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Pekka Pankka

Publications and source records attributed to Pekka Pankka.

At least 19 recordsLinked to original sources
arXiv2026

Unveiling topology in imaging problems via quasi-isometry and persistent homology

We show that the topological structures, such as loops, voids, and higher-dimensional holes of unknown objects (of flow of an object in space-time) can be recovered from noisy and indirect measurements. More precisely, we describe how the part of the persistent homology of a space can be determined from a noise-prone and discretized model space when there is a quasi-isometry between the original space and the space modeling indirect measurements. The result not only guarantees the existence of the structures but also provides size bounds for them. The structure is studied using persistent homology, and the results assume the existence of a quasi-isometry between a model space and the noisy measurements. We explore imaging problems, particularly X-ray imaging and EIT, that are well-suited to this framework.

math.AT↗
arXiv2026

On Lakes of Wada

There exist Lakes of Wada in $\mathbb S^n, n\ge 3,$ which are quasiconformally equivalent to a Euclidean ball and are John domains.

math.CV↗
arXiv2025

The Heinonen-Semmes problems after thirty years

We survey the current status of the questions posed by Juha Heinonen and Stephen Semmes in `Thirty-three yes or no questions about mappings, measures, and metrics' (Conformal Geometry and Dynamics, 1997).

math.MG↗
arXiv2025

Nodal resolution of quasiregular curves via bubble trees

We prove a version of Gromov's compactness theorem for quasiregular curves into calibrated manifolds with bounded geometry. In our main theorem, given an $n$-dimensional calibration $\omega$ on manifold $N$, we associate to a weak-$\star$ limit $\mu = \lim_{k \to \infty} \star F_k^*\omega$ of measures induced by a sequence $(F_k \colon X\to N)_{k\in \mathbb{N}}$ of $K$-quasiregular $\omega$-curves on a nodal manifold $X$, a bubble tree $\widehat X$ over $X$, a sequence of mappings $(\widehat F_\ell \colon X \to N)_{\ell \in \mathbb{N}}$ converging locally uniformly to a quasiregular curve $\widehat F\colon \widehat X\to N$ which realizes the measure $\mu$, that is, $\mu = \pi_*(\star \widehat F^*\omega)$, where $\pi \colon \widehat X\to X$ is the natural projection. We call the sequence $(\widehat F_\ell)_{\ell \in \mathbb{N}}$ a nodal resolution of the sequence $(F_k)_{k\in \mathbb{N}}$. As a corollary we obtain a normality criterion for families of quasiregular curves. Classic interpretations of bubbling via Gromov--Hausdorff convergence and pinching maps also follow.

math.DG↗
arXiv2024

Rigidity and quasisymmetric uniformization of Thurston-type maps

We prove the No Invariant Line Fields conjecture for a class of generalized postcritically-finite branched covers on higher-dimensional Riemannian manifolds. Moreover, we establish a quasisymmetric uniformization theorem for this class of generalized postcritically-finite maps.

math.DS↗
arXiv2024

Liouville's theorem in calibrated geometries

We consider the following extension of the classical Liouville theorem: A calibration $\omega \in \Lambda^n \mathbb{R}^m$, where $3 \le n \le m$, has the Liouville property if a Sobolev mapping $F\colon \Omega \to \mathbb{R}^m$, where $\Omega \subset \mathbb{R}^n$ is a domain, in $W^{1,n}_{loc}( \Omega, \mathbb{R}^m )$ satisfying $\|DF\|^n = \star F^{*}\omega$ almost everywhere is a restriction of a M\"obius transformation $\mathbb{S}^m \to \mathbb{S}^m$. We show that, for $m\ge 5$, every calibration in $\Lambda^{m-2} \mathbb{R}^m$ has the Liouville property and, in low dimensions, a calibration $\omega \in \Lambda^n \mathbb{R}^m$ has the Liouville property for $3 \le n \le m \le 6$ unless $\omega$ is face equivalent to the Special Lagrangian. In these cases, the Liouville property stems from isoperimetric rigidity of these mappings together with a classification of calibrations whose conformally flat calibrated submanifolds are flat. We also show that, for $3 \leq n \leq m$, the calibrations with the Liouville property form a dense $G_\delta$ set in the space of calibrations. As an application, we consider factorization of more general quasiregular curves and stability of quasiregular curves of small distortion.

math.DG↗
arXiv2024

Quasiregular cobordism theorem

In this article we prove that, for an oriented PL $n$-manifold $M$ with $m$ boundary components and $d_0\in \mathbb N$, there exist mutually disjoint closed Euclidean balls and a $\mathsf K$-quasiregular mapping $M \to \mathbb S^n \setminus \mathrm{int}(B_1\cup \cdots \cup B_m)$ of degree at least $d_0$. The result is quantitative in the sense that the distortion $\mathsf K$ of the mapping does not depend on the degree. As applications of this construction, we obtain Rickman's large local index theorem for quasiregular maps to all dimensions $n\ge 4$. We also construct, in dimension $n=4$, a version of a wildly branching quasiregular map of Heinonen and Rickman, and a uniformly quasiregular map of arbitrarily large degree whose Julia set is a wild Cantor set. The existence of a wildly branching quasiregular map yields an example of a metric $4$-sphere $(\mathbb S^4,d)$, which is not bilipschitz equivalent to the Euclidean $4$-sphere $\mathbb S^4$ but which admits a BLD-map to $\mathbb S^4$. For the proof of the main theorem, we develop a dimension-free deformation method for cubical Alexander maps. For cubical and shellable Alexander maps this completes the $2$-dimensional deformation theory originated by S.~Rickman in 1985.

math.CV↗
arXiv2023

De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean

We show that, if a closed, connected, and oriented Riemannian $n$-manifold $N$ admits a non-constant quasiregular mapping from the Euclidean $n$-space $\mathbb R^n$, then the de Rham cohomology algebra $H_{\mathrm{dR}}^*(N)$ of $N$ embeds into the exterior algebra ${\bigwedge}^*\mathbb R^n$. As a consequence, we obtain a homeomorphic classification of closed simply connected quasiregularly elliptic $4$-manifolds.

math.CV↗
arXiv2023

TILT: topological interface recovery in limited-angle tomography

A novel reconstruction method is introduced for the severely ill-posed inverse problem of limited-angle tomography. It is well known that, depending on the available measurement, angles specify a subset of the wavefront set of the unknown target, while some oriented singularities remain invisible in the data. Topological Interface recovery for Limited-angle Tomography, or TILT, is based on lifting the visible part of the wavefront set under a universal covering map. In the space provided, it is possible to connect the appropriate pieces of the lifted wavefront set correctly using dual-tree complex wavelets, a dedicated metric, and persistent homology. The result is not only a suggested invisible boundary but also a computational representation for all interfaces in the target.

eess.IV↗
arXiv2022

Deep Invertible Approximation of Topologically Rich Maps between Manifolds

How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use tools such as persistent homology to learn topological signatures of data and stabilize training but may not be universal approximators or have stable inverses. Other architectures universally approximate data distributions on submanifolds but only when the latter are given by a single chart, making them unable to learn maps that change topology. By exploiting the topological parallels between locally bilipschitz maps, covering spaces, and local homeomorphisms, and by using universal approximation arguments from machine learning, we find that a novel network of the form $\mathcal{T} \circ p \circ \mathcal{E}$, where $\mathcal{E}$ is an injective network, $p$ a fixed coordinate projection, and $\mathcal{T}$ a bijective network, is a universal approximator of local diffeomorphisms between compact smooth submanifolds embedded in $\mathbb{R}^n$. We emphasize the case when the target map changes topology. Further, we find that by constraining the projection $p$, multivalued inversions of our networks can be computed without sacrificing universality. As an application, we show that learning a group invariant function with unknown group action naturally reduces to the question of learning local diffeomorphisms for finite groups. Our theory permits us to recover orbits of the group action. We also outline possible extensions of our architecture to address molecular imaging of molecules with symmetries. Finally, our analysis informs the choice of topologically expressive starting spaces in generative problems.

cs.LG↗
arXiv2021

Quasiregular Curves of Small Distortion in Product Manifolds

We consider, for $n\ge 3$, $K$-quasiregular $\operatorname{vol}_N^\times$-curves $M\to N$ of small distortion $K\ge 1$ from oriented Riemannian $n$-manifolds into Riemannian product manifolds $N=N_1\times \cdots \times N_k$, where each $N_i$ is an oriented Riemannian $n$-manifold and the calibration $\operatorname{vol}_N^\times\in Ω^n(N)$ is the sum of the Riemannian volume forms $\operatorname{vol}_{N_i}$ of the factors $N_i$ of $N$. We show that, in this setting, $K$-quasiregular curves of small distortion are carried by quasiregular maps. More precisely, there exists $K_0=K_0(n,k)>1$ having the property that, for $1\le K\le K_0$ and a $K$-quasiregular $\operatorname{vol}_N^\times$-curve $F=(f_1,\ldots, f_k) \colon M \to N_1\times \cdots \times N_k$ there exists an index $i_0\in \{1,\ldots, k\}$ for which the coordinate map $f_{i_0}\colon M\to N_{i_0}$ is a quasiregular map. As a corollary, we obtain first examples of decomposable calibrations for which corresponding quasiregular curves of small distortion are discrete and admit a version of Liouville's theorem.

math.DG↗
arXiv2021

Transformation Optics for the Modelling of Waves in a Universe with Nontrivial Topology

We consider how transformation optics and invisibility cloaking can be used to construct models in subsets $\mathbb{R}^3$ with a varying metric, where the time-harmonic waves for a given angular wavenumber $k$, are equivalent to the waves in some closed orientable manifold. The obtained models could in principle be physically implemented using a device built from metamaterials. In particular the measurements in the metamaterial device given by the Helmholtz source-to-solution operator are equivalent to Helmholtz source-to-solution measurements in a universe given by $(\mathbb{R}_+\times M, -dt^2 +g)$, where $(M,g)$ is a closed, orientable, $C^\infty$-smooth, 3-dimensional Riemannian manifold. Thus the obtained construction could be used to simulate cosmological models using metamaterial devices.

math.AP↗
arXiv2020

Quasiregular curves: Hölder continuity and higher integrability

We show that a $K$-quasiregular $ω$-curve from a Euclidean domain to a Euclidean space with respect to a covector $ω$ is locally $(1/K)(\lVert ω\rVert/|ω|_{\ell_1})$-Hölder continuous. We also show that quasiregular curves enjoy higher integrability.

math.CV↗
arXiv2020

Quasiregular curves

We extend the notion of a pseudoholomorphic vector of Iwaniec, Verchota, and Vogel to mappings between Riemannian manifolds. Since this class of mappings contains both quasiregular mappings and (pseudo)holomorphic curves, we call them quasiregular curves. Let $n\le m$ and let $M$ be an oriented Riemannian $n$-manifold, $N$ a Riemannian $m$-manifold, and $ω\in Ω^n(N)$ a smooth closed non-vanishing $n$-form on $N$. A continuous Sobolev map $f\colon M \to N$ in $W^{1,n}_{\mathrm{loc}}(M,N)$ is a $K$-quasiregular $ω$-curve for $K\ge 1$ if $f$ satisfies the distortion inequality $(\lVertω\rVert\circ f)\lVert Df\rVert^n \le K (\star f^* ω)$ almost everywhere in $M$. We prove that quasiregular curves satisfy Gromov's quasiminimality condition and a version of Liouville's theorem stating that bounded quasiregular curves $\mathbb R^n \to \mathbb R^m$ are constant. We also prove a limit theorem that a locally uniform limit $f\colon M \to N$ of $K$-quasiregular $ω$-curves $(f_j \colon M\to N)$ is also a $K$-quasiregular $ω$-curve. We also show that a non-constant quasiregular $ω$-curve $f\colon M \to N$ is discrete and satisfies $\star f^*ω>0$ almost everywhere, if one of the following additional conditions hold: the form $ω$ is simple or the map $f$ is $C^1$-smooth.

math.CV↗
arXiv2020

Entropy in uniformly quasiregular dynamics

Let $M$ be a closed, oriented, and connected Riemannian $n$-manifold, for $n\ge 2$, which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f\colon M\to M$, the topological entropy $h(f)$ is $\log \mathrm{deg}( f )$. This proves Shub's entropy conjecture in this case.

math.DS↗
arXiv2020

A Neohookean model of plates

This article is about hyperelastic deformations of plates (planar domains) which minimize a neohookean type energy. Particularly, we investigate a stored energy functional introduced by J.M. Ball in his seminal paper "Global invertibility of Sobolev functions and the interpenetration of matter". The mappings under consideration are Sobolev homeomorphisms and their weak limits. They are monotone in the sense of C. B. Morrey. One major advantage of adopting monotone Sobolev mappings lies in the existence of the energy-minimal deformations. However, injectivity is inevitably lost, so an obvious question to ask is: what are the largest subsets of the reference configuration on which minimal deformations remain injective? The fact that such subsets have full measure should be compared with the notion of global invertibility which deals with subsets of the deformed configuration instead. In this connection we present a Cantor type construction to show that both the branch set and its image may have positive area. Another novelty of our approach lies in allowing the elastic deformations be free along the boundary, known as frictionless problems.

math.AP↗
arXiv2019

Vertical quasi-isometries and branched quasisymmetries

We introduce a class of mappings called vertical quasi-isometries and show that branched quasisymmetries $X\to Y$ of Guo and Williams between compact, bounded turning metric doubling spaces admit natural vertically quasi-isometric extensions $\widehat X\to \widehat Y$ between hyperbolic fillings $\widehat X$ and $\widehat Y$ of $X$ and $Y$, respectively. We also give a converse for this result by showing that a finite multiplicity vertical quasi-isometry $\widehat X \to \widehat Y$ between hyperbolic fillings induces a branched quasisymmetry $X \to Y$.

math.MG↗