Analysseminariet 2008, löpande planering

Vårterminen

Onsdagen den 23 januari 13.15-14.15. Kristian Bjerklöv (KTH): Universality in hyperbolicity breakdown.
Abstract: In a recent publication, Haro and de la Llave have found spectacular scaling laws when numerically investigating the loss of uniform hyperbolicity in quasi-periodically perturbed systems. We present rigorous results showing that some of these power law asymptotics indeed exist. This is a joint project with M. Saprykina.

Onsdagen den 30 januari 13.15-14.15. Anders Karlsson (KTH): The Liouville property and the escape rate of Brownian motion.

Onsdagen den 6 februari 13.45-14.45 (OBS tiden!). Bergfinnus Durhuus (Köpenhamn) : The spectral dimension of random trees.
Abstract: We consider statistical ensembles of infinite trees obtained as limits of finite trees on $N$ verices as $N\to\infty$. One characteristic exponent of simple random walk on an infinite recurrent is the spectral dimension, defined as the power decay rate of its return probability as a function of time. The main result that will be explained is that, under rather general conditions, the spectral dimension of trees in the ensembles mentioned is $4/3$. This is joint work with T. Jonsson (Reykjavik) and J. Wheater (Oxford).

Onsdagen den 13 februari 13.15-14.15. Joaquim Ortega Cerda (Barcelona): Close encounters with Bloch functions of the third kind.
Abstract: Landau in the 30's estimated the univalent Bloch-Landau constant U , i.e., the biggest radius R that such that f (D(0, 1)) always contains a disk of radius R for any univalent f normalized with |f'(0)| = 1. Although the exact value of U is not known, many authors have provided upper and lower bounds. In a joint work with T. Carroll we have studied fine properties of the extremal functions (they are known as Bloch functions of the third kind) and shown the connection with another well-studied question, the P\'olya-Chebotarev problem. This relationship has been exploited to improve (very slightly) the upper bound for the constant.

Onsdagen den 20 februari 13.15-14.15. Jean-Pierre Rosay (Madison): Almost complex structures and $\bar\partial$ inequalities.

Onsdagen den 27 februari 13.15-14.15. Ragnar Sigurdsson (Reykjavik): Siciak-Zahariuta extremal functions and polynomial hulls.
Abstract: The lecture is a report on joint work with Finnur Larusson, University of Adelaide, Australia. We have proved so-called disc envelope formulas for the Siciak-Zahariuta functions of open domains in affine space and we are able to use these formulas to characterize polynomial hulls of compact connected subsets of complex affine space.

Onsdagen den 5 mars 13.15-14.15. Masha Saprykina (KTH): Superexponential growth of the number of periodic points and other unexpected phenomena close to a homoclinic tangency.

Onsdagen den 12 mars 13.15-14.15. Petter Bränden (KTH): P\'olya-Schur theorems and beyond.
Abstract: The problem of characterizing linear operators preserving real-rootedness of polynomials has a long history that goes back to the work of Laguerre and P\'olya-Schur. Recently we (joint with J. Borcea and B. Shapiro) completely characterized all linear operators on univarite polynomials preserving the property of having all zeros in a prescribed closed circular domain or its boundary. In particular this solves the problem of preserving real-rootedness. The generalization (joint with J. Borcea) to several variables has applications to the Lee-Yang program in statistical mechanics and unifies theorems of Lee-Yang, Lieb-Sokal, Grace-Walsh-Szeg\"o and Hinkkanen.

Måndagen (!) 17 mars 13.15-14.15. Ramona Anton (Baltimore and Paris): Non-linear Schr\"odinger equations on domains with boundary.
Abstract: We are interested in proving global existence results in the energy space for the semi-linear Schr\"odinger equation on domains of dimension 2 or 3. The main ingredients are generalized Strichartz inequalities adapted to the domains, which have some loss of derivatives. We present the results and the strategy for three types of domains.

Onsdagen 26 mars 13.15-14.15. Utgår p g a påsk.

Onsdagen den 2 april 13.15-14.15. Laszlo Lempert (Purdue): Complex analysis in infinite dimensions..
Abstract: I will start by motivating the subject of infinite dimensional complex analysis. Then I will review the main notions and move to the central question of the talk, the so called Cousin problem, as well as its generalization involving analytic cohomology groups. The subject saw much progress in the last ten years, but, as is often the case, many open problems remain.

Onsdagen den 9 april 13.15-14.15. Alfonso Montes Rodriguez (Sevilla): Invariant subspaces for parabolic self-maps .
Abstract: We show a description of invariant subspaces of composition operators induced by parabolic self-maps in several spaces of analytic functions.

Onsdagen den 16 april 13.15-14.15. Bo Berndtsson (Chalmers): Asymptotics of Bergman kernels.
Abstract:

Onsdagen den 23 april 13.15-14.15. Johan Thim (Linköping): A Fixed Point Theorem in Locally Convex Spaces.

Onsdagen den 30 april 13.15-14.15. Denis Gaidashev (KTH): .

Onsdagen den 7 maj 13.15-14.15. Mats Andersson (Chalmers): Ideals of holomorphic functions and residue currents.
Abstract: Any analytic variety in $\C^n$, or more generally any ideal of holomorphic functions, can be represented as the annihilator of an analytic object, a so-called residue current. This was proved independently by Dickenstein-Sessa and Passare some 20 years ago for the case when the ideal is given as a so-called complete intersection. The general case was recently obtained in a joint work with E Wulcan. We will indicate the construction of such residue currents, and discuss some applications; for instance we obtain (in a joint work with H Samuelsson) new existence results for the $\dbar$-equation on analytic varieties.

Onsdagen den 14 maj 13.15-14.15. Andreas Axelsson (SU): A new approach to solvability of some elliptic PDEs with square integrable boundary data.
Abstract: I will survey recent progress in the study of second order elliptic divergence form equations with complex measurable coefficients $A$. The main result is that the set of $A\in L_\infty(\R^n;\C^{n+1})$ for which boundary value problems with $L_2$ Dirichlet or Neumann data are well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when $A$ is either Hermitean, block or constant. This is joint work with P. Auscher and A. McIntosh.

Onsdagen den 21 maj 13.15-14.15. Yacin Ameur (KTH): On fluctuations of eigenvalues of random normal matrices.
Abstract: I will discuss a theorem about the joint (weighted) asymptotic distribution of eigenvalues of random normal matrices when the order of those matrices increases indefinitely. The result applies to fairly general weight functions on the complex plane. This reports on joint work with Haakan Hedenmalm and Nikolai Makarov.

Onsdagen den 28 maj 13.15-14.15. Jens Hoppe (KTH): Membranes and Singularities.

Höstterminen

Onsdagen den 10 september 13.15-14.15. Ana Rodrigues (Indianapolis): Tongues and cusps.
Abstract: We study the shape of the boundaries of the tongues, and the behaviour close to their tips, for the family of double standard maps $f_{a,b}(x)=2x+a+(b/\pi)\sin(2\pi x)$ (mod 1). It turns out that the shape is fairly regular, mainly due to the real analyticity of the maps. We present a new method to classify non-generic cusp bifurcations for two-parameter families of one-dimensional systems. We illustrate this method in the generic case. This reports on joint work with M. Misiurewicz.

Onsdagen den 17 september 13.15-14.15. Nir Lev (KTH): Span of translates, and zeros of Fourier transform.
Abstract: A function $f$ in $l^p(Z)$ or $L^p(R)$ is called a "generator" if the set of all translates of $f$ spans the whole space. How to decide whether a given function is a generator or not? We shall discuss this problem including joint work with A. Olevskii.

Onsdagen den 24 september 13.15-14.15. Giovanni Felder (ETH Z\"urich): Configuration spaces and quantization.
Abstract: Kontsevich's construction of a universal quantization of the algebra of functions on a manifold is given in terms of integrals of differential forms on configuration spaces of points on the disk. In this talk we will review this result and discuss recent extensions to S1-equivariant differential forms and their relation with cyclic homology and traces in deformation quantization.

Onsdagen den 1 oktober 13.15-14.15. Dani Blasi Babot (Barcelona): Interpolating sequences in analytic Besov spaces.
Abstract: We will describe the interpolating sequences for some weighted analytic Besov spaces and the corresponding multiplier spaces.

Onsdagen den 22 oktober 13.15-14.15. Håkan Eliasson (Paris): Dynamical localization for the discrete one-dimensional quasi-periodic Schrödinger equation.
Abstract: The quasi-periodic Schrödinger equation in one space-dimension has been intensively studied both as a finite-dimensional and as an infinite-dimensional dynamical system since the work of Dinaburg & Sinai in the middle 70's and of Fröhlich & Spencer & Wittwer and Sinai in the late 80's. We shall discuss the property of dynamical localization for this equation in the strong coupling regime.

Onsdagen den 5 november 13:15-14:15. Neil Dobbs (KTH): Ergodic properties of some maps from the exponential family
Abstract: We consider entire maps of the form $z\mapsto \lambda exp(z)$, where $\lambda\in \mathbb{C}$ is such that the orbit of zero is bounded and such that all periodic poits are repellinf. For such maps it was known that a $\sigma$-finite absolutely continuous invariant measure exists. However, even in the simplest case where $\lambda=2\pi i$, it was an open question whether the measure could be finite. We show it cannot, i.e., for the class of such maps considered, no absolutely continuous invariant measure can exist. This is a joint work with B. Skorulski.

Onsdagen den 19 november 13:15 - 14:15. Michael Benedicks: Nonlinear evolution equations, Coupled Map Lattices and noninvertible dynamics in two dimensions.
Abstract: The talk is inspired by recent work of Pesin and Yurchenko, who discovered nonlinear evolution equations which seem to have "strange attractors". More specifically, after discretization one obtains a Coupled Map Lattice (CML), whose local maps have strange attractors. We will also try to relate this to Sinai-Ruelle-Bowen measures for CML's (work by Bunimovich-Sinai and Kupiainen-Bricmont).

Onsdagen den 26 november 13:15-14:15. Nicolae Mihalache (KTH): John regularity of Fatou components.
Abstract: A famous result of Carleson, Jones and Yocoz shows that the basin of attraction of infinity of a polynomial P is a John domain if and only if P is semi-hyperbolic (i.e. the critical points are not recurrent and there are no parabolic orbits). We show that all Fatou components of a semi-hyperbolic rational map are John domains. The converse does not hold. We show local connectivity of connected Julia sets for a large class of non-uniform hyperbolic rational maps, including semi-hyperbolic maps and topological Collet-Eckmann maps.

Onsdagen den 17 december 13:15-14:15. Pavel Kurasov (Lund). Inverse problems for graphs with cycles
Abstract: The talk is devoted to the inverse problem for Schrödinger operators on metric graphs in the presence of a magnetic field. It is claimed that the knowledge of the corresponding Titchmarsh-Weyl (matrix) function for different values of the magnetic field may help to solve the inverse problem, i.e. to reconstruct the metric graph and real (electric) potential on it. This approach is fully developed for graphs with Euler characteristic zero but without loops. It is proven that this reconstruction is possible if a certain non-resonant condition is satisfied. The problem is handled using the solution to the inverse problem for the periodic Schrödinger operator investigated by Marchenko-Ostrovskii and MacKean- Trubowitz in the 1970’s. We hope that our study sheds a new light on this classical theorem.