# Jyväskylä Analysis Seminar

The seminar usually takes place on Wednesdays 14:15-16:00 in the lecture room MaD380 at the Department of Mathematics and Statistics. Everyone is welcome!

# Upcoming Talks

## Fall 2015

 Wednesday 26.8.2015 14:15-16:00 TBA

# Past Talks

## Fall 2014

 Wednesday 10.12.2014 14:15-16:00 Lukáš Malý Sobolev-type spaces based on linear lattices of measurable functions in metric spaces Show abstract. Hide abstract. Hajłasz and Shanmugalingam pioneered two different approaches to first-order Sobolev-type functions in metric spaces. The theory has been built several times, based on various Banach function spaces (e.g., on Lebesgue $L^p$, Orlicz $L^{\Psi}$, Musielak $L^{p(\cdot)}$ spaces). I will present a theory that covers all these base function spaces and goes much further. Some pathological phenomena that may emerge in such a generality will be discussed. Minimal weak upper gradients and minimal Hajłasz gradients will be proven to exist in a highly general setting. Boundedness and continuity of Sobolev-type functions based on “integrability” of their gradients are also to be looked into. Wednesday 3.12.2014 14:15-16:00 Juan Souto (Université de Rennes 1) Ergodicity of the mapping class group action on a component of the character variety Show abstract. Hide abstract. Goldman proved that the variety Xg of characters of representations of the fundamental group of a surface of genus g into PSL2ℝ has precisely 4g-3 connected components Xg(2-2g),...,Xg(2g-2) where moreover the component Xg(k) consists of those representations with Euler number k. The two extremal component Xg(2-2g) and Xg(2g-2) are Teichmueller spaces and hence the mapping class group acts discretely on them. On the other hand Goldman conjectured that the action of the mapping class group on each one of the remaining components. I will prove that Goldman's conjecture holds true for the component Xg(0) corresponding to representations with vanishing Euler number. Wednesday 26.11.2014 14:15-16:00 Martí Prats (Universitat Autònoma de Barcelona) Quasiconformal mappings with Beltrami coeficient in Sobolev spaces of domains Download abstract (PDF). Wednesday 19.11.2014 14:15-16:00 Sita Benedict Hardy-Orlicz spaces of conformal densities Wednesday 12.11.2014 14:15-16:00 Esa Vesalainen Corner scattering Wednesday 5.11.2014 14:15-16:00 Juhana Siljander (University of Helsinki) Everywhere differentiability of viscosity solutions to Aronsson's equations Wednesday 29.10.2014 14:15-16:00 Laurent Moonens (Université Paris-Sud, Orsay) Continuous and bounded solutions to the equation div v=F : existence and singularities Download abstract (PDF). Wednesday 22.10.2014 14:15-16:00 Lizaveta Ihnatsyeva Hardy inequalities in Triebel-Lizorkin spaces Wednesday 15.10.2014 14:15-16:00 Martin Kell (Institut des Hautes Études Scientifiques) Heat and entropy flows Show abstract. Hide abstract. Ambrosio, Gigli and Savaré developed a calculus for an abstract heat flow on general metric measure spaces and showed that one can identify this flow with the gradient flow of the Boltzmann entropy in the 2-Wasserstein space. In this talk I will show how extend their result to cover the q-heat flow, the gradient flow of the q-Cheeger energy, which is a solution of the parabolic q-Laplace equation in the smooth setting. I will give a sufficient condition for mass preservation of the flow and show that it solves a generalized gradient flow of the Rényi entropy in the p-Wasserstein space where p is the Hölder conjugate of q. In case p is between 1 and 2 one can show that this gradient flow has at most one solution and thus the two flows, the q-heat flow and the Rényi entropy flow, can be identified. Wednesday 8.10.2014 14:15-16:00 Zhuomin Liu Sobolev isometric immersions and related problems Show abstract. Hide abstract. An isometric immersion of co-dimension k is a mapping from a domain of ℝn into ℝn+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 C2 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 C1 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. A natural question arises in this context for analysts: what about isometric immersions of intermediate regularity, say of Hölder space C1,α or Sobolev space W2,p? These are areas with many open questions, especially concerning the critical regularity that distinguishes developability and flexibility. In this talk, we will focus on the development and unsolved gaps of isometric immersions in the Sobolev class, as well as their related problems. Wednesday 1.10.2014 14:15-16:00 Vasilis Chousionis (University of Helsinki) Square functions, uniform rectifiability and Wolff potentials Download abstract (PDF). Wednesday 10.9.2014 14:15-16:00 Antti Vähäkangas Fractional Sobolev spaces on domains: zero extension and Hardy inequality Wednesday 3.9.2014 14:15-16:00 Vyron Vellis Quasisymmetric spheres over Jordan domains Show abstract. Hide abstract. Let Ω be a planar Jordan domain and α > 0. We consider double-dome-like surfaces Σ defined by graphs of dist(·,∂Ω)α over Ω. The goal is to find the right conditions on the geometry of the base Ω and the growth of tα so that Σ is a quasisphere, or quasisymmetric to S2. An internal uniform quasicircle condition on the constant distance sets to ∂Ω, coupled with a weak chord-arc condition on ∂Ω, gives a sharp answer. Our method also produces new examples of quasispheres in ℝn , for any n ≥ 3. This talk is based on a joint work with Jang-Mei Wu. Wednesday 20.8.2014 14:15-16:00 Congwen Liu (University of Science and Technology of China) p-norms of the Bergman projection and the Cauchy transform

## Spring 2014

 Wednesday 19.6.2013 14:15-16:00 Vincent Millot (Université Paris Diderot - Paris 7) Fractional Ginzburg-Landau Vs fractional harmonic maps: asymptotics, regularity, and defect measures Tuesday 18.6.2013 14:15-16:00 Bruce Hanson (St. Olaf College)Lipschitz Conditions and Differentiability Wednesday 22.5.2013 14:15-16:00 Massimiliano Morini (Università degli Studi di Parma) Wednesday 8.5.2013 14:15-16:00 Marco Barchiesi (University of Naples "Federico II") Stability of the isoperimetric inequality and Polya-Szego inequality under Steiner and Schwarz rearrangements Show abstract. Hide abstract. Abstract: We shall start with a quick review of the basic properties of Steiner and Schwartz symmetrizations of sets and functions. Through some recently developed analytical techniques, we give a characterization of the cases of equality. Then, we prove a sharp quantitative version of the inequalities, in the case of convex sets and log-concave functions. Wednesday 24.4.2013 14:15-16:00 Frank Duzaar (Universität Erlangen-Nürnberg) Global weak solutions to the heat flow for prescribed mean curvature surface Show abstract. Hide abstract. Abstract:In the talk we present results concerning the existence of global weak solutions to some geometric motivated flows, such as the heat flow for prescribed mean curvature disk-type surfaces or the m-harmonic map heat flow for maps from a compact m-dimensional Riemannian manifold Ω with non-empty boundary ∂Ω into a compact Riemannian manifold N without boundary. We consider either Cauchy-Dirichlet data or a Plateau type boundary condition. This is joint work with Verena Bögelein (Erlangen) and Christoph Scheven (Duisburg). Monday 22.4.2013 14:15-16:00 Verena Bögelein (Universität Erlangen-Nürnberg) A quantitative isoperimetric inequality on the sphere Download abstract (PDF). Wednesday 17.4.2013 14:15-16:00 Pilar Silvestre (Aalto-yliopisto) Connections between resistance conditions and the geometry of a metric measure space Show abstract. Hide abstract. Abstract: This talk studies analytic and geometrical aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed PoincarÃ© inequality, but however, they are sufficiently strong to imply several useful facts in analysis on metric measure spaces. We show that under a p-Resistance conductor inequality, any discretely quasiconvex space is annuli discretely quasiconvex. Wednesday 10.4.2013 14:15-16:00 Zhuomin Liu (Charles University in Prague) The Liouville theorem under second order differentiability assumption Dowload abstract (PDF). Wednesday 3.4.2013 14:15-16:00 Lizaveta Ihnatsyeva Characterization of traces of smooth functions on Ahlfors regular sets Wednesday 27.3.2013 14:15-16:00 Joonas Ilmavirta Broken ray tomography in the disk Show abstract. Hide abstract. Abstract: The fundamental question in X-ray imaging turns out to be: Can one reconstruct a function from its line integrals? The answer is affirmative and the theory is well understood, but much less is known if one only knows the integrals over lines with reflections (broken rays). Answers to such questions in the broken ray context are related to inverse problems in PDE. After reviewing the background and motivation, I will present two reconstruction results for broken ray tomography in the Euclidean disk. Wednesday 20.3.2013 14:15-16:00 Hiroaki Aikawa (Hokkaido University) Intrinsic ultracontractivity and capacitary width Show abstract. Hide abstract. Abstract: Intrinsic ultracontractivity for a heat kernel has been extensively studied by probabilistic methods and logarithmic Sovolev inequalities. In this talk, we give an elementary analytic proof for intrinsic ultracontractivity with the aid of capacitary width and a parabolic box argument. Joint work with Tsubasa Itoh. Wednesday 13.3.2013 14:15-16:00 Benny Avelin (Uppsala University) The Quest for a Boundary Comparison Principle for the Parabolic p-Laplace Equation Show abstract. Hide abstract. Abstract: In this talk I will present a new result for parabolic equations of p-Laplace type, namely the Carleson estimate. I will discuss mostly the degenerate case, and talk about the exotic differences between the nonlinear case and the linear (Heat equation) case, and how this affects the techniques used to prove estimates at the boundary. Wednesday 6.3.2013 14:15-16:00 Nicola Fusco (University of Naples and University of Jyväskylä) Almegren's isoperimetric inequality in quantitative form Show abstract. Hide abstract. Abstract: In 1986 F. Almgren proved a deep and beautiful version of the classical isoperimetric inequality for the higher co-dimensional case. After reviewing various reviewing various quantitative formulations of the standard isoperimetric inequality I shall discuss a recent result obtained in collaboration with V.Boegelein and F.Duzaar that extends to this more general inequality the stability estimates known in the classical case. Wednesday 27.2.2013 14:15-16:00 Jani Onninen Beyond the Riemann Mapping Problem Thursday 21.2.2013 12:15-14:00 Davoud Cheraghi (University of Warwick) Dynamics of complex quadratic polynomials with an irrationally indifferent fixed point Show abstract. Hide abstract. Abstract: The study of the dynamics of quadratic polynomials with an irrationally indifferent fixed point has been one of the major challenges in complex dynamics. Recently, there has been major progress in the study of the dynamics of such maps, mainly due to the introduction of a sophisticated renormalization technique by Inou and Shishikura. In the fist part of the talk we introduce the renormalization technique and outline how one uses this method to describe the fine scale geometric properties of the dynamics of such maps. In the second part of the talk we discuss the methods of quasi-conformal mappings that is used to obtain some sharp estimates on conformal mappings that appear in this study. Wednesday 20.2.2013 14:15-16:00 Karl-Theodor Sturm (University of Bonn) The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces Wednesday 13.2.2013 14:15-16:00 Nicola Gigli (Université de Nice) Remarks about the differential structure of metric measure spaces and applications Show abstract. Hide abstract. Abstract: In the first half of the talk I'll review the standard definition of Sobolev space over a metric measure space in light of the results obtained in collaboration with Ambrosio and Savaré. In the second I will discuss more recent results about their differential structure, in particular in connection with the problem of integration by parts. Wednesday 6.2.2013 14:15-16:00 Pekka KoskelaBoundary blow up under Sobolev mappings Wednesday 30.1.2013 14:15-16:00 Tuomo Ojala Thin and Fat (Cantor-) sets in metric spaces. Show abstract. Hide abstract. Abstract: I will discuss on fatness and thinness for doubling measures. Symmetric Cantor sets in real line have simple characterization of fatness/thinness in terms of the defining sequence. I will explain this and prove similar result in uniformly perfect metric spaces. While doing so I will also show some nice connections to quasisymmetric maps. Wednesday 23.1.2013 14:15-16:00 Thomas Zürcher Space fillings from a turtle's perspective Wednesday 16.1.2013 14:15-16:00 Mark Veraar (Delft University of Technology)Maximal regularity for SPDE Show abstract. Hide abstract. Abstract: In this talk I will give an introduction to recently developed regularity theory for stochastic evolution equations of parabolic type. The time/space-regularity of solutions of SPDES is important for e.g. numerical approximation schemes. Moreover, it can be used to prove well-posedness results for nonlinear SPDEs arising in filtering theory. The proofs of the regularity estimates are based on results from harmonic and stochastic analysis in an infinite dimensional framework combined with functional calculus techniques. Wednesday 9.1.2013 14:15-16:00 Naotaka Kajino (Universität Bielefeld) Analysis and geometry of the measurable Riemannian structure on the Sierpi\'{n}ski gasket (and other fractals) Show abstract. Hide abstract. Abstract: On the Sierpi\'{n}ski gasket, Kigami [Math. Ann. 340 (2008), 781--804] has introduced the notion of the measurable Riemannian structure, with which the gradient vector fields" of functions, the Riemannian volume measure" and the geodesic metric" are naturally associated. Kigami has also proved in the same paper the two-sided Gaussian bound for the corresponding heat kernel, and I have further shown several detailed heat kernel asymptotics, such as Varadhan's asymptotic relation, in a recent paper [Potential Anal. 36 (2012), 67--115]. In the talk, Weyl's Laplacian eigenvalue asymptotics is presented for this case. In the limit of the eigenvalue asymptotics we obtain a constant multiple of the Hausdorff measure (of the appropriate dimension) with respect to the geodesic metric", which is in fact singular to the Riemannian volume measure". A complete characterization of geodesics is also presented, and as an application it is shown that the curvature-dimension condition of Sturm and Lott-Villani and the measure contraction property of Ohta and Sturm are NOT satisfied in this setting. For most of the results it is quite essential that the underlying topological space is the 2-dimensional Sierpi\'{n}ski gasket. It seems that extensions to other fractals will be only partially possible and a similar result may or may not be true depending on each fractal. If time permits I would like to explain this subtlety in generalization to other fractals.
 Wednesday 19.12.2012 15:15-16:00 Valentino Magnani (Pisa University) Exterior differentiation through blow-up and some applications in sub-Riemannian Geometry Show abstract. Hide abstract. Abstract: We establish a low rank property'' for Sobolev mappings that almost everywhere solve a special nonlinear system of PDEs. This system, associated to a nonintegrable tangent distribution, implies the so-called contact property of its solutions. The proof of this property relies on a "special weakly exterior differentiation'' performed through a blow-up procedure. As an application, we give a complete solution to a question raised in a paper by Z. M. Balogh, R. Hoefer-Isenegger and J. T. Tyson. These results are a joint work with J. Malý and S. Mongodi. Wednesday 28.11.2012 15:15-16:00 Kai Rajala Optimal assumptions for discreteness Wednesday 28.11.2012 14:15-15:00 Kai Rajala An upper gradient approach to weakly differentiable cochains Wednesday 21.11.2012 14:15-16:00 Sergey Repin Estimates of deviations from exact solutions of PDE's Wednesday 14.11.2012 14:15-16:00 David Dos Santos Ferreira (IECN, Nancy) Stability estimates for the Radon transform with restricted data Wednesday 7.11.2012 14:15-16:00 Alden Waters A parametrix construction for the wave equation with low regularity coefficients using a frame of gaussians Show abstract. Hide abstract. Abstract: We show how to construct frames for square integrable functions out of odulated Gaussians. Using the frame representation of the Cauchy data, we show that we can build a suitable approximation to the solution for low regularity, time dependent wave equations. The talk will highlight the relationship of the construction to harmonic analysis and will explore the differences of the new construction to the standard Gaussian beam ansatz. Wednesday 31.10.2012 14:15-16:00 Katrin Fässler (University of Helsinki) Examples of uniformly quasiregular mappings on sub-Riemannian manifolds Wednesday 24.10.2012 14:15-16:00 Stefan Geiss Gradient and Hessian estimates for semi-linear parabolic PDEs Wednesday 17.10.2012 14:15-16:00 Ville Tengvall Differentiability in the Sobolev space W1,n-1 Wednesday 10.10.2012 14:15-16:00 Francis Chung A Partial Data Result for the Magnetic Schrödinger Inverse Problem Show abstract. Hide abstract. Abstract: I will give an introduction to the magnetic Schrödinger inverse problem, and describe a recent partial data result for it. The proof relies on establishing a Carleman estimate for the magnetic Schrödinger operator, and I will explain a little bit why that is and how the estimate is proved. Wednesday 3.10.2012 14:15-16:00 Pekka Pankka Distributional limits of sequences of quasiconformally equivalent manifolds Wednesday 26.9.2012 14:15-16:00 Tuomo Kuusi (Aalto-yliopisto) Linear potentials in nonlinear potential theory Show abstract. Hide abstract. Abstract: We give an update to some aspects of theory for solutions to nonlinear elliptic or parabolic, possibly degenerate, equations involving p-Laplacean type operators and datum, which, in full generality, can be a measure. The main focus is to describe recent pointwise potential estimates for solutions' gradients. Wednesday 19.9.2012 14:15-16:00 Camille Petit Boundary behavior of harmonic functions on Gromov hyperbolic graphs and manifolds Wednesday 12.9.201214:15-16:00 Paweł Goldstein (University of Warsaw) Weakly and approximately differentiable homeomorphisms of a cube Wednesday 5.9.2012 14:15-16:00 Tapio Rajala Optimal mass transportation, Ricci-curvature and branching geodesics Wednesday 29.8.2012 14:15-16:00 Matthew Rudd (University of the South, Sewanee) Statistical functional equations and p-harmonic functions, 1 ≤ p ≤ ∞ Show abstract. Hide abstract. Abstract: I will discuss recent work on functional equations that generalize the mean value property of harmonic functions and whose continuous solutions approximate p-harmonic functions. Most of the talk will focus on $p$ between 1 and 2 (including the endpoints); toward the end, I will discuss how the techniques apply to a problem studied by Parviainen et al. when $p \geq 2$. Thursday 16.8.2012 15:15-16:15 Scott Armstrong (University of Wisconsin, Madison) Stochastic homogenization of convex Hamilton-Jacobi equations Show abstract. Hide abstract. Abstract: I will give an overview of the problem of homogenizing a first-order Hamilton-Jacobi equation in a random (stationary-ergodic) environment and explain some connections to the theory of first-passage percolation Thursday16.8.201214:00-15:00 Sylvia Serfaty (Université Pierre et Marie Curie Paris 6 and Courant Institute of Mathematical Sciences) 2D Coulomb gas, Abrikosov lattice and renormalized energy Show abstract. Hide abstract. Abstract: In joint work with Etienne Sandier, we studied the statistical mechanics of a classical two-dimensional Coulomb gas, particular cases of which also correspond to random matrix ensembles. We connect the problem to the “renormalized energy" W, a Coulombian interaction for an inﬁnite set of points in the plane that we introduced in connection to the Ginzburg-Landau model, and whose minimum is expected to be achieved by the “Abrikosov" triangular lattice. I will brieﬂy allude to the results obtained on Ginzburg-Landau and focus mostly on the Coulomb gas system. Results include a next order asymptotic expansion of the partition function, and various characterizations of the behavior of the system at the microscopic scale. When the temperature tends to zero (the limit also corresponds to “weighted Fekete sets") we show that the system tends to “crystallize" to a minimizer of W. Keywords: Coulomb gas, Ginzburg-Landau model, Abrikosov lattice, crystallization, random matrices, Ginibre ensemble. [1] E. Sandier, S. Serfaty, From the Ginzburg-Landau Model to Vortex Lattice Problems, Comm. Math. Phys., 2012, available online. [2] E. Sandier, S. Serfaty, 2D Coulomb Gases and the Renormalized Energy, http://arxiv.org/abs/1201.3503v1