GeometricMeasureTheory

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Talks

Talks

Matthew Badger: Geometry of Radon measures via Hölderparameterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

David Bate: Rectifiability in metric and Banach spaces viaarbitrarily small perturbations. . . . . . . . . . . . . . . . . . . . . . . . . 5

Alan Chang: Small unions of affine subspaces and skeletonsvia Baire category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Marianna Csörnyei: The Kakeya needle problem and theexistence of Besicovitch and Nikodym sets for rectifiablesets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Guy C. David: Differentiability and rectifiability on metricplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Guido De Philippis and Filip Rindler: On the converse ofRademacher theorem and the rigidity of measures in Lip-schitz differentiability spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Michael Dymond: Mapping n grid points onto a squareforces an arbitrarily large Lipschitz constant . . . . . . . . . . . 7

Sylvester Eriksson-Bique: Classification of Poincaré inequal-ities and PI-rectifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

De-Jun Feng: Dimension of invariant measures for affineiterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Kornélia Héra: Hausdorff dimension of union of affine sub-spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Tamás Keleti: Fubini type result for Hausdorff dimension. 8

Bruce Kleiner: Sobolev mappings between Carnot groups,the exterior derivative, and rigidity . . . . . . . . . . . . . . . . . . . . 8

Miklós Laczkovich: The story of squaring the circle . . . . . . . 9

Panu Lahti: Fine potential theory in metric spaces for p = 1 9

Sean Li: Geometric characterizations of RNP differentiabil-ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Valentino Magnani: Remarks on Hausdorff measure and dif-

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Talks

ferentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Olga Maleva: Where can real-valued Lipschitz functions onRn be non-differentiable? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Jan Malý: Measuring families of curves by approximationmodulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Lukáš Malý: Poincaré inequalities that fail to constitute anopen-ended condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Andrea Marchese: One-rectifiable representations of Eu-clidean measures and applications. . . . . . . . . . . . . . . . . . . . . . 11

Annalisa Massaccesi: On the rank-one theorem for BV func-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

András Máthé: A covering theorem for closed covers of met-ric spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Ulrich Menne: Sobolev-type spaces on varifolds. . . . . . . . . . . 12

Dali Nimer: Geometry of uniform measures . . . . . . . . . . . . . . 12

Tuomas Orponen: Sharpening Marstrand’s projection theo-rem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Andrea Pinamonti and Gareth Speight: Maximal direc-tional derivatives and universal differentiability sets inCarnot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Tapio Rajala: On density of Sobolev functions on Euclideandomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Severine Rigot: Besicovitch covering property on gradedgroups and applications to measure differentiation . . . . . 13

Filip Rindler and Guido De Philippis: On the converse ofRademacher theorem and the rigidity of measures in Lip-schitz differentiability spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Elena Riss: A set of positive Gaussian measure with uni-formly zero density everywhere . . . . . . . . . . . . . . . . . . . . . . . . 14

Andrea Schioppa: Calculus on Metric Spaces: Beyond thePoincaré Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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TalksNageswari Shanmugalingam: In the setting of metric mea-

sure spaces, what is Dirichlet problem for functions ofleast gradient?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Gareth Speight and Andrea Pinamonti: Maximal direc-tional derivatives and universal differentiability sets inCarnot groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Dario Trevisan: On level sets in the Heisenberg group . . . . 15

Jeremy Tyson: Tube formulas, uniform measures and heatcontent in the Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . 15

Davide Vittone: Existence of tangent lines to sub-Riemannian geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Thomas Zürcher: Sets where lip is infinite . . . . . . . . . . . . . . . . 16

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Matthew Badger: Geometry of Radon measures via Hölder parameteriza-tions

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

David Bate: Rectifiability in metric and Banach spaces via arbitrarilysmall perturbations

This talk presents a new characterisation of compact, purelyd-unrectifiable metric spaces X with finite d-dimensional Hausdorff mea-sure. 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 atmost ε. The key point of our construction is that the Lipschitz constantof such a perturbation is independent of ε, which allows us to prove theconverse statement.In fact, we construct such a perturbation σ : X ⊂ B → B for any Banachspace B with an unconditional basis. The result for metric spaces isobtained 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 provedthe characterisation for Ahlfors regular subsets of Euclidian space. Hisproof crucially relies on the Besicovitch–Federer projection theorem, whichcompletely fails in the infinite dimensional setting, and our techniquesare different. However, our results can be considered as an alternative tothe projection theorem in this setting.

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Alan Chang: Small unions of affine subspaces and skeletons via Bairecategory

Our aim is to find the minimal Hausdorff dimension of the union of scaledand/or rotated copies of the k-skeleton of a fixed polytope centered at thepoints of a given set. For many of these problems, we show that a typicalarrangement in the sense of Baire category gives minimal Hausdorffdimension. In particular, this proves a conjecture of R. Thornton.Our results also show that Nikodym sets are typical among all sets whichcontain, for every x ∈ Rn, a punctured hyperplane H \ x through x.With similar methods we also construct a Borel subset of Rn of Lebesguemeasure zero containing a hyperplane at every positive distance fromevery point.

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

We show that the classical results about rotating a line segment inarbitrarily small area, and the existence of a Besicovitch and a Nikodymset hold if we replace the line segment by an arbitrary rectifiable set.This is a 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 dif-ferentiable structure in the sense of Cheeger and certain quantitativetopological control; specifically, their blowups are topological planes.We show that any differentiable structure on such a space is at most2-dimensional, and furthermore that if it is 2-dimensional the space is 2-rectifiable. This is partial progress on a question of Kleiner and Schioppa,and is joint work with Bruce Kleiner.

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

Rademacher Theorem asserts that every Lipschitz function is differentiablealmost everywhere with respect to the Lebesgue measure. Results in thisspirit have been established by Pansu in Carnot groups and by Cheeger inabstract metric measure spaces. A natural question is then the rigidity ofthose measures for which every Lipschitz function is differentiable almosteverywhere.Aim of the talk is to discuss this issue in increasing generality. In particularwe will present a proof of the fact that Rademacher Theorem can holdfor a measure if and only if it is absolutely continuous with respect tothe Lebesgue measure. The theorem is based on a new structural resultfor measures satisfying a PDE constraint.We will also show some consequences of this fact concerning the structureof measures in Lipschitz differentiability spaces and discuss some ongoingwork concerning the converse of Pansu Theorem.

Michael Dymond: Mapping n grid points onto a square forces an arbi-trarily large Lipschitz constant

We prove that the regular n×n square grid of points in the integer latticeZ2 cannot be recovered from an arbitrary n2-element subset of Z2 viaa mapping with prescribed Lipschitz constant (independent of n). Thisanswers negatively a question of Feige. Our resolution of Feige’s questiontakes place largely in a continuous setting and is based on new results forLipschitz mappings falling into two broad areas of interest, which we studyindependently. Firstly we discuss Lipschitz regular mappings on Euclideanspaces, with emphasis on their bilipschitz decomposability in a sensecomparable to that of the well known result of Jones. Secondly, we buildon work of Burago and Kleiner and McMullen on non-realisable densities.We verify the existence, and further prevalence, of strongly non-realisabledensities inside spaces of continuous functions. This is joint work withVojtěch Kaluža and Eva Kopecká.

Sylvester Eriksson-Bique: Classification of Poincaré inequalities andPI-rectifiability

I’ll discuss conditions equivalent to possessing Poincaré inequalities, andnew constructions of PI-spaces by using hyperbolic fillings.

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De-Jun Feng: Dimension of invariant measures for affine iterated functionsystems

Iterated function system (IFS) is a broad scheme for generating fractalsets and measures. In this talk, we discuss the dimensional properties ofcertain fractal measures associated with affine IFS. We prove the exactdimensionality of ergodic stationary measures for any average-contractiveaffine IFS. Applications are given to the dimensions of self-affine sets, aswell 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 followingheuristic principle can be applied: an s-Hausdorff-dimensional collectionof 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 in Rn has Hausdorff dimensions+ 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 affinesubspaces (1 ≤ k ≤ n−1) in place of hyperplanes. We also proved a lowerestimate for the Hausdorff dimension of generalized Furstenberg-type sets:sets intersecting every element of a given family of k-dimensional affinesubspaces in a set of Hausdorff dimension 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.

Tamás Keleti: Fubini type result for Hausdorff dimension

It is well known that Fubini theorem does not hold for Hausdorff dimen-sion: there exist compact subsets of the plane with Hausdorff dimension 2such that all vertical sections contain at most one point. Our main resultstates that from every Borel set one can remove a small (Gamma-null) setsuch that Fubini holds for the rest of the set. We also study the specialcase when our set is the union of lines. It turns out that this case isrelated to the Kakeya problem and to a 1-parameter family of orthogonalprojections. Joint results with Kornélia Héra and András Máthé.

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Bruce Kleiner: Sobolev mappings between Carnot groups, the exteriorderivative, and rigidity

I will discuss some rigidity theorems for quasiconformal and Sobolevmappings between Carnot groups, in particular, for mappings betweenproducts. A key role is played by the pullback of differential forms usingthe Pansu derivative, and its interaction with distributional exteriorderivatives.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 AlfredTarski asked whether a disc in the plane is equidecomposable with asquare 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 thatthe disc is equidecomposable with a square with Borel pieces and onlyusing translations. In the talk I will outline the main steps and sketchthe ideas 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 doublingmeasure and supports a Poincaré inequality. I will discuss some recentresults on functions of least gradient, i.e. BV functions that locallyminimize the total variation, and related superminimizers. In particular,I will consider the (semi)continuity properties of such functions. Then Iwill show how certain notions of fine potential theory, such as thinnessand fine (semi)continuity, can be extended from the case p > 1 to thecase p = 1. Finally, I will consider a weak Cartan property, and therelationship 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 aCheeger-type differentiability theorem for Lipschitz maps into all Banachspaces with the Radon–Nikodym property. Specifically, we show thatthe space admits families of fragmented curves (Alberti representations)along which the space is infinitesimally "quasiconvex". Time permitting,we also discuss a connection with the Poincaré inequality.

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Valentino Magnani: Remarks on Hausdorff measure and differentiability

In noncommutative homogeneous groups, standard methods to establishthe area formula by Rademacher’s theorem fail to apply. We show how the“adapted” Rademacher’s type theorem in these groups plays an unexpectedrole in computing the spherical measure of subsets. This is part of a workin 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 Lipschitzfunction f defined on the whole of Rn but non-differentiable at everypoint of N . Of course, by the Rademacher theorem any such set N isLebesgue null, however, due to a celebrated result of Preiss from 1990 notevery Lebesgue null subset of Rn gives rise to such a Lipschitz function f .In this talk, I discuss sufficient conditions on a setN for such f to exist. Asa corollary of the main result we show that every purely unrectifiable setU possesses a Lipschitz function non-differentiable on U in the strongestpossible 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 spaces or in quasiconformal theory and itsgeneralizations. 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 metric measure spaces). We show that a Lorentzspace version is suitable for neglecting some families of curves naturallyrelated to sets of a given dimension. Also, we show an application tothe Stokes theorem. This is a joint work with Olli Martio and VendulaHonzlová Exnerová.

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

It was proven by Keith and Zhong that a p-Poincaré inequality is anopen-ended condition. Specifically, if a complete metric space endowedwith a doubling measure admits a p-Poincaré inequality with p > 1,then the metric space admits a q-Poincaré inequality for some q < p.This result was later refined by DeJarnette who provided some sufficientconditions for Orlicz-type Poincaré inequalities to self-improve in a similarfashion. However, optimality of these conditions was left unsolved.I will discuss self-improvement of more general types of Poincaré in-equalities that are, in a way, close to the 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 ofa complete Ahlfors regular metric space that supports a Lorentz-typePoincaré inequality that cannot be improved. In other words, a Lorentz-type Poincaré inequality can fail to be an open-ended condition. As aby-product, the sufficient conditions in DeJarnette’s paper are shown tobe optimal.

Andrea Marchese: One-rectifiable representations of Euclidean measuresand applications

Rademacher’s Theorem asserts that Lipschitz functions on Rd are differen-tiable Lebesgue-a.e. Questions about its sharpness naturally arise. Givena 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 of one-rectifiable representations of (piecesof) µ and it encodes the information about the differentiablity of Lipschitzfunctions with respect to µ. More precisely, given a Lipschitz functionon Rd, its restriction to the affine subspace x+ V (µ, x) is differentiableat x, for µ-a.e. x; moreover it is possible to construct Lipschitz functionswhich do not admit any directional derivative outside the bundle, µ-a.e.I will also show that it is possible to find Lipschitz functions attainingin the set of blowups, at many points, any prescribed function that islinear along the decomposability bundle. In the last part, I will discussapplications related to a result of De Philippis–Rindler: we deduce acovering result for singular measures via Lipschitz slabs of arbitrarilysmall total width and we use it to prove non-continuity properties of theJacobian determinants for singular measures. Based on joint works withAlberti, Csörnyei, Preiss, and Schioppa.

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

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

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

The Besicovitch covering theorem states that if a collection of nonde-generate closed balls in Rd with uniformly bounded diameters is givenand E is the set of all centres then there exist cd subcolllections suchthat each subcollection consists of countably many disjoint balls and theunion of all these balls covers E. As a corollary, a large portion of E iscovered by a family of disjoint balls. Obviously, even the corollary mayfail if we replace balls by more general sets with given “centres”. It turnsout that for certain applications (differentiation of measures) a weakerconclusion is enough: that a large portion of E is covered by sets whichdo not contain each other’s centres. A general theorem of this form isthe 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 hasmeasure at least µ(X \Y )/5, where Y has the property that µ(C∩Y ) = 0for every C with centre in Y .In the talk I will sketch the ideas behind the proof of this theorem, someof which are combinatorial. (Based on joint work with David Preiss.)

Ulrich Menne: Sobolev-type spaces on varifolds

Even on stationary integral varifolds, there is no satisfactory class ofSobolev type spaces, that is closed with 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 with a linear subclass tailored for functionalanalytical considerations. Possible agreement of the latter with metric-measure-theoretic Sobolev spaces then is a sign of regularity.

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Dali Nimer: Geometry of uniform measures

A measure in Rd is called n-uniform if the measure of a ball of radius rcentered at a point x of its support is given 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 uniformmeasures.

Tuomas Orponen: Sharpening Marstrand’s projection theorem

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

Andrea Pinamonti and Gareth Speight: Maximal directional derivativesand universal 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 arecalled universal differentiability sets and their construction relies on thefact that existence of an (almost) maximal directional derivative impliesdifferentiability. We will see that maximality of directional derivatives im-plies differentiability in all Carnot groups where the Carnot-Caratheodorydistance is suitably differentiable, which include all step 2 Carnot groups(in particular the Heisenberg group). Further, one may construct a mea-sure zero universal differentiability set in any step 2 Carnot group. Finally,we will observe that in the Engel group, a Carnot group of step 3, thingscan 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)inW k,p(Ω), when 1 ≤ p < q ≤ ∞ for domains Ω in the Euclidean space. Iwill also discuss removability of sets of measure zero for Sobolev functionsand extension operators from W 1,p(Ω) to W 1,p(Rn) when 1 ≤ p ≤ ∞.The talk is based on joint works with P. Koskela, D. Nandi, T. Schultzand Y. Zhang.

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Severine Rigot: Besicovitch covering property on graded groups andapplications to measure differentiation

In this talk we give a characterization of graded groups admitting ho-mogeneous distances for which the Besicovitch covering property (BCP)holds. In particular it follows from this characterization that a stratifiedgroup admits homogeneous distances for which BCP holds if and only ifit has step 1 or 2. We will illustrate this result with explicit examples ofhomogeneous distances satisfying BCP on the Heisenberg group. We willalso discuss applications to measure differentiation which is one of themotivations for considering such covering properties. Most of this talk isbased on a joint work with E. Le Donne.

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

See Guido De Philippis.

Elena Riss: A set of positive Gaussian measure with uniformly zerodensity everywhere

Gaussian measures in infinite dimensional spaces are often consideredto be a suitable replacement for the (non-existent) Lebesgue measure.For example, Rademacher’s Theorem holds for real-valued Lipschitzfunctions in separable Banach spaces when "differentiable" is understoodas "directionally differentiable" and "almost everywhere" as "almosteverywhere with respect to every non-degenerated Gaussian measure".However, many properties of Gaussian measures do not mimic thoseof the Lebesgue measure. In this talk we show they fail the DensityTheorem in an extremely strong sense: for a suitable Gaussian measureon a separable Hilbert space there are sets of almost full measure whoseball density ratio tends to zero uniformly on the whole space

Andrea Schioppa: Calculus on Metric Spaces: Beyond the PoincaréInequality

We discuss a framework introduced by J. Cheeger (1999) to differentiateLipschitz maps defined on metric measure spaces which admit Poincaréinequalities, and discuss (the first) examples on which it is still possible todifferentiate despite the infinitesimal geometry being incompatible withthe Poincaré inequality.

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Nageswari Shanmugalingam: In the setting of metric measure spaces,what is Dirichlet 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 problemshould be when the boundary data is continuous. Such a problem willalways have a solution (and the solution is unique if the domain of interestis bounded).In the setting of functions of bounded variation, the differential operatoris the 1-Laplacian, and solutions are 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. In this talk we will explore variousnotions of Dirichlet problem, and give a curvature condition on domainsthat guarantee that all these various notions agree and that the solutionwill exist.

Gareth Speight and Andrea Pinamonti: Maximal directional derivativesand universal 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 wherewe introduce a new parametrization of level sets of regular maps on theHeisenberg group taking values in R2, which allow for a “good calculus”on these nonsmooth objects. If time allows, we discuss some features ofthe corresponding problem for higher dimensional level sets.

Jeremy Tyson: Tube formulas, uniform measures and heat content in theHeisenberg group

I will discuss several problems of geometric measure theory, differentialgeometry and subelliptic PDE in the sub-Riemannian Heisenberg group.The common theme is the appearance of local power series expansionsfor geometric or analytic quantities (such as volumes of extrinsic andintrinsic neighborhoods of submanifolds, short time asymptotics of thesubelliptic heat content, etc.) featuring curvature invariants of (smooth)boundaries. Motivation comes from the study of densities of Radonmeasures and its connection to rectifiability via uniform and uniformlydistributed measures, as well as ongoing efforts to establish Weyl tubeformulas for the Carnot-Caratheodory metric.

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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 length among all Lipschitz curves with fixedendpoints and whose derivatives lie in a fixed bundle (of non-maximalrank) of admissible directions. We prove that, at any point, lengthminimizers possess at least one tangent curve (i.e., a blow-up limit inthe “tangent” Carnot group) equal to a straight horizontal line. This is ajoint work with R. Monti and A. Pigati.

Thomas Zürcher: Sets where lip is infinite

In this talk, I will present joint work with Zoltán Buczolich, BruceHanson, David Preiss, and Martin Rmoutil. Given a continuous functionf : [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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