## Programme

The Semester Seminar will take place:

**9.IX-22.XI:** Tuesday and Thursday **14.00-17.00** (with coffee breaks around 15.30) in room 403, IMPAN

In addition, the following rooms have been reserved for the purpose of the Semester:

**2.IX-30.IX:** Monday-Friday, **10.00-17.00**, room 403

**1.X-22.XI:** Monday-Friday, **10.00-12.00**, room 405

**Schedule of the Semester Seminar (details to be announced gradually).**

September

**Tue 10 IX,** **14.00: Alan Chang (University of Chicago)**

** ****Title: **The Kakeya needle problem for rectifiable sets

**Abstract:** We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is joint work with Marianna Csörnyei.

**15.30: Antoine Julia (Universita degli Studi di Padova)**

**Title: **Stokes' Theorem and integration on integral currents

** Abstract: **Non absolutely convergent integrals yield very general versions of the Fundamental Theorem of Calculus and of the Divergence Theorem. Notable applications are linked to the removability of singularities for PDEs in divergence forms. I will present the integral of Henstock and Kurzweil, as well as that of Pfeffer. I will then show how the latter can be transposed to yield a Generalized Stokes' Theorem on singular integral currents in Euclidian space. In particular, I will discuss the condition for removability of singularities on these currents.

**Thu 12 IX,** **14.00-16.30 (coffee break 15-15.30):** **Riikka Korte**

**Title: **Two notions of functions of bounded variation and the Semmes pencil of curves

** Abstract: **We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide, if the measure is doubling and supports a 1-Poincaré inequality. We will also discuss the benefits of the new approach. In proving the result, we also show that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure. This is joint work with E. Durand-Cartagena, S. Eriksson-Bique and N. Shanmugalingam. The second result was obtained independently by K. Fässler and T. Orponen

**Mon 16 IX**, **14.00: Hubert Sidler** **(****University of Fribourg)**

**Title: **Harmonic quasi-isometric maps from Hadamard manifold into Gromov-hyperbolic CAT(0) spaces, part 1

** Abstract: **The Schoen-Li-Wang conjecture asserts that for every quasi-isometric map between rank-one symmetric spaces there is a unique energy minimizing harmonic map within bounded distance. Several breakthroughs by Markovic, Lemm-Markovic and Benoist-Hulin finally led to an affirmative answer to this conjecture. Later Benoist-Hulin generalized the existing results to the case of Hadamard manifolds with negatively pinched curvature. In a joint work with Stefan Wenger, we were able to generalise the existence part of Benoist-Hulin to the case where the target is only assumed to be CAT(0), Gromov-hyperbolic and proper.

In the** first part**, I will introduce the definitions and the motivation. In the **second part**, I will outline the proof.

**Tue 17 IX,** **14.00: Hubert Sidler (****University of Fribourg)**

**Title: ** Harmonic quasi-isometric maps from Hadamard manifold into Gromov-hyperbolic CAT(0) spaces, part 2

**Abstract: **The Schoen-Li-Wang conjecture asserts that for every quasi-isometric map between rank-one symmetric spaces there is a unique energy minimizing harmonic map within bounded distance. Several breakthroughs by Markovic, Lemm-Markovic and Benoist-Hulin finally led to an affirmative answer to this conjecture. Later Benoist-Hulin generalized the existing results to the case of Hadamard manifolds with negatively pinched curvature. In a joint work with Stefan Wenger, we were able to generalise the existence part of Benoist-Hulin to the case where the target is only assumed to be CAT(0), Gromov-hyperbolic and proper.

In the **first part,** I will introduce the definitions and the motivation. In the** second part**, I will outline the proof.

** 15.30: Luigi D’Onofrio (****Universita degli Studi di Napoli Parthenope, Italy)**

**Title: **Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces

**Thu 19 IX,**** 14.00-17.00 (coffee break 15-15.30) Tuomas Orponen**

** Title:** Cheeger’s theorem, Alberti representations, and the multilinear Kakeya inequality

** Abstract:** I will outline a relatively short proof of Cheeger’s generalisation of Rademacher’s theorem to nice metric spaces. On the way, I will talk about Alberti representations, and the connection between Cheeger’s theorem and the multilinear Kakeya inequality.

**Tue 24 IX,** **14.00: Katrin Faessler**

** Title: **The Riesz transform in the Heisenberg group

** Abstract: **A prototypical singular integral operator is the Riesz transform defined by convolution with the kernel x/|x|^3 on 2-dimensional subsets in Euclidean space R^3. This talk concerns a similar operator in a noncommutative group, the Heisenberg group H, equipped with a non-Euclidean metric. I will present a class of 1-codimensional subsets of H on which the Heisenberg Riesz transform is bounded in L^2, and I will explain some connections to harmonic functions and geometric measure theory in the Heisenberg group. The sets we consider are certain intrinsic Lipschitz graphs which satisfy a vertical oscillation condition inspired by recent work of A. Naor and R. Young. The talk is based on collaboration with T. Orponen.

** 15.30: Tuomas Orponen**

** Title:** On two notions of rectifiability in the Heisenberg group

** Abstract:** I will discuss two notions of rectifiability of surfaces in the first Heisenberg group. The first one is built around Lipschitz images of the parabolic plane, while the second one is the “intrinsic” notion, based on H-regular surfaces. The (possible) equivalence of these notions is not fully understood, but I will discuss what is known, and mention some open problems.

**Thu 26 IX, ****14.00: Giona Veronelli (University of Milano)**

** Title:** Distance-like functions and Sobolev spaces on manifolds

** Abstract:** Let (M,g) be a complete non-compact Riemannian manifold. The distance function r(x) from a fixed reference point in general fails to be everywhere differentiable. We seek for geometric assumptions which guarantee the existence of a function H on M which is smooth, distance-like (i.e. r(x)/C < H(x) < Cr(x) outside a compact set) and whose derivatives are bounded up to a certain order. We will present classical results and some more recent answers to this problem. As we will see, distance-like functions permit to prove the density of smooth compactly supported functions in Sobolev spaces on manifolds, and to generalize to M other analytic tools and properties which are well-known in the Euclidean space.

This is a joint project with Debora Impera and Michele Rimoldi.

**15:30: Martin Fitzi (University of Fribourg)**

** Title:** ** **Morrey's* epsilon*-conformality lemma in metric spaces

October

**Tue 1 X,** **14.00-17.00 Sebastiano Golo **

**Title: **Minimal surfaces in the Heisenberg group

**Abstract:**** **The Heisenberg group has a well studied sub-Riemannian structure. Geometric measure theory in the sub-Riemannian setting is still in development and several fundamental questions are still open. One reason is that sets of finite sub-Riemannian perimeter may have fractal behaviours.

I will present some of the most recent results on minimal surfaces in this setting, in particular: a solution to the Bernstein problem for intrinsic graphs with Euclidean Lipschitz regularity, an example of a stable surface that is not area minimizer, and some remarks about contact variations of the area functional.

**Title: **Tangent spaces

**Abstract: **This is an overview of notions of tangent space, from the classical Euclidean case to metric spaces.

**Wed 2 X,** **14.00 Sebastiano Golo **

**Title: **Metric Lie groups

**Abstract: **We are interested in classifications of locally compact metric groups up to quasi-isometry, or bi-Lipschitz equivalence or isometry. I will introduce the subject, present the most important conjectures and say what is known at the moment.

**15.30 Zhuang Wang (University of Jyvaskyla)**

**Title: **Dyadic energies on the boundaries of regular trees and the related trace results on regular trees.

**Abstract: **The regular tree is a special metric measure space whose boundary is a Cantor-type set. I will talk about the function spaces defined via dyadic energies on the boundaries of regular trees and I will show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces. If there is time left, I would like to present some trace results on regular trees with more general metrics and measures.

**Thu 3 X,** **at 14.00, ****r. 321 IMPAN,**** ****IMPAN-PTM Colloquium, Pekka Koskela**

** ** **Title: **Functional properties of Sobolev extensions

**Abstract:** A given domain (open and connected set) in a Euclidean space may or may not have the property that each function in a Sobolev space defined over the domain is actually the

restriction of a function in the analogous Sobolev space to the domain in question. In the positive case, there is a bounded extension operator. I will discuss results related to the dependence of this extension property on the integrability degree and on the number of derivatives in the definition of the chosen Sobolev space. I will also give the other known functional properties for

this kind of extendability. A number of open problems and conjectures get stated.

**Sun 6 X-18. X**, Workshop in Geometry and Analysis

**Tue 22 X,** **14.00 Matthew Badger (University of Connecticut)**

**Title: **Recent Developments in Fractional Rectifiability

**Abstract: **One of the special features of one-dimensional metric geometry is the compatibility of intrinsic and extrinsic measurements of length of a curve. In fact, a theorem of Ważewski from the 1920s asserts that every curve with finite one-dimensional Hausdorff measure (extrinsic length) admits a parameterization by a Lipschitz map with finite total variation (intrinsic length). For higher-dimensional curves, this phenomenon breaks down. Every curve of finite *s*-variation has finite *s*-dimensional Hausdorff measure, but not conversely. This situation deserves further investigation.

In this talk, I will discuss recent progress on understanding the geometry of *(1/s)*-Hölder curves, which are reasonable models for *s*-dimensional curves when *s>1*, obtained in joint work with L. Naples and V. Vellis. Among other results, we prove a higher-dimensional Ważewski-type theorem for almost flat curves and establish a version of the Analyst's Traveling Salesman Theorem for Hölder curves. While this work has increased our understanding of higher-dimensional curves, there are still more questions than answers.

** 15.30 Nageswari Shanmugalingam**

**Title: **DeGiorgi measure and a double obstacle problem

**Abstract: **We will discuss a double obstacle problem related to BV theory, and a DeGiorgi measure associated with this problem. The setting is metric measure spaces equipped with a doubling measure supporting a 1-Poincarè inequality.

**Thu 24 X,** **14.30 **Futoshi Takahashi (Osaka City University)

**Title: **SHARP HARDY-LERAY INEQUALITIES FOR CURL-FREE VECTOR FIELDS

**Tue 29 X,** **14.00 Andreas Minne (KTH)**

**Title: **Obstacle-Type Problems in the Subelliptic Setting

**Abstract: **One of the most classical problems in the field of free boundaries is the obstacle problem. It asks for the properties of an energy minimizer in a domain, but with the additional constraint that it has to lie above some given function. In this seminar we briefly present some long-established results of the optimal regularity of solutions, and then show our recent theorem: the optimal regularity can be achieved also in the subelliptic setting for a more general type of problems. Key ideas are the use of BMO-estimates, and a decay estimate for the so-called coincidence set. These results have been obtained in collaboration with Valentino Magnani (Università di Pisa).

** 15.30 David Tewodrose (Université de Cergy-Pontoise)**

**Title: **AMV Laplacian on Metric Measure Spaces

**Abstract: **In the Euclidean space, a C^{2} function is harmonic if and only if it satisfies the mean-value property. Since the mean-value property makes sense on any general metric measure space, this observation could serve to define the notion of an harmonic function in this broad context. This approach was pursued in the recent years by Adamowicz, Gaczkowski and Górka, leading to the definition of strongly/weakly harmonic functions. In this talk I will present a joint work with Andreas Minne (KTH) in which we use the mean-value property in another (asymptotic) way to define a pointwise Laplacian on general metric measure spaces. I will provide some examples, results and perspectives on this new notion.

**Wed 30 X, 14.00** **Hiroshi Ohtsuka (Kanazawa University)**

**Title: **On the linear response of equilibrium vortices.

**Abstract: **Motivated by the experimental facts, we are interested in the precise structure of the correlation function for equilibrium states of vortices. Our strategy is the combination of the linear response theory of Green-Kubo and the mean field approximation. Unfortunately, we are not yet able to reach the end of the story mathematically rigorously, but we think that there are a lot of mathematically interesting problems in it. In this talk, we will present this story and several mathematical facts we have got up to now, e.g., the mean field equation of perturbed system of vortices based on the argument of Messer-Spohn ('82 J. Statist. Phys.) and Caglioti-Lions-Marchioro-Pulvirenti (’92 Comm. Math. Phys.), and an example of an expected mean field limit of the correlation function derived from the Prajapat-Tarantello entire space solution for the two dimensional singular Liouville equation ('01 Proc. Roy. Soc. Edinburgh Sect. A). This is based on the joint works work with physicists, Y. Yatsuyanagi (Shizuoka University) and T. Hatori (National Institute for Fusion Science).

**Thu 31 X,** **14.00 Tadeusz Iwaniec (Syracuse University)**

**Title: **LIMITS OF SOBOLEV HOMEOMORPHISMS AND ENERGY-MINIMAL DEFORMATIONS

** 15.30 Jani Onninen (Syracuse University & Jyvaskyla University)**

**Title: **Monotone Hopf-harmonics

**Abstract: **I introduce the concept of monotone Hopf-harmonics in 2D as an alternative to harmonic homeomorphisms. It opens a new area of study in Geometric Function Theory. Much of the foregoing is motivated by the principle of non-interpenetration of matter in the mathematical theory of Nonlinear Elasticity. In particular, the question we are concerned with is whether or not a Dirichlet energy-minimal mapping between Jordan domains with a prescribed boundary homeomorphism remains injective in the domain. This talk is based on my joint work with Tadeusz Iwaniec.

November

**Tue 5 XI, 14.00 Jose Llorente (Autonomous University of Barcelona)**

**Title: **Regularity of measures satisfying the annular decay condition

**15.30 Yi Ru-Ya Zhang (ETH)**

**Title: ** Strong stability for the Wulff inequality with a crystalline norm

**Wed 6 XI, 14.00, (joint meeting with Functional Analysis Seminar) ****Piotr Hajłasz (University of Pittsburgh)**

** Title: **Implicit Function Theorem for Lipschitz mappings into metric spaces

**Thu 7 XI, 14.00 Ilmari Kangasniemi (University of Helsinki)**

**Title: **On automorphic quasiregular maps

**Abstract: **We present an obstruction result of Martio for periodic quasiregular maps, and discuss its generalization to quasiregular maps which are automorphic under a discrete group of Euclidean isometries. While initally motivated by the study of Lattès-type uniformly quasiregular maps, the generalization instead runs into a question due to a technical assumption used in the proof.

** 15.30 Maria J. Gonzalez (University of Cadiz)**

**Title: **HAUSDORFF MEASURES, DYADIC APPROXIMATIONS AND DOBINSKI SET

**Abstract: **Dobinski set D is an exceptional set for a certain infinite product identity, whose points are characterized as having exceedingly good approximations by dyadic rationals. We study the Hausdorff dimension and logarithmic measure of D by means of the Mass Transference Principle and by the construction of certain appropriate Cantor-like sets, termed willow sets, contained in D. We will end this presentation with an open question relative to the logarithmic capacity of this set.

**Tue 12 XI, 14.00 Pekka Pankka (University of Helsinki)**

**Title: **Quasiregular curves

**Abstract: **The analytic definition of quasiregular mappings requires that the domain and range of the map have the same dimension. This equidimensionality presents itself also in the fundamental topological results on quasiregular mappings. In this talk, I will discuss an extension of quasiregular mappings, called quasiregular curves, for which the range may have higher dimension than the domain.

** 15.30 Elefterios Soultanis (University of Fribourg)**

**Title: **Metric currents and polylipschitz forms (part I)

**Abstract: **Euclidean k-currents are functionals on differential k-forms. Ambrosio-Kirchheim gave a far reaching generalization, introducing k-currents on metric spaces as multilinear functionals on (k+1)-tuples of Lipschitz functions. I will give a brief overview of both Euclidean and metric currents, after which I explain the notion of polylipschitz forms on metric spaces, and describe how they form a predual to metric currents, in the Euclidean spirit. Time permitting, I will describe an application to geometric mapping theory: defining the pull-back of metric currents by BLD-maps, and homological boundedness of metric spaces admitting a BLD-map from Euclidean space. This new approach and the application is joint work with Pekka Pankka (https://arxiv.org/abs/1902.06106 and https://arxiv.org/abs/1809.03009). ** **

**Thu 14 XI, 14.00-16.30 Elefterios Soultanis (University of Fribourg)**

**Title: **Metric currents and polylipschitz forms (parts II-III)

**Abstract: **Euclidean k-currents are functionals on differential k-forms. Ambrosio-Kirchheim gave a far reaching generalization, introducing k-currents on metric spaces as multilinear functionals on (k+1)-tuples of Lipschitz functions. I will give a brief overview of both Euclidean and metric currents, after which I explain the notion of polylipschitz forms on metric spaces, and describe how they form a predual to metric currents, in the Euclidean spirit. Time permitting, I will describe an application to geometric mapping theory: defining the pull-back of metric currents by BLD-maps, and homological boundedness of metric spaces admitting a BLD-map from Euclidean space. This new approach and the application is joint work with Pekka Pankka (https://arxiv.org/abs/1902.06106 and https://arxiv.org/abs/1809.03009).

**Fri 15 XI, 14.00 (room 321) Alexander Lytchak (Universität Köln)**

**Title: **Minimal discs in singular spaces and applications

**Abstract: **In the talk I will describe applications to metric geometry of the classical solution of Plateau's problem. The presented results are obtained jointly with Stefan Wenger and Stephan Stadler.

**15.30 (room 321) Gianmarco Giovanardi (University of Bologne & University of Granada)**

**Title: **Variations for submanifolds in graded manifolds

**Abstract:**** **The aim of this talk is to present the deformability properties of submanifolds immersed in graded manifolds that are a generalization of Carnot manifolds. We consider an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. In the one-dimensional case, the integrability of compact supported vector fields depends on the surjection of the holonomy map at the endpoints. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. This talk is based on my joint work with G. Citti and M. Ritoré.

**Mon 18 XI, 14.00 Renjin Jiang (Tianjin University)**

** Title:** Some developments of the Riesz transform via heat kernels and harmonic functions

**Abstract:** Let $(X,d,\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\mathscr{E}$ deriving from a ``carr\'e du champ". Assume the related heat semi-group admits a heat kernel satisfying an upper Gaussian bounds.

Recently, we studied the boundedness of the Riesz transform and regularity of heat kernels and harmonic functions: In particular, we established equivalences between: $L^p$-estimate for the gradient of the associated heat semigroup, $p>2$, $L^p$-reverse H\"older inequality for the gradients of harmonic functions, $L^p$-boundedness of the Riesz transform ($p<\infty$) .

The equivalence is proved for all $2<p<\infty$ under the requirement of an $L^2$ Poincare inequality, or for $p$ in 2 and the reverse doubling index of the measure under the requirement of an $L^2$ Poincare inequality on the ends of $X$. The stability of the boundedness of the Riesz transform under gluing operation and perturbation is also studied.

**Tue 19 XI, 14.00-16.30, Sylvester Eriksson-Bique (UCLA)**

**Title: **The Poincaré inequality as a quantitative notion of connectivity, and its relation to embeddability of carpets (parts I-II).

**Abstract: **The Poincaré inequality is an old friend to an analyst working in analysis on metric spaces. However, until recently, many basic questions have remained unanswered and this object has remained somewhat mysterious. Many of the questions revolve around how one might hope to prove such an inequality in different cases. In the planar case, these inequalities have further both positive and negative implications for the embeddability of a metric measure space.

I will give a series of three lectures, where I start from the characterization of the Poincaré inequality, and how on the one hand this condition is a notion of ``quantitative connectivity’’, and on the other hand how precisely (via convex duality) this can be characterized by a so called Semmes family of curves. I will outline some consequences of these new characterizations, and how it gives a new perspective on old results such as the seminal result of Keith and Zhong. These characterizations are also related to a resolution of a question on the relationship between differentiability and Poincaré inequalities. The final lecture will then discuss the special case of carpets, and planar metric spaces, and show how possessing such an inequality, or something similar, is related to both embeddability and non-embeddability of a metric measure space. This last part will solve a question from the list of Heinonen and Semmes on the (non) bi-Lipschitz embeddability of certain spaces admitting regular mappings to Euclidean space.

**Thu 21 XI, 14.00, Sylvester Eriksson-Bique (UCLA) **

** ** **Title: **The Poincaré inequality as a quantitative notion of connectivity, and its relation to embeddability of carpets (part III)

**Abstract: **See 19.XI

** ****15.30 Eden Prywes (UCLA)**

** Title: **Characterization of Branched Covers with Simplicial Branch Sets

** Abstract: **A branched covering $f \colon \mathbb R^n \to \mathbb R^n$ is an open and discrete map. Branched coverings are topological generalizations of quasiregular and holomorphic mappings. The branch set of $f$ is the set where $f$ fails to be locally injective. It is well known that the image of the branch set of a PL branched covering between PL $n$-manifolds is a simplicial $(n-2)$-complex.

I will discuss a recent result that the reverse implication also holds. More precisely, a branched covering with the image of the branch set contained in a simplicial $(n-2)$-complex is equivalent up to homeomorphism to a PL mapping. This result is classical for $n=2$ and was shown by Martio and Srebro for $n = 3$. This is joint work with Rami Luisto.

**24-30.XI** Conference in Będlewo

**Tue 3 XII**,** 14.00, Valentino Magnani (Pisa University)**

** Title:** Recent results on the intrinsic area of submanifolds in homogeneous groups

** Abstract:** We present the problem of computing the area of smooth submanifolds in a homogeneous group, equipped with its natural distance. This distance is not bi-Lipschitz equivalent to the Euclidean distance, not even locally, so the classical tools for the area formula in Euclidean spaces do not work. We discuss a different approach to the problem, along with recent results and open questions.