Fässler: Quantitative rectifiability in the Heisenberg group

In geometric measure theory of the sub-Riemannian Heisenberg group, the class of intrinsic Lipschitz graphsplays an important role, in particular in connection with codimension 1 rectifiable sets. The goal of this talkis to present a quantitative notion of rectifiability in the Heisenberg group. Extending a result of David andSemmes from Euclidean spaces, we characterize sets with big pieces of intrinsic Lipschitz graphs as those setswhich satisfy a weak geometric lemma for vertical beta-numbers and which have big vertical projections. Theproof has connections to nonlinear PDEs, which do not play a role in the Euclidean theory. This is based onjoint work with V. Chousionis and T. Orponen.

De-Jun Feng: Dimension of invariant measures for affine iterated function sys-tems

Iterated function system (IFS) is a broad scheme for generating fractal sets and measures. In this talk, wediscuss the dimensional properties of certain fractal measures associated with affine IFS. We prove the exactdimensionality of ergodic stationary measures for any average-contractive affine IFS. Applications are givento the dimensions of self-affine sets, as well as their projections and slices.

Kornélia Héra: Hausdorff dimension of union of affine subspaces

It might be interesting to know for which type of sets the following heuristic principle can be applied: ans-Hausdorff-dimensional collection of d-dimensional sets in Rn must have positive measure if s+ d > n andHausdorff dimension s+ d if s+ d ≤ n.Falconer and Mattila proved that the union of any s-Hausdorff-dimensional family of affine hyperplanes inRn has Hausdorff dimension s+ n− 1 if s ∈ [0, 1], and positive Lebesgue-measure if s > 1.We generalized the result for the range s ∈ [0, 1], for k-dimensional affine subspaces (1 ≤ k ≤ n− 1) in placeof hyperplanes. We also proved a lower estimate for the Hausdorff dimension of generalized Furstenberg-typesets: sets intersecting every element of a given family of k-dimensional affine subspaces in a set of Hausdorffdimension at least α, where 0 < α ≤ k. To obtain these results, we used an L2 estimation technique.Joint work with Tamás Keleti and András Máthé.

11

Tamás Keleti: Fubini type result for Hausdorff dimension

It is well known that Fubini theorem does not hold for Hausdorff dimension: there exist compact subsets ofthe plane with Hausdorff dimension 2 such that all vertical sections contain at most one point. Our mainresult states that from every Borel set one can remove a small (Gamma-null) set such that Fubini holds forthe rest of the set. We also study the special case when our set is the union of lines. It turns out that thiscase is related to the Kakeya problem and to a 1-parameter family of orthogonal projections. Joint resultswith Kornélia Héra and András Máthé.

Bruce Kleiner: Sobolev mappings between Carnot groups, the exterior deriva-tive, and rigidity

I will discuss some rigidity theorems for quasiconformal and Sobolev mappings between Carnot groups, inparticular, for mappings between products. A key role is played by the pullback of differential forms usingthe Pansu derivative, and its interaction with distributional exterior derivatives.This is joint work with Stefan Muller and Xiangdong Xie.

Miklós Laczkovich: The story of squaring the circle

The story of measure theoretic circle squaring started in 1924 when Alfred Tarski asked whether a disc in theplane is equidecomposable with a square of the same area. The latest development in the topic is the resultof A. S. Marks and S. T. Unger, to appear in the Annals, stating that the disc is equidecomposable with asquare with Borel pieces and only using translations. In the talk I will outline the main steps and sketch theideas that led to this result.

Panu Lahti: Fine potential theory in metric spaces for p = 1

Consider a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality.I will discuss some recent results on functions of least gradient, i.e. BV functions that locally minimize thetotal variation, and related superminimizers. In particular, I will consider the (semi)continuity propertiesof such functions. Then I will show how certain notions of fine potential theory, such as thinness and fine(semi)continuity, can be extended from the case p > 1 to the case p = 1. Finally, I will consider a weakCartan property, and the relationship between finely open and quasiopen sets in the case p = 1.

Sean Li: Geometric characterizations of RNP differentiability

We give a geometric characterization of metric measure spaces satisfying a Cheeger-type differentiabilitytheorem for Lipschitz maps into all Banach spaces with the Radon–Nikodym property. Specifically, weshow that the space admits families of fragmented curves (Alberti representations) along which the space isinfinitesimally "quasiconvex". Time permitting, we also discuss a connection with the Poincaré inequality.

12

Valentino Magnani: Remarks on Hausdorff measure and differentiability

In noncommutative homogeneous groups, standard methods to establish the area formula by Rademacher’stheorem fail to apply. We show how the “adapted” Rademacher’s type theorem in these groups plays anunexpected role in computing the spherical measure of subsets. This is part of a work in progress.

Olga Maleva: Where can real-valued Lipschitz functions on Rn be non-differentiable?

There are subsets N of Rn for which one can find a real-valued Lipschitz function f defined on the whole ofRn but non-differentiable at every point of N . Of course, by the Rademacher theorem any such set N isLebesgue null, however, due to a celebrated result of Preiss from 1990 not every Lebesgue null subset of Rn

gives rise to such a Lipschitz function f .In this talk, I discuss sufficient conditions on a set N for such f to exist. As a corollary of the main resultwe show that every purely unrectifiable set U possesses a Lipschitz function non-differentiable on U in thestrongest possible sense. This is joint work with D. Preiss.

Jan Malý: Measuring families of curves by approximation modulus

Moduli of families of curves are widely used as a tool in potential theory, in the theory of Sobolev spacesor in quasiconformal theory and its generalizations. Recently, Olli Martio invented a new kind of modulus,so called approximation modulus (AM), which fits well in the BV theory (including the setting of metricmeasure spaces). We show that a Lorentz space version is suitable for neglecting some families of curvesnaturally related to sets of a given dimension. Also, we show an application to the Stokes theorem. This is ajoint work with Olli Martio and Vendula Honzlová Exnerová.

Lukáš Malý: Poincaré inequalities that fail to constitute an open-ended condition

It was proven by Keith and Zhong that a p-Poincaré inequality is an open-ended condition. Specifically, if acomplete metric space endowed with a doubling measure admits a p-Poincaré inequality with p > 1, then themetric space admits a q-Poincaré inequality for some q < p. This result was later refined by DeJarnette whoprovided some sufficient conditions for Orlicz-type Poincaré inequalities to self-improve in a similar fashion.However, optimality of these conditions was left unsolved.I will discuss self-improvement of more general types of Poincaré inequalities that are, in a way, close tothe usual p-Poincaré inequality. Such a setting allows us to pinpoint where the proof of open-endednesscould go wrong, which clears the way for finding a simple example of a complete Ahlfors regular metric spacethat supports a Lorentz-type Poincaré inequality that cannot be improved. In other words, a Lorentz-typePoincaré inequality can fail to be an open-ended condition. As a by-product, the sufficient conditions inDeJarnette’s paper are shown to be optimal.

13

Andrea Marchese: One-rectifiable representations of Euclidean measures andapplications

Rademacher’s Theorem asserts that Lipschitz functions on Rd are differentiable Lebesgue-a.e. Questions aboutits sharpness naturally arise. Given a positive measure µ on Rd, we construct a map V (µ, ·) : Rd → Gr(Rd),called the decomposability bundle of µ, whose values are vector subspaces. Such bundle is defined in terms ofone-rectifiable representations of (pieces of) µ and it encodes the information about the differentiablity ofLipschitz functions with respect to µ. More precisely, given a Lipschitz function on Rd, its restriction to theaffine subspace x+ V (µ, x) is differentiable at x, for µ-a.e. x; moreover it is possible to construct Lipschitzfunctions which do not admit any directional derivative outside the bundle, µ-a.e. I will also show that it ispossible to find Lipschitz functions attaining in the set of blowups, at many points, any prescribed functionthat is linear along the decomposability bundle. In the last part, I will discuss applications related to a resultof De Philippis–Rindler: we deduce a covering result for singular measures via Lipschitz slabs of arbitrarilysmall total width and we use it to prove non-continuity properties of the Jacobian determinants for singularmeasures. Based on joint works with Alberti, Csörnyei, Preiss, and Schioppa.

Annalisa Massaccesi: On the rank-one theorem for BV functions

We will present a recent elementary proof of the rank-one property for the singular part of the derivative of avector-valued BV function. Time permitting, the rank-one theorem for BV functions in Heisenberg groupswill be discussed. These results are based on joint works with S. Don and D. Vittone.

András Máthé: A covering theorem for closed covers of metric spaces

The Besicovitch covering theorem states that if a collection of nondegenerate closed balls in Rd with uniformlybounded diameters is given and E is the set of all centres then there exist cd subcolllections such that eachsubcollection consists of countably many disjoint balls and the union of all these balls covers E. As a corollary,a large portion of E is covered by a family of disjoint balls. Obviously, even the corollary may fail if we replaceballs by more general sets with given “centres”. It turns out that for certain applications (differentiation ofmeasures) a weaker conclusion is enough: that a large portion of E is covered by sets which do not containeach other’s centres. A general theorem of this form is the following:Let X be a separable metric space and µ a finite Borel measure on X. Consider any family of pairs (C, x)where C is closed and x (“centre” of C) is any point in X. Then we can choose finitely many sets C suchthat they do not cover each other’s centres and that their union has measure at least µ(X \ Y )/5, where Yhas the property that µ(C ∩ Y ) = 0 for every C with centre in Y .In the talk I will sketch the ideas behind the proof of this theorem, some of which are combinatorial. (Basedon joint work with David Preiss.)

14

Ulrich Menne: Sobolev-type spaces on varifolds

Even on stationary integral varifolds, there is no satisfactory class of Sobolev type spaces, that is closedwith respect to addition and post-composition, and that is well-behaved with respect to decompositions.Nevertheless, there is a coherent theory based on a purely geometric, non-additive class of functions witha linear subclass tailored for functional analytical considerations. Possible agreement of the latter withmetric-measure-theoretic Sobolev spaces then is a sign of regularity.

Dali Nimer: Geometry of uniform measures

A measure in Rd is called n-uniform if the measure of a ball of radius r centered at a point x of its support isgiven by crn for some fixed constant c. In this talk, I will present some results on their geometry including adescription of their support and a family of new examples of uniform measures.

Tuomas Orponen: Sharpening Marstrand’s projection theorem

I will discuss the problem of sharpening Marstrand’s projection theorem, and associated exceptional setestimates, in two and three dimensions.

Andrea Pinamonti and Gareth Speight: Maximal directional derivatives anduniversal differentiability sets in Carnot groups

Rademacher’s theorem asserts that Lipschitz functions from Rn to Rm are differentiable almost everywhere.Such a theorem may not be sharp: if n > 1 then there exists a Lebesgue null set N in Rn containing a pointof differentiability for every Lipschitz mapping f : Rn → R. Such sets are called universal differentiabilitysets and their construction relies on the fact that existence of an (almost) maximal directional derivativeimplies differentiability. We will see that maximality of directional derivatives implies differentiability in allCarnot groups where the Carnot-Caratheodory distance is suitably differentiable, which include all step 2Carnot groups (in particular the Heisenberg group). Further, one may construct a measure zero universaldifferentiability set in any step 2 Carnot group. Finally, we will observe that in the Engel group, a Carnotgroup of step 3, things can go badly wrong. . . Based on joint work with Enrico Le Donne.

Tapio Rajala: On density of Sobolev functions on Euclidean domains

I will present recent results on the density of the Sobolev space W k,q(Ω) in W k,p(Ω), when 1 ≤ p < q ≤ ∞for domains Ω in the Euclidean space. I will also discuss removability of sets of measure zero for Sobolevfunctions and extension operators from W 1,p(Ω) to W 1,p(Rn) when 1 ≤ p ≤ ∞. The talk is based on jointworks with P. Koskela, D. Nandi, T. Schultz and Y. Zhang.

15

Séverine Rigot: Besicovitch covering property on graded groups and applicationsto measure differentiation

In this talk we give a characterization of graded groups admitting homogeneous distances for which theBesicovitch covering property (BCP) holds. In particular it follows from this characterization that a stratifiedgroup admits homogeneous distances for which BCP holds if and only if it has step 1 or 2. We will illustratethis result with explicit examples of homogeneous distances satisfying BCP on the Heisenberg group. Wewill also discuss applications to measure differentiation which is one of the motivations for considering suchcovering properties. Most of this talk is based on a joint work with E. Le Donne.

Filip Rindler and Guido De Philippis: On the converse of Rademacher theoremand the rigidity of measures in Lipschitz differentiability spaces

See Guido De Philippis.

Elena Riss: A set of positive Gaussian measure with uniformly zero densityeverywhere

Gaussian measures in infinite dimensional spaces are often considered to be a suitable replacement forthe (non-existent) Lebesgue measure. For example, Rademacher’s Theorem holds for real-valued Lipschitzfunctions in separable Banach spaces when "differentiable" is understood as "directionally differentiable"and "almost everywhere" as "almost everywhere with respect to every non-degenerated Gaussian measure".However, many properties of Gaussian measures do not mimic those of the Lebesgue measure. In this talkwe show they fail the Density Theorem in an extremely strong sense: for a suitable Gaussian measure on aseparable Hilbert space there are sets of almost full measure whose ball density ratio tends to zero uniformlyon the whole space

Andrea Schioppa: Calculus on Metric Spaces: Beyond the Poincaré Inequality

We discuss a framework introduced by J. Cheeger (1999) to differentiate Lipschitz maps defined on metricmeasure spaces which admit Poincaré inequalities, and discuss (the first) examples on which it is still possibleto differentiate despite the infinitesimal geometry being incompatible with the Poincaré inequality.

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Nageswari Shanmugalingam: In the setting of metric measure spaces, what isDirichlet problem for functions of least gradient?

In studying the Dirichlet problem in metric measure spaces for the (non-linear) p-Laplacian with 1 < p <∞,it is clear what the Dirichlet problem should be when the boundary data is continuous. Such a problem willalways have a solution (and the solution is unique if the domain of interest is bounded).In the setting of functions of bounded variation, the differential operator is the 1-Laplacian, and solutionsare called functions of least gradient. Unlike in the setting of p-Laplacian for p > 1, the 1-Laplacian is lesswell-behaved, and therefore depending on the notion of Dirichlet problem, solutions may or may not exist. Inthis talk we will explore various notions of Dirichlet problem, and give a curvature condition on domains thatguarantee that all these various notions agree and that the solution will exist.

Gareth Speight and Andrea Pinamonti: Maximal directional derivatives anduniversal differentiability sets in Carnot groups

See Andrea Pinamonti.

Dario Trevisan: On level sets in the Heisenberg group

We present a recent joint work with V. Magnani and E. Stepanov where we introduce a new parametrizationof level sets of regular maps on the Heisenberg group taking values in R2, which allow for a “good calculus”on these nonsmooth objects. If time allows, we discuss some features of the corresponding problem for higherdimensional level sets.

Jeremy Tyson: Tube formulas, uniform measures and heat content in the Heisen-berg group

I will discuss several problems of geometric measure theory, differential geometry and subelliptic PDEin the sub-Riemannian Heisenberg group. The common theme is the appearance of local power seriesexpansions for geometric or analytic quantities (such as volumes of extrinsic and intrinsic neighborhoods ofsubmanifolds, short time asymptotics of the subelliptic heat content, etc.) featuring curvature invariants of(smooth) boundaries. Motivation comes from the study of densities of Radon measures and its connection torectifiability via uniform and uniformly distributed measures, as well as ongoing efforts to establish Weyltube formulas for the Carnot-Caratheodory metric.

Davide Vittone: Existence of tangent lines to sub-Riemannian geodesics

We consider the regularity problem for sub-Riemannian length minimizers, i.e., for curves of minimal lengthamong all Lipschitz curves with fixed endpoints and whose derivatives lie in a fixed bundle (of non-maximalrank) of admissible directions. We prove that, at any point, length minimizers possess at least one tangentcurve (i.e., a blow-up limit in the “tangent” Carnot group) equal to a straight horizontal line. This is a jointwork with R. Monti and A. Pigati.

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Thomas Zürcher: Sets where lip is infinite

In this talk, I will present joint work with Zoltán Buczolich, Bruce Hanson, David Preiss, and Martin Rmoutil.Given a continuous function f : [0, 1] → R, I will talk about characterizing the sets where the followingquantity is infinite:

lipf(x) = lim infr→0

supy∈B(x,r)

|f(y)− f(x)|r

.

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Participants

Giovanni Alberti Università di PisaMatthew Badger University of ConnecticutMarco Baffetti University of NottinghamDavid Bate University of HelsinkiAlan Chang University of ChicagoMiroslav Chlebík University of SussexAntoine Choffrut University of WarwickVasileios Chousionis University of ConnecticutMarianna Csörnyei University of ChicagoGuy C. David New York UniversityGuido De Philippis SISSAMichael Dymond Universität InnsbruckSylvester Eriksson-Bique

UCLA

Katrin Fässler University of FribourgDe-Jun Feng The Chinese University of Hong KongKornélia Héra Eötvös Loránd University, Institute of MathematicsWenshuai Jiang University of WarwickTomasz Kania University of WarwickTamás Keleti Eötvös Loránd UniversityBruce Kleiner Courant Instititute of Mathematical SciencesJan Kolář University of BirminghamKamil Kosiba University of WarwickKristýna Kuncová Charles University, Faculty of Mathematics and PhysicsMiklós Laczkovich Eötvös Loránd UniversityPanu Lahti Linköping UniversitySean Li University of ChicagoValentino Magnani University of PisaOlga Maleva University of BirminghamJan Malý Charles UniversityLukáš Malý University of CincinnatiAndrea Marchese University of ZurichAnnalisa Massac-cesi

Universität Zürich

András Máthé University of WarwickUlrich Menne University of ZurichAndrea Merlo Scuola Normale SuperioreAbdalla Dali Nimer University of WashingtonTuomas Orponen University of HelsinkiAndrea Pinamonti University of TrentoDavid Preiss University of Warwick

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Tapio Rajala University of JyväskyläSéverine Rigot Université Nice Sophia AntipolisFilip Rindler University of WarwickElena Riss Russian State Pedagogical UniversityMartin Rmoutil University of WarwickAndrea Schioppa Booking.com BVNageswari Shanmu-galingam

University of Cincinnati

Jack Skipper University of WarwickGareth Speight University of CincinnatiJaroslav Tišer Czech Technical UniversityDario Trevisan Università degli studi di PisaJeremy Tyson University of IllinoisMichele Villa MIGSAA (Maxwell Institute, Edinburgh)Davide Vittone University of PadovaPolina Vytnova University of WarwickPaul Wileman University of NottinghamThomas Zürcher University of Warwick

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