# Analysis Seminar

Analysis seminar is held on Wednesdays at 14:15-16:00 in the lecture room MaD380. There is a 15 minute coffee break between the lecture hours. Everyone is welcome!

## Schedule of the fall term 2012

Sylvia Serfaty (Université Pierre et Marie Curie Paris 6 and Courant Institute of Mathematical Sciences): 2D COULOMB GAS, ABRIKOSOV LATTICE AND RENORMALIZED ENERGY.

Abstract: click

Scott Armstrong (University of Wisconsin, Madison): Stochastic homogenization of convex Hamilton-Jacobi equations.

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.

Matthew Rudd (University of the South, Sewanee): Statistical functional equations and $p$-harmonic functions, $1 \leq p \leq \infty$.

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 $p \geq 2$.

Tapio Rajala: Optimal mass transportation, Ricci-curvature and branching geodesics.

Paweł Goldstein (University of Warsaw): Weakly and approximately differentiable homeomorphisms of a cube.

Camille Petit: Boundary behavior of harmonic functions on Gromov hyperbolic graphs and manifolds.

Tuomo Kuusi (Aalto-yliopisto): Linear potentials in nonlinear potential theory.

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.

Pekka Pankka: Distributional limits of sequences of quasiconformally equivalent manifolds.

Francis Chung: A Partial Data Result for the Magnetic Schrödinger Inverse Problem.

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.

Ville Tengvall: Differentiability in the Sobolev space W^{1,n-1}.

Stefan Geiss: Gradient and Hessian estimates for semi-linear parabolic PDEs.

Katrin Fässler (University of Helsinki): Examples of uniformly quasiregular mappings on sub-Riemannian manifolds.

Alden Waters: A parametrix construction for the wave equation with low regularity coefficients using a frame of gaussians.

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.

David Dos Santos Ferreira (IECN, Nancy): Stability estimates for the Radon transform with restricted data.

Sergey Repin: Estimates of deviations from exact solutions of PDE's.

Kai Rajala: An upper gradient approach to weakly differentiable cochains.

Kai Rajala: Optimal assumptions for discreteness.

Cancelled!!!

Cancelled!!!

Valentino Magnani (Pisa University): Exterior differentiation through blow-up and some applications in sub-Riemannian Geometry.

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.

## Schedule of the spring term 2013

Naotaka Kajino (Universität Bielefeld): Analysis and geometry of the measurable Riemannian structure on the Sierpi\'{n}ski gasket (and other fractals).

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.

Mark Veraar (Delft University of Technology): Maximal regularity for SPDE.

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.

Thomas Zürcher: Space fillings from a turtle's perspective.

Tuomo Ojala: Thin and Fat (Cantor-) sets in metric spaces.

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.

Pekka Koskela: Boundary blow up under Sobolev mappings.

Nicola Gigli (Université de Nice): Remarks about the differential structure of metric measure spaces and applications.

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.

Karl-Theodor Sturm (University of Bonn): The space of spaces: curvature bounds and gradient flows on the space of metric measure spaces.

Thu 21.2.2013, 12-14, MaA102
Davoud Cheraghi (University of Warwick): Dynamics of complex quadratic polynomials with an irrationally indifferent fixed point.

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.

Jani Onninen: Beyond the Riemann Mapping Problem.

Nicola Fusco (University of Naples and University of Jyväskylä): Almegren's isoperimetric inequality in quantitative form.

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.

Benny Avelin (Uppsala University): The Quest for a Boundary Comparison Principle for the Parabolic p-Laplace Equation.

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.

Hiroaki Aikawa (Hokkaido University): Intrinsic ultracontractivity and capacitary width.

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.

Joonas Ilmavirta: Broken ray tomography in the disk.

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.

Lizaveta Ihnatsyeva: Characterization of traces of smooth functions on Ahlfors regular sets.

Zhuomin Liu (Charles University in Prague): THE LIOUVILLE THEOREM UNDER SECOND ORDER DIFFERENTIABILITY ASSUMPTION.

Abstract: click

Pilar Silvestre (Aalto-yliopisto): Connections between resistance conditions and the geometry of a metric measure space.

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 Poincaré 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.

Verena Bögelein (Universität Erlangen-Nürnberg): A quantitative isoperimetric inequality on the sphere.

Abstract: click

Frank Duzaar (Universität Erlangen-Nürnberg): Global weak solutions to the heat flow for prescribed mean curvature surface.

Abstract: click

Cancelled!!!