Junior Seminar
Elefterios Soultanis (University of Helsinki)
Title: Lifts and the existence of p-energy minimizers in homotopy classes of Newtonian maps.
Abstract: This talk ties together some of the aspects in this reseach term by considering classes of Newtonian (Sobolev type) maps between a (compact) doubling Poincare space and a metric space of nonpositive curvature (locally CAT(0)). I define what it means for two Newtonian maps to be homotopic, and consider the question of minimizing the p-energy in a given homotopy class. The nonpositive curvature is used to reduce the issue to some questions concerning the existence of lifts of Newtonian maps.
Li Chen (ICMAT)
Title: Sobolev inequalities and Riesz transforms on Vicsek graphs
Abstract: Consider graphs with polynomial volume growth, we give the generalised \(L^p\) Poincar\'e inequality and Sobolev inequality. In particular, they are optimal on Vicsek graphs. We also study the Riesz transform on graphs with sub-Gaussian heat kernel estimates. We prove the \(L^p\) boundedness of quasi Riesz transforms for \(1
2\).
Colleen Ackerman (University of Iilinois at Urbana-Champaign)
Title: Quasiconformal and Quasisymmetric Mappings on the Grushin Plane.
Abstract: I will give a brief introduction to planar quasiconformal and quasisymmetric mappings, and then explain how we can expand the theory to a class of generalized Grushin planes. These Grushin spaces are of interest to us, because they lack the regularity properties of other spaces where quasiconformal mappings have been studied in the past.
Jordi López Abad (ICMAT)
Title: The Ramsey property of finite dimensional normed spaces.
Abstract: It is well known that the approximate structural Ramsey property characterizes the fixed point property of the automorphism group of “Rich” metric structures. Recall that a topological group G has the fixed point property (extremely amenable) when every continuous action of G on a compactum has a fixed point. The group of linear isometries of the separable infinite dimensional Hilbert space or the group of the isometries of the universal Urysohn space, with their pointwise topologies, are extremely amenable. This is a consequence of the approximate Ramsey property (ARP) of those structures. A family F of metric structures of the same sort has the ARP when for every X; Y 2 F and every " > 0 there is Z 2 C such that for every real-valued 1-Lipschitz mapping f defined on the set of linear isometric embeddings Emb(X;Z) from X into Z there exists 2 Emb(Y;Z) such that Osc(f; Emb(X; Y )) < ": In a joint work with D. Bartosova and B. Mbombo (U. Sao Paulo) we prove
Matthew Romney (University of Iilinois at Urbana-Champaign)
Title: Bi-Lipschitz embedding of spaces with Whitney curvature bounds.
Abstract: When does a metric space admit a bi-Lipschitz embedding into some Euclidean space? This question in difficult in general and has been studied by many authors. In this talk, we give a new sufficient condition for embeddability in the context of Alexandrov spaces, based on the idea of curvature bounds on each cube of a Whitney decomposition. This condition is motivated by the classical Grushin plane and leads to a simpler proof of the fact (first proved by Seo) that the Grushin plane is bi-Lipschitz embeddable in Euclidean space.
Tomasz Kostrzewa (Warsaw University of Technology)
Title: Sobolev spaces on locally compact abelian groups.
Abstract: In this talk I will recall so basic concepts of locally compact abelian groups (LCA) such as the Haar measure, the dual group and the Fourier transform. Using this tools I will define Sobolev spaces on any LCA group and discuss some of their properties, like continuous and compact embeddings. Furthermore, we will discuss what happens when the group is metrizable. This talk is based on papers written together with P.Górka.
Rami Luisto (University of Helsinki)
Title: Local and global properties of BLD-mappings
Abstract: In this talk we define mappings of Bounded Length Distortion and introduce ourselves to their basic structure. The main theme of the talk is to study how the local behaviour of BLD mappings gives rise to global properties. The structure of the talk is based on the speakers preprint "Note on local-to-global properties of BLD-mappings”.
Scott Zimmerman (University of Pittsburgh)
Title: Geodesics in the Heisenberg Group \(\mathbb{H}^n\)
Abstract: In this talk, we will explore the basic geometry of the Heisenberg Group \(\mathbb{H}^1\), and we will discuss a simple argument which provides an explicit formulation of geodesics in the group. Further, we will examine the higher dimensional Heisenberg Groups \(\mathbb{H}^n\) and see a new proof (involving Fourier Series) which provides parametric equations for geodesics in these groups. If time permits, we will also discuss a corollary showing that the distance function in each Heisenberg Group \(\mathbb{H}^n\) is real analytic. The talk will be based on a paper submitted for publication by the speaker and Dr. Piotr Hajłasz.
Vyron Vellis (University of Jyvaskyla)
Title: Quasiconformal non-parametrizability of almost smooth spheres
Abstract: What properties of a metric \(d\) ensure that \((\mathbb{S}^n,d)\) is quasiconformal to \(\mathbb{S}^n\)? We show that, for each \(n\ge 4\), there exists a smooth Riemannian metric \(g\) on a punctured sphere \(\mathbb{S}^n\setminus \{x_0\}\) for which the associated length metric extends to a length metric \(d\) of \(\mathbb{S}^n\) with the following properties: the metric sphere \((\mathbb{S}^n,d)\) is Ahlfors \(n\)-regular and linearly locally contractible but there is no quasiconformal homeomorphism \((\mathbb{S}^n,d)\to \mathbb{S}^n\). This is a joint work with Pekka Pankka.
Sebastiano Nicolussi Golo (Jyvaskyla & Trento)
Title: Regularity of spheres in metric groups
Abstract: We study the possible regularities of spheres in graded groups for distances that are 1-homogeneous with respect to a family of dilations.
SubRiemannian and subFinsler Carnot groups are examples of graded groups.
In fact, we are interested in rectifiability properties, presence of cusps, and finiteness of Riemannian surface area of such spheres.
In particular, we present a more precise analysis for the Heisenberg group.
Kostiantyn Drach (V.N. Karazin Kharkiv National University; and Sumy State University)
Title: Reverse isoperimetric inequality for strictly convex curves
Abstract: The classical isoperimetric inequality on the Euclidean plane states that among all simple closed curves of fixed length the largest area is enclosed by a circle. What about the smallest possible area in this case? Obviously, it can be arbitrary close to zero. At the same time, sticking to a natural class of closed strictly convex curves, it appears that there is a non-trivial solution for such a reverse-type isoperimetric problem. In the talk, we will see how the mixture of differential geometry with Pontryagin's Maximum Principle helps to get the result in \(2\)-dimensional constant curvature spaces.
Furthermore, we will discuss how to extend it to CBB(\(\kappa\)) spaces.
This is a joint work with prof. Alexander Borisenko.
Isidro Humberto Munive Lima (SISSA)
Title: Curvature of sub-Riemannian manifolds
Abstract: In this talk we study the curvature of sub-Riemannian manifolds. In particular, we study the curvature of the Goursat distribution that naturally contains the three-dimensional Heisenberg group and the Engel group. We begin our study with the Heisenberg group by means of the generalized curvature-dimension inequality recently introduced by Baudoin and Garofalo. We then compute the sub-Riemannian sectional curvature of the Goursat distribution. This notion of sub-Riemannian sectional curvature was recently introduced by Agrachev, Barilari and Rizzi.
Martina Aaltonen (Helsinki)
Title: Monodromy representations of branched coverings
Abstract: A covering map \(f \colon X \to Z\) between manifolds is a factor of such a normal covering map \(q \colon Y \to Z\) that the deck-transformation group of \(q\) is isomorphic to the monodromy group of \(f.\) In this talk we see a ramification of this construction for a class of branched coverings between manifolds.
Aapo Kauranen (Jyvaskyla)
Title: Images of porous sets under Sobolev mappings
Abstract: Let \(\Omega\subset \mathbb{R}^n.\) I t is well known that mappings in \(W^{1,n}(\Omega, \mathbb{R}^n)\) may map sets of Lebesgue measure zero to sets of positive measure. This may happen even for porous sets.
On the other hand, under some assumptions on the modulus of continuity of the mapping it is true that images of porous sets will be sets of measure zero.
I will also mention the corresponding results in the context of \(Q\)-Ahlfors regular metric measure spaces.
Sita Benedict (Jyvaskyla)
Title: \(H^\psi\) spaces of conformal mappings on the unit disk
Abstract: By replacing the distance \(|f(z)|\) in the classical definition of Hardy spaces with the intrinsic path distance between \(f(z)\) and \(f(0)\), we obtain a new type of Hardy space. Since a conformal mapping belonging to the intrinsic Hardy space will automatically belong to the classically defined space, it is natural to ask for which \(0 < p < \infty\) these spaces coincide. Using Modulus of Curve Families and a number of tools from Harmonic Analysis, we give a complete answer to this question in the context of more general growth functions \(\psi\) in place of the functions \(t^p\). Joint work with P.Koskela.