# Quasiweekend II - Ten years after

• #### Aalto UniversityUniversity of HelsinkiUniversity of Jyväskylä

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# SCIENTIFIC PROGRAMME

The lectures take place in Lecture Room PIV (Suomen Laki-sali) at Porthania (Yliopistonkatu 3) in the City Center Campus.

Thursday

10:00Registration
11:00Opening
11:10 – 12:00Drasin
-- lunch --
13:30 – 14:20Liu
14:30 – 15:20Prause
-- coffee --
16:00 – 16:50Meyer
17:00 – 17:50Onninen
FridaySaturday
9:00 – 9:50BaloghFässler
10:00 – 10:50KorteKleiner
--  coffee ---- coffee --
11:20 – 12:10

Mackay

Bourdon
-- lunch ---- lunch --
13:30 – 14:20AzzamRajala
14:30 – 15:20JääskeläinenFaraco
-- coffee ---- coffee --
16:00 – 16:50IwaniecTyson

19:00 -- Conference dinner
Sunday

10:00 – 10:50Wenger
11:00 – 11:50Shanmugalingam

-- lunch --

13:30 – 14:20Astala
14:30 – 15:20Martin

## Chairpersons

ThursdayFridaySaturdaySunday
Morning ChairOlli MartioKari HagNageswari ShanmugalingamJussi Väisälä
Afternoon ChairGaven MartinJani OnninenDavid DrasinJussi Väisälä

## Titles and Abstracts

### Thursday

David Drasin - A metamorphasis in the second century of value-distribution theory.
Abstract: Value-distribution theory had its birth within walking distance of our meeting location. Its roots were in the French school of function theory, but Nevanlinna's viewpoint soon provided
the standard orientation for its study.

Nevanlinna theory traditionally studied the relation between omitted values, branching and the growth of a meromorphic function.  Very early on, Nevanlinna recognized it as a transcendental version of the Riemann-Hurwitz relation, something which has become more apparent from recent work of K. Yamanoi. While the structure of the branching term is not fully understood in dimension greater than two, beginning in the late 1970s, Seppo Rickman obtained remarkable advances concerning omitted values, although some work was valid only for dimension 3.

We discuss some recent progress (with P. Pankka) and also show that Rickman's successes showed remarkable mastery of special tools from the classical theory.

Zhuomin Liu - Isometric immersion below the critical Sobolev regularity.
Abstract:  An isometric immersion of co-dimension-$k$ is a mapping from a domain $\rn$ to $\er^{n+k}$ that preserves the angle between any two curves passing through each point of the domain, as well as their lengths. It has been well-known that any $C^2$ isometric immersions of a flat domain is developable, that is, passing through each point there is at least one line segment on which the map is affine. As a surprising contrast, Nash and Kuiper established the existence of $C^1$ isometric immersions of any flat domain into balls of any higher dimension and of arbitrarily small radius. In particular, it cannot be affine on any line segments. It is then natural to consider isometric immersion in the intermediate class $W^{2,p}$. Actually, the study of $W^{2,p}$ isometric immersions is also important in nonlinear elasticity. It has been known that the critical Sobolev exponent which guarantees developability depends on the co-dimension rather than the dimension of the domain. In this talk, we will mainly focus on $W^{2,p}$ isometric immersions when $p$ is below the critical exponent. In particular, we show that in this case, the Gaussian curvature vanishes in the almost everywhere sense, but may remain a singular measure, as contrast to the critical case where the Gaussian curvature vanishes as a measure. This result thus connects isometric immersion to the $W^{2,p}$ solution of the Monge-Ampere equation $\det D^2u=0$ a.e. Indeed, if $p$ is below the same critical exponent, the solution can be nowhere affine, but once $p$ is above the critical exponent, it must be developable.

Istvan Prause - Quasidisks and twisting of the Riemann map.
Abstract: Consider a conformal map from the unit disk onto a quasidisk. We determine a range of critical complex powers with respect to which the derivative is integrable. Our method is effective in the presence of strong expansion. The results are consistent with the so-called circular Brennan conjecture.

Daniel Meyer - Fractal spheres, rational maps, and growth of groups
Abstract: In this talk, I will give an overview how some seemingly separate questions are connected. First we consider quasispheres. The general question is when a given metric sphere is quasisymmetrically equivalent to a quasisphere. It turns out, for certain self similar fractal spheres this can be answered by constructing a certain rational map.  Then we consider iterated monodromy groups. These are groups that are naturally assiciated rational maps (or more general dynamical systems). These have received a lot of attention lately, since they exhibit highly interesting behavior, for example in terms of growth and amenability. The vast majority of these results have however been obtained for polynomials. Here we show that certain non-polynomial rational maps have exponential growth. These include rational maps where the Julia set is the whole sphere, a Sierpinski carpet, as well as obtructed maps. These are the first known examples of this kind. This is joint work with with Mikhail Hlushchanka.

Jani Onninen - Variational Approach to Geometric Function Theory
Abstract: One of the reasons for studying variational problems of geometric nature is to generalize the Riemann Mapping Theorem to a larger class of homeomorphims. While the rigidity of conformal maps may prevent solutions of such problems from being conformal, the goal is to fin?d functionals whose minimizers resemble that of a conformal mapping. There is a growing literature on minimizers which, in general, do not satisfy the Euler-Lagrange equation. This leads us to investigate the energy-minimal solutions based on the inner-variational equation only. Even in the basic case of the Dirichlet integral there are new surprising phenomena. This talk is based on joint work with J. Cristina, T. Iwaniec, N-T. Koh and L. Kovalev.

### Friday

Zoltan Balogh - Hausdorff dimension distortion by quasiconformal and Sobolev maps

Absract: We give results on the distortion of Hausdorff dimension by Sobolev and quasiconformal mappings and indicate the sharpness of the results by examples.

Riikka Korte - Quasiconformal mappings, BMO and density conditions
Abstract: We discuss the relation between quasiconformality, mappings preserving BMO and mappings preserving certain density properties. The talk is based on joint work with Juha Kinnunen, Niko Marola, Nageswari Shanmugalingam and Olli Saari.

John MacKay - Metric properties of boundaries of relatively hyperbolic groups
Abstract: A key invariant of a hyperbolic group is its boundary at infinity, which is a metric space canonically defined up to quasisymmetry. In this talk I will discuss on-going work with Alessandro Sisto looking at the metric structure of the boundary at infinity of relatively hyperbolic groups, and applications to embedding problems.

Jonas Azzam - A characterization of rectifiable measures
Abstract: The analyst's traveling salesman theorem of Peter Jones gives a necessary and sufficient condition for when one can construct a curve that contains {\it all} of a given set. The condition is given in terms of $\beta$-numbers, which measure the flatness of the set at various scales and locations. In this talk, we will survey some old and recent results that deal with the problem of classifying when one can construct curves (or surfaces) that trace out {\it most} of a set, where by {\it most} we mean up to a set of measure zero with respect to some specified measure. In particular, using a variant of Jones' original $\beta$-numbers, we have a characterization of when this happens assuming the given measure is absolutely continuous with respect to Hausdorff measure. This is joint work with Xavier Tolsa.

Jarmo Jääskeläinen - Nonlinear Beltrami equations and positive Jacobians.
Abstract: We show that a homeomorphic solution to the nonlinear Beltrami equation with Hölder continuous coefficients has a positive Jacobian. The key is that, when the Jacobian of the solution is small, we can find a nonlinear Beltrami equation with Hölder continuous coefficients for the inverse, too. Then the derivative of the inverse is actually Hölder continuous, by Schauder estimates. This gives us enough regularity to conclude that the Jacobian of the solution must be positive everywhere. This is a joint work with Kari Astala, Albert Clop, Daniel Faraco and Aleksis Koski.

Tadeusz Iwaniec  - The Principle of Non-interpenetration of Matter
Abstract: The original theory of Nonlinear Elasticity has been based on the assumption that the energy-minimal displacement field h : X ->> Y is a homeomorphism. From the mathematical point of view, this is highly oversimplified precondition. One quickly runs into a serious difficulty when passing to the limit of an energy-minimizing sequence of homeomorphisms; injectivity may be lost. Therefore, we must accept weak and strong limits of Sobolev homeomorphisms  to Continuum Mechanics and Nonlinear Elasticity as realistic deformations. In two-dimensional models of hyperelastic plates and thin films such limits are none other than Monotone Sobolev Mappings. It is characteristic of monotone deformations to squeeze some parts of the plate (to a point or an arc) but not to fold it. It is exactly these parts of the body in which the Lagrange-Euler equation fails. Monotone Sobolev mappings can be approximated with diffeomorphisms. This is a far reaching extension of Youngs’ theorem in topology. Our approximation result plays a fundamental role in establishing the Existence Theorems of traction free deformations in Nonlinear Elasticity and in the variational approach to Quasiconformal Theory.

### Saturday

Katrin Fässler - Distortion and radial curves in the Heisenberg group
Abstract: The Heisenberg group endowed with a sub-Riemannian distance was among the first spaces where quasiconformal mappings have been studied outside the Euclidean and Riemannian framework. While the Heisenberg group carries a rich geometric structure that allows to transfer many results from Euclidean spaces, constraints coming from its sub-Riemannian geometry may cause new rigidity phenomena. I will discuss the role played by the family of 'radial curves' in this context. It can be used both to prove uniqueness of a minimizer for a specific mean distortion functional (contrasting the situation in the Euclidean plane), and to establish certain covering properties for quasiregular mappings (agreeing with the Euclidean situation).

Bruce Kleiner - PI spaces: some new examples and open problems.
Abstract: After reviewing some background on spaces satisfiying Poincare inequalities, I will discuss some new examples and open questions.

Marc Bourdon - Hyperbolic spaces and conformal dimension

Abstract : The conformal dimension of the boundary of a (Gromov) hyperbolic space is a quasi-isometry invariant of the space. It is bounded by below by the topological dimension of the boundary, and by above by its Hausdorff dimension. The conformal dimension will be presented, and we will be interested in determining the hyperbolic groups for which the conformal dimension of the boundary is equal to the topological dimension.

Kai Rajala - Uniformization of metric surfaces
Abstract: We discuss uniformization theorems for metric spaces with locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a geometric characterization for the existence of a uniformizing quasiconformal map.

Daniel Faraco - Conformal Parallel Vector Fields
Abstract: Smooth Beltrami Systems in higher dimensions can be solved if and only if the Weyl and Cotton tensor vanishes. On the other hand inverse problems in manifolds can be solved in many situations when the manifold is CTA, conformally transversally anisotropic, by the pionering work of Dos Santos Ferreira, Kenig Salo and Uhlmann.  Since this is a conformal invariant notion we give a necessary condition in terms of the Weyl and Cotton tensor. In this way, we can clarify for which  Thurston geometries the Calderón problem can be solved. This is a joint work with Pablo Angulo, Luis Guijarro and Alberto Ruiz from Madrid.

Jeremy Tyson - Conformal dynamical systems in Carnot groups
Abstract: I will discuss self-similar and self-conformal iterated function systems in the setting of Carnot groups equipped with a sub-Riemannian metric. Hausdorff and packing dimensions of their invariant sets can be computed via sub-Riemannian versions of the classical Moran-Hutchinson formula or the thermodynamic formalism. Self-similar sets in Carnot groups have been used to establish sharpness in the Dimension Comparison Theorem relating the families of sub-Riemannian and Euclidean Hausdorff measures. The Heisenberg group and other Carnot groups of Iwasawa type support a rich family of conformal self-maps which are not similarities. Conformal iterated function systems in Iwasawa Carnot groups arise as models in the theory of complex Kleinian groups as well as in the theory of Heisenberg continued fractions (Lukyanenko and Vandehey). This talk is based on joint work with Zoltan Balogh, Ben Warhurst, Vasilis Chousionis and Mariusz Urbanski.

### Sunday

Stefan Wenger - Minimal discs in metric spaces
Abstract: The classical Problem of Plateau asks to find a disc-type surface of minimal area with prescribed Jordan boundary. This problem was first solved by Douglas and Rado in the setting of Euclidean space and then extended by Morrey to the setting of Riemannian manifolds. In this talk, I will present a generalization to metric spaces. Precisely, I will show that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization (in a weak sense). If the space admits a local quadratic isoperimetric inequality for curves then such a disc is locally Hölder continuous in the interior and continuous up to the boundary. I will then discuss some applications: results on the regularity of quasi-harmonic discs in metric spaces, on the intrinsic structure of area minimizing discs, and a characterization of non-positive curvature in the sense of Alexandrov via isoperimetric inequalities. The talk is based on joint work with Alexander Lytchak.

Nageswari Shanmugalingam: Geometry behind Poincaré inequalities in the metric setting
Abstract: With the emergence of non-smooth analysis the role of Poincaré inequalities became linked to the geometry of metric measure spaces. The work of Heinonen and Koskela showed that an Ahlfors $Q$-regular space supports a $Q$-Poincaré inequality if and only if the space satisfies a geometric property related to $Q$-moduli of condensers in the space. In this talk we will show more links between geometry and $p$-Poincaré inequality for other values of $p$, including $p=\infty$.

Kari Astala - Families of quasiconformal mappings and non-linear Beltrami equations
Abstract: We discuss different properties of the non-linear Beltrami equations, in particular the manifold structure and the uniqueness properties of normalised solutions. The talk is based on joint works with Albert Clop, Daniel Faraco, Jarmo Jääskeläinen, Aleksis Koski and Laszlo Szekelyhidi.

Gaven Martin - Germs, Geodesics and Quasicircles.
Abstract: A germ $(U,\mu)$ is a doubly connected open set $U$ in the unit disk one of whose boundary components is the unit circle, together with a Beltrami coefficient $\mu$ supported on $U$. Each germ determines a unique quasisymmetric homeomorphism $g_0$ of the circle (up to rotation). Given this data, it is an interesting problem to determine the smallest maximal distortion of a quasiconformal homeomorphism of the disk with boundary values $g_0$. Even for the germs $(U,0)$ of asymptotically conformal quasisymmetric maps the problem is interesting and the answer depends on new geometric invariants - a hyperbolic modulus measuring the roundness'' of the inner boundary component. This is then used to solve the general problem with sharp'' estimates. We use these clean estimates on the maximal distortion of extensions, together with some geometry to prove the following: If $\Omega$ is a planar domain and $\gamma$ is a simple closed hyperbolic-geodesic in $\Omega$, then $\gamma$ is a $K$-quasicircle with $K$ depending only on the length of $\gamma$. The bound we find is (better than) $K \leq 1+\frac{5\ell}{\pi} \sqrt{e^\ell+1}$. This has interesting applications relating the conformal structures of domains. The first part of this talk is joint work with Riku Klén.

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