nthca10 New Trends in Harmonic and Complex Analysis June 29 - July 3, 2010Jacobs University, Bremen, Germany

Abstracts (ordered by name)
 "Beyond analiticity: time-frequency analysis of Bergman and Fock-type spaces" Abreu, Luis DanielUsing Gabor and wavelet frames, we will study Bergman and Fock-type spaces when the elements of the space are not necessarily analytic. The study is based on the observation that Bergman-type spaces (of non-analytic functions) have an orthogonal decomposition in several subspaces (of polyanalytic functions), where the first subspaces are the classical Bergman and Fock space. In particular, this approach leads to new sampling and interpolation results. As in the collaboration of Gr?nig with Borichev and Lyubarskii, it shows how a blend of frame and complex analysis methods can lead to new and unexpected results. "Boundary Forelli theorem" Agranovsky, MarkForelli theorem on holomorphicity on slices can be regarded as a variation of the classical Hartogs theorem on separate analyticity of functions of several complex variables. Forelli theorem says that if a function is holomorphic on each complex line passing through a point and is infinitely smooth at this point then it is holomorphic. The boundary analog of Forelli theory assumes charcterization of boundary values of holomorphic functions by one-dimensional holomorphic extension into slices: the cross-sections of the domain by complex lines. It is a quite old question what are minimal varieties of affine complex Grassmanian for boundary Forelli theorem to be true. Recent progress in solving this problem will be presented. "Extension of vector fields in the disk and conformal welding" Airault, HeleneReferences: "Modululus of continuity of the canonic Brownian motion on the group of diffeomorphisms of the circle" by Airault-Ren, J. Funct. Anal. 196 (2002) 395-426. "Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows" by Airault-Malliavin-Thalmaier, J. Math. Pures Appl. 83 (2004) 955-1018. "Remarks on the Bohr and Rogosinski phenomena for power series" Aizenberg, LevThe following problems are discussed in this work. 1.The Bohr and Rogosinski radii for Hardy classes of functions holomorphic in the disk. 2.Nether Bohr and Rogosinski radius exists for functions holomorphic in an annulus, with natural basis. 3.Asymptotics of the majorant function in the Reinhargt domains in $\mathbb C^n$. 4.The Bohr and Rogosinski radii for the mappings of Reinhardt Domains into Reinhardt domains. "Discrete Hilbert transform on sparse sequences" Belov, YuriiLet $H((a_n)) = \sum_n a_n*v_n/(z-t_n)$ be a discrete Hilbert transform. We are looking for necessary and sufficient conditions for $H$ being bounded operator from $l^2(v_n)$ to $L^2(\mu,C)$ where $\mu$ is some measure. The special interest for us is the case when $\mu$ is a discrete measure. For fast increasing sequences $|t_n|$ we are able to find such conditions. They are similar to classical Muckenhoupt condition. Discrete Hilbert transform naturally appears when we study Hilbert spaces of entire functions with Riesz basis of reproducing kernels (Paley-Wiener spaces, de Branges spaces, some Fock-type spaces). From our results we can obtain the description of all Carleson measures (and , in particular, Bessel sequences) and complete interpolating sequences for such spaces. As an application we verify Feichtinger conjecture for such spaces (and reproducing kernels) and give a counterexample for Baranov's conjecture about Bessel sequences in de Branges spaces. "A uniqueness theorem for harmonic functions" Borichev, Alexander "The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups" boscain, ugoWe present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp?s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares. We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane. Our study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation. "Loewner's theory in the abstract" Bracci, FilippoIn this talk I am going to describe an abstract approach to Loewner's theory which work also on complex hyperbolic manifolds. In particular I will show how to relate the evolution familes, Herglotz vector fields and Loewner's chains in a abstract way and which consequences this implies. "Parabolic Attitude" Bracci, FilippoIn this talk I will explain how apparently non-parabolic germs of holomorphic diffeomorphisms might have parabolic-like dynamics and will describe some invariants which determine the existence of parabolic basins of attraction in presence of resonances. "Recovery of data matrices from incomplete and corrupted entries: Theory and algorithms" Candes, EmmanuelA problem of considerable interest concerns the recovery of a data matrix from a sampling of its entries. In partially filled out surveys, for instance, we would like to infer the many missing entries. In the area of recommender systems, users submit ratings on a subset of entries in a database, and the vendor provides recommendations based on the user's preferences. Because users only rate a few items, we would like to infer their preference for unrated items (this is the famous Netflix problem). Formally, suppose that we observe a few entries selected uniformly at random from a matrix. Can we complete the matrix and recover the entries that we have not seen? This talk discusses two surprising phenomena. The first is that one can recover low-rank matrices exactly from what appear to be highly incomplete sets of sampled entries; that is, from a minimally sampled set of entries. Further, perfect recovery is possible by solving a simple convex optimization program, namely, a convenient semidefinite program. The second is that exact recovery via convex programming is further possible even in situations where a positive fraction of the observed entries are corrupted in an almost arbitrary fashion. In passing, this suggests the possibility of a principled approach to principal component analysis that is robust vis a vis outliers and corrupted data. We discuss algorithms for solving these optimization problems emphasizing the suitability of our methods for large scale problems. Finally, we present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces. "The Kadison-Singer problem in harmonic analysis" Casazza, PeteWe will see that the 1959 Kadison-Singer Problem in C*-Algebras generates several fundamental open problems in Harmonic Analysis including the Feichtinger Conjecture and Paving Conjecture for Laurant Operators. We will look at these open problems and their relationship to the Kadison-Singer Problem and recent advances on these problems. "Large global regular solutions of 3-D incompressible Navier-Stokes" Chemin, Jean-YvesThe purpose of this talk is to construct a class of large initial data which generates global smooth solutions of the incompressible Navier-Stokes system in three space dimension without boundary. We shall first recall what means large initial data, using the Koch and Tataru theorem. The initial data we consider are slowly varying vector fields. After rescaling in one direction, we studied a problem on profiles (i.e. rescalded vector fields) which seems ill-posed. We prove a global Cauchy-Kovalevska theorem on this sytem which requires smallness in a norm which measure the radius of analyticity of the initial data for this problem. Going back to the original Navier-Stokes problem, the initial data turns out to be very large. "Holomorphic self-maps of the disk intertwinig two linear fractional maps" Contreras, Manuel D.In this talk we characterize the holomorphic self-maps of the unit disk that intertwine two given linear fractional self-maps of the disk. The proofs are based on iteration and a detailed analysis of the solutions of Schroeder's and Abel's equations. In particular, we characterize the maps that commute with a given linear fractional map. This is a joint work with S. D?-Madrigal, D. Vukotic, and M.J. Mart? "A new calculus for two dimensional ideal fluid dynamics" Crowdy, DarrenIn classical fluid dynamics, an important problem arising in a variety of applications is to understand how vorticity interacts with solid objects (e.g. aerofoils, obstacles or stirrers). For planar flows, a variety of powerful mathematical results exist (complex variable methods, conformal mapping, Kirchhoff-Routh theory) that have been used to study such problems but the constructions are usually restricted to problems with just one, or perhaps two, objects. Expressed another way, most studies deal only with fluid regions that are simply or doubly connected. There has been a general and longstanding perception that problems involving fluid regions of higher connectivity are too challenging to be tackled analytically. The talk will show that there is a way to formulate the theory so that the relevant fluid dynamical formulae are exactly the same irrespective of the connectivity of the domain. This provides a flexible and unified tool for modelling the fluid dynamical interaction of multiple objects/aerofoils/obstacles/stirrers in ideal flow and their interaction with free vortices. The approach rests on use of a special transcendental function associated with multiply connected planar domains called the Schottky-Klein prime function. While the presentation will be in the context of ideal fluid dynamics, the same methods provide a constructive calculus for solving general potential theory problems in planar multiply connected domains. "The continuous shearlet transform in arbitrary space dimensions" Dahlke, StephanIn this talk, we shall be concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. It turns out that the associatated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, one can verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals. This is joint work with G. Steidl (Mannheim) and G. Teschke (Neubrandenburg) "Anisotropic Delaunay triangulations and denoising" Demaret, LaurentAnalysing the nature of boundaries between objects in a scene is one of the main tasks in image processing. Isotropic approximation methods -like wavelets- do not provide optimal approximation rates when applied to functions with singularities along curves. The solution to this problem is a challenging mathematical issue. Anisotropic triangulations have proved to be a fruitful ansatz to solve this problem: they provide sparse and flexible image representations, as well as a good reconstruction quality. Adaptive thinning is a greedy refinement technique which proceeds to the removal of the points leading to minimal reconstruction error: the output is a content-adapted Delaunay triangulation, with elongated triangles in areas where gradients are highly directed, in particular along singularities. These methods have already been successfully used in the context of terrain modelling, image compression and more recently of compression of video sequences. In this talk we present a new application field of these methods: image denoising. The key observation is the following: the approximation produced by Delaunay based adaptive thinning acts as a highly nonlinear filtering of images, but are simultaneously edge- and shape-preserving. After a short general introduction to the topic of anisotropic Delaunay triangulations for image approximation, the focus is on the practical and theoretical questions related to the problem of denoising based on these schemes. "A non-autonomous version of the Denjoy-Wolff theorem" Diaz-Madrigal, SantiagoThe aim of this work is to establish the celebrated Denjoy-Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, omega-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior. "Geometry behind chordal Loewner chains" Diaz-Madrigal, SantiagoLoewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of the famous Bieberbach's conjecture and the well-celebrated Schramm's Stochastis Loewner Evolution. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. "Time-frequency localization in phase-space" Doerfler, MonikaInspired by the classical work of Slepian/Pollak/Landau we aim at giving similar results for eigenfunctions and eigenvalues of localization operators defined directly in phase space. We will give basic definitions, some existing results on localization operators and some first results concerning localization in phase space. Possible consequences and connections to other problems in harmonic analysis will be discussed. "The decoration theorem and symmetries of the Mandelbrot set" Dudko, Dzmitry "The probabilistic frame potential and random tight fusion frames" Ehler, MartinA unit norm frame for R^d is a finite sampling of the unit sphere that spans R^d. We generalize this concept to probability distributions on the sphere. In fact, we introduce probabilistic p-frames and a probabilistic p-th frame potential. Its minimizers then are characterized by means of probabilistic p-frames that are “tight”. A fusion frame can be considered as a finite sampling of the Grassmannian manifold that “spans” R^d. We generalize this concept to probability distributions. Finally, we prove that the random sampling of the Grassmannian manifold converges (with growing sample size) towards a tight fusion frame. "Boundary behavior and rigidity of semigroups of holomorphic mappings" Elin, MarkThe asymptotic behavior of semigroups of holomorphic mappings and their boundary behavior have attracted considerable attention of many mathematicians during a long period, but especially in the last decade. In this talk we discuss some recent quantitative characteristics of the boundary asymptotic behavior of such semigroups acting on the open unit disk of the complex plane. In particular, we present new results on the limit curvature of semigroup trajectories at the boundary Denjoy-Wolff point. These results enable us to establish an asymptotic rigidity property for semigroups of parabolic type. This is joint work with D. Shoikhet. "Singular perturbation of polynomial potentials with application to PT-symmetric families" Eremenko, AlexandreIn the first part, eigenvalue problems of the form -w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on two rays in the complex plane are discussed. Sufficient conditions for continuity of the spectrum when the leading coefficient of P tends to 0 are given. In the second part, these results are applied to the study of topology and geometry of the real spectral loci of PT-symmetric families with P of degree 3 and 4, and prove several related results on the location of zeros of their eigenfunctions. "3D geometrical sparse representations with application to inverse problems and video processing" Fadili, JalalThis work describes several 3D multiscale transforms among which two new 3D curvelet transforms. The latter are built as 3D extensions of the first generation 2D curvelets and then extend their sparsifying properties depending on the geometrical content of the data. The first one, called BeamCurvelet transform, is well designed for representing 1D filamentary structures in a 3D space, while the second, the RidCurvelet transform, is designed to efficiently capture 2D surfaces. Thus, the first one will sparsify smooth content away from singularities along lines, and the second will provide near optimal sparse representations of piecewise volumes away from smooth 2D boundaries. We finally demonstrate the potential applicability of these transforms to a variety of tasks such as object detection, denoising, inpainting and video processing. "Weak conformality at a point" Golberg, AnatolyThe condition of conformality in a domain is satisfied by a rich but rather narrow class of mappings in the plane. In higher dimensions the class is in a sense even narrower as the well-known theorem of Liouville states that the only conformal mappings, even in $\mathbb R^3$, are of M\"obius transformations. On the other hand a class that naturally extends conformality in the plane and higher dimensions - the class of quasiconformal mappings guarantees real differentiability almost everywhere. A notion closely related to real differentiability is "local weak conformality". In our talk we discuss local sufficient conditions, in plane and/or in higher dimensions, for a homeomorphism to be real differentiable at a point, weakly conformal at a point, to be H\"older continuous, to preserve angles, etc. Such properties have applications in Mechanics and Physics. A joint work with Melkana Brakalova, University of Fordham, US. "Spectral models for orthonormal wavelets and multiresolution analysis" Gomez-Cubillo, FernandoSpectral representations of the dilation and translation operators on $L^2({\mathbb R})$ are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions defined on the functional spectral spaces. The approach is useful for computational purposes. "Gabor frames and complex analysis" Groechenig, KarlheinzGabor analysis offers many problems at the intersection of harmonic analysis and complex analysis. One of the goals of Gabor analysis is to characterize those lattices in R^d that generate a Gabor frame with a given basis function. At this time the anwer is known only for the Gaussian (Lyubarski, Seip), for the hyperbolic cosine (Janssen-Strohmer), and for the one-sided exponential (Janssen). In this talk we will explain some problems and results about Gabor frames with Gaussians and Hermite functions. (a) For the characterization of Gaussian Gabor frames in dimension 1 one needs precise growth estimates for certain classical elliptic functions, the Weierstrass sigma-function. (b) The investigation of Gabor frames with Hermite functions leads to a new type of interpolation problem for entire functions that is not at all understood. (c) In higher dimensions the study of Gabor frames with the Gaussian leads to an interpolation problem for entire functions of several complex variables. We explain some preliminary results and some challenges to complex analysts. The talk will cover joint work with Ya. Lyubarskii and A. Borichev. "A regularity criterion for the 3D NSE vorticity in a local version of $BMO$" Grujic, ZoranA spatio-temporal localization of a BKM-type Kozono-Taniuchi regularity criteria for the solutions to the 3D Navier-Stokes equations; namely, the time-integrability of the $BMO$-norm of the vorticity, will be presented. The localization is based on an explicit localization of the vortex-stretching term and a version of local nonhomogeneous Div-Curl Lemma. This is a joint work with Rafaela Guberovic. "Localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3D NSE" Guberovic, RafaelaIn '95 daVaiga has shown that $\|D u\|^\frac{2q}{2q-3}_q$ $\in L^1(0,T)$ is a regularity class for the Navier-Stokes Equations $\mbox{for any}$ $3 \le q < \infty$ . Previously, Beale-Kato-Majda proved the regularity when the time-integrability of the $L^\infty$-norm of the vorticity holds. In this talk we will show the localized versions of the afforementioned conditions imply local enstrophy remains bounded. The geometric conditions on the vorticity direction field are important in studying the regularity and the localized version has been recently obtained by Grujic. The special scaling invariant regularity class of weighted $L^pL^q$ type for the vorticity magnitude with the coherence factor as a weight will be demonstrated in a localized setting. "Loewner evolution in doubly connected domains" Gumenyuk, PavelLoewner Theory constitutes an essential part of Geometric Function Theory and is known to have applications both in Complex Analysis and Mathematical Physics. From the modern point of view Loewner Theory can be described by the relations and interplay of three concepts: Loewner chains, Herglotz vector fields and evolution families. In the talk we will extend this general approach for doubly connected case. In contrast to the simply connected case, evolution families do not in general consist of self-mappings of any xed reference domain and one has to consider families of holomorphic mappings between an expanding system of annuli. Accordingly, Herglotz vector elds in doubly connected case cannot be described as mappings from [0;+1) to the set Gen(D) of all in nitesimal generators of one-parametric semigroups. In the talk we will give de nitions of evolution families and Herglotz vector elds in doubly connected setting and establish one-to-one correspondence between these two classes of objects. "Spectral theory of the Fourier operator trunkated on the positive half-axis" Katsnelson, VictorThe spectral theory of the Fourier operator truncated on the positive half-axis is developed. "The Cauchy problem for the weakly dissipative mu-DP equation" Kohlmann, MartinWe discuss the Cauchy problem for the periodic $\mu$DP equation with weak dissipation and prove local well-posedness in the Sobolev spaces $H^s$ for $s>3/2$. For $s=2$ we establish the precise blow-up setting and consider solutions with finite existence time and global solutions. "Parametric representation and Loewner chains in several complex variables" Kohr, GabrielaIn this talk we shall present recent results related to the theory of Loewner chains in several complex variables. Applications to extreme and support points for mappings which have parametric representation on the unit ball will be also given. "Boundary value problems for Brinkman operators on Lipschitz domains. Applications" Kohr, MirelaIn this talk we present recent results in the area interfacing potential theory and boundary value problems for Brinkman operators on Lipschitz domains in $\mathbb{R}^n$. Extension to boundary value problems for Brinkman operators on Lipscjitz domains in Riemannian manifolds are also considered. "Bogoliubov functionals and the dynamics of particles in continuum" Kozicki, JurijMarkov evolution of infinite systems of particles in R^d is described by means of generating functionals. It is shown that such functionals can be continued to holomorphic functions on Banach spaces. As a result, the evolution is described in spaces of such functions. A number of theorems are proven which establish the properties of such dynamical systems. "Accuracy and stability of the butterfly fast Fourier transform" Kunis, StefanA straightforward discretization of high dimensional problems often leads to a serious growth in the number of degrees of freedom and thus even efficient algorithms like the fast Fourier transform have high computational costs. Utilizing sparsity allows for a severe decrease of the problem size but asks for the customization of efficient algorithms to these thinner discretization. We discuss recent generalizations of the FFT based on a multilevel approximation scheme and present bounds on the approximation error as well as on its numerical stability. "Characterization of singularities in multidimensions using the continuous shearlet transform" Labate, DemetrioDirectional multiscale systems such as the Shearlet Transform were recently introduced to overcome the limitations of the traditional wavelet transform in dealing with multidimensional data. In particular, while the continuous wavelet transform is able to identify the location of singularities of functions and distributions through its asymptotic behavior at fine scale, it lacks the ability to capture the geometry of the singularity set. In this talk, we show that the Shearlet Transform has the ability to precisely characterize both the location and orientation of the set of singularities of 2D and 3D functions. This approach leads to very competitive imaging applications such as edge detection and feature extraction. "Optimally sparse approximations of 3D data using shearlets" Lemvig, JakobMany important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds. To analyze the ability of representation systems to reliably capture and sparsely represent anisotropic structures, Donoho introduced the model situation of so-called cartoon-like images, i.e., two-dimensional functions which are $C^2$ smooth apart from a $C^2$ discontinuity curve. In the past years, it was shown that curvelets, contourlets, and shearlets all have the ability to essentially optimal sparsely approximate cartoon-like images measured by the $L_2$-error of the (best) n-term approximation. Traditionally, this type of results has only been available for band-limited generators, but recently Kutyniok and Lim showed that optimal sparsity also holds for spatial compactly supported shearlet generators under weak moment conditions. In this talk, we introduce three-dimensional cartoon-like images, i.e., three-dimensional functions which are $C^2$ except for discontinuities along $C^2$ surfaces, and consider sparse approximations of such. We first derive the optimal rate of approximation which is achievable by exploiting information theoretic arguments. Then we introduce three-dimensional pyramid-adapted shearlet systems with compactly supported generators and prove that such shearlet systems indeed deliver essentially optimal sparse approximations of three-dimensional cartoon-like images. Finally, we even extend this result to the situation of surfaces which are $C^2$ except for zero- and one-dimensional singularities, and again derive essential optimal sparsity of the constructed shearlet frames. This is joint with G. Kutyniok and W.-Q Lim (University of Osnabr\"uck). "A disk rotating around a black hole" Lenells, JonatanTwo of the most well-studied solutions of the axisymmetric Einstein equations are the Kerr black hole and the Neugebauer-Meinel disk. In this talk I will present exact solutions of a class of boundary value problems for the Einstein equations which combine the Kerr and Neugebauer-Meinel spacetimes. Thus, the new solutions involve a disk (modeled by a pressureless perfect fluid) rotating uniformly around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus four. "Representation of a function via the absolutely convergent Fourier integral" Liflyand, ElijahNew sufficient conditions for representation of a function via the absolutely convergent Fourier integral are obtained. In one dimension, a new feature is that this is controlled by the behavior near infinity of both the function and its derivative. Most of such and related results are extended to any dimension $d\ge2.$ The sharpness of the obtained theorems is checked on the known multipliers (joint work with R.M. Trigub). "Construction and applications of compactly supported shearlets" Lim, Wang-QIt is now widely acknowledged that analyzing the intrinsic geometrical features of the underlying object is essential in many applications. In order to achieve this, several directional representation schemes have been proposed in the past. Of those, shearlet tight frames have been extensively studied during the last years due to their optimal approximation properties of 2D data governed by curvilinear singularities and their unified treatment of the continuum and digital setting. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial. In this talk, we will discuss a novel approach to construct shearlets which not only have compact support in spatial domain but can also provide optimally sparse approximation of curvilinear singularities. We will provide sufficient conditions for compactly supported shearlet frames for both the 2D and 3D situation and present concrete examples. Finally, we will discuss some applications of compactly supported shearlets for imaging science. This is joint with G. Kutyniok and J. Lemvig. "Laplacian growth: toward exact solutions in dimensions higher than two" Lundberg, ErikThe Laplacian Growth (or Hele-Shaw) Problem is a nonlinear moving boundary problem which has attracted wide and growing attention mainly for two reasons (to be brief): (1) it is ubiquitous as an idealized model of many growth processes in nature and industry (2) it admits an abundance of explicit exact solutions in two dimensions. In stark contrast, ellipsoids are the only exact solutions known in higher dimensions. It is tempting to blame this on the lack of conformal maps in space. However, exact solutions can be understood without the time-dependent conformal map by studying dynamics of singularities of the Schwarz function. We will lift this point of view to higher dimensions using the "Schwarz potential" introduced by D. Khavinson and H.S. Shapiro. We give a new exact solution in R^4, and we discuss possible future directions to take in this program. "Radial behavior of functions in the Korenblum class" Lyubarskii, Yurii "Radial explorer" Makarov, NikolaiI will define two random curve models which could be thought of as radial versions of the well-known Schramm-Sheffield construction of the (chordal) harmonic explorer. The scaling limit law, the radial SLE(4), is the same in both cases but the underlying field theories are different (Ramond and Neveu-Schwatz boundary conditions in physical terminology). The talk is based on the joint work with Nam-Gyu Kang and Dapeng Zhan. "Virasoro algebra and distribution in the space of univalent functions" Markina, Irina "On particle trajectories in linear water waves" Matioc, Anca-VoichitaWe determine the phase portrait of a Hamiltonian system of equations describing the motion of the particles in linear water waves. The particles experience in each period a forward drift which is minimal on the flat bed. "On a three phase Muskat-like problem" Matioc, Bogdan-VasileIn this talk we present a model for the evolution of two fluid phases in a porous medium. The fluids are separated from each other and the wetting phases from air by interfaces which evolve in time. Within parabolic theory we establish existence and uniqueness of classical solutions for the problem and study the stability properties of the finger-shaped equilibria. "Linearisers of entire maps" Mihaljevic-Brandt, Helena "Classification of postcritically finite Newton maps" Mikulich, Yauhen "Multi-operator colligations and multi-variate characterisctic functions" Neretin, YuryIn the spectral theory of non-self-adjoint operators there is a well-known operation of product of operator colligations. Many similar operations appear in the theory of infinite-dimensional groups as multiplications of double cosets. We construct characteristic functions for such double cosets and get semigroups of matrix-valued functions in matrix balls. "Beurling-Helson theorem for modulation spaces" Okoudjou, KassoThe main problem I would like to address in this talk is the following. Given a function $f$ with certain properties, what are the operations we can perform on the function, that will result in new functions possessing the same properties as $f$? In particular, I will focus on what change of variables $\phi$ will guarantee that $f$ and $f(\phi)$ have exactly the same properties. I will first discuss this problem in the context of Fourier transforms and show how a theorem of Beurling and Helson in this setting generalizes to functions defined by time-frequency methods. "Sparse geometric processing of images" Peyre, GabrielIn this talk, I will review recent works on the sparse representations of natural images. I will in particular focuss on the application of these emerging models for the resolution of various imaging problems, which include compression, denoising and super-resolution of images, as well as compressive sensing and compressive wave computations. Natural images exhibit a wide range of geometric regularities, such as curvilinear edges and oscillating textures. Adaptive image representations select bases from a dictionary of orthogonal or redundant frames that are parameterized by the geometry of the image. If the geometry is well estimated, the image is sparsely represented by only a few atoms in this dictionary. The resolution of ill-posed inverse problems in image processing is then regularized using sparsity constraints in these adapted representations. I will also discuss the application of these ideas for the acceleration of computing problems, such as the reverse time migration in seismic imaging. "Curvelet-wavelet regularized split Bregman iteration for compressed sensing" Plonka, GerlindCompressed sensing is a new concept in signal processing. Assuming that a signal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the Shannon-Nyquist theory requires. Many images can be sparsely approximated by expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent point-like features while curvelets represent line-like features well. For a suitable recovery of images, we propose a model that contains weighted sparsity constraints in two different frames as e.g. wavelets and curvelets. We present an efficient iteration method to solve the corresponding sparsity-constrained optimization problem, based on Alternating Split Bregman algorithm. The convergence of the proposed iteration scheme will be proved by showing that it can be understood as a special cases of the Douglas-Rachford Split algorithm. Numerical experiments for compressed sensing based Fourier-domain random imaging show good performances of the proposed curvelet-wavelet regularized split Bregman (CWSpB) method, where we particularly use a combination of wavelet and curvelet coefficients as sparsity constraints. "Travelling bubbles in Hele-Shaw flows with kinetic undercooling" Prokert, GeorgWe discuss a two-dimensional free boundary problem for the Laplacian with Robin boundary conditions in an exterior domain. This problem describes so-called Hele-Shaw flows with kinetic undercooling and is also discussed in the context of models for gas ionization processes. The formulation of the problem as evolution equation leads to a nonlinear, degenerate transport equation on the circle line with a nonlocal lower order term. We identify the solutions that correspond to uniformly translating bubbles near trivial (circular) ones. It is shown that locally there is precisely one such nontrivial bubble for any translation velocity. In particular, the trivial solutions are unstable in a co-moving frame of reference. The degenerate character of the problem is reflected in a loss of regularity for the solutions of the linearized problem. Moreover, there is an upper bound on the regularity of the nontrivial solutions which depends smoothly on the regularization parameter. The proof of the results uses quasilinearization by differentiation, index theorems for degenerate ordinary differential operators on the circle line, and perturbation arguments for unbounded Fredholm operators. (Joint work with M. Guenther, Leipzig) -- "Integrability of the Loewner equation" Prokhorov, DmitryWe consider a special integrable case for the chordal version of the Loewner differential equation and describe its singular trajectories. "Wavelets and random power series: what is in common ?" Protasov, VladimirThe problem of distributions of power random series has been known since the famous works of P.Erdos of 1939-40. It has found many applications in the study of functional equations, PDE, random walks, Bernoulli convolutions, etc. Suppose we have a power series, whose coefficients are independent identically distributed random variables. The problem is to determine, whether such a series has a density, or is it singular? Surprisingly, one special case of this problem leads to the so-called refinement equation (functional difference equation with the contraction of the argument), which is applied in the study of compactly-supported wavelets. This phenomenon explains many common properties of wavelets and distributions of random series. We will also discuss applications of joint spectral characteristics of linear operators (joint spectral radius, the p-radius, the Lyapunov exponent, etc.) to this problem. "The identification problem for time-frequency localizing operators" Rashkov, PeterThe talk will discuss the study of general classes of Hilbert-Schmidt operators with time-frequency localizing properties. Identification of incompletely known linear operators based on the observation of restricted input and output signals is quite important in communications engineering. We develop a general setup for identification based on a Gabor series discretization of the operator. A study by Kozek and Pfander has suggested that the identification problem might be dual to the statement of density theorem for Gabor frames for a particular class of operators. In the general case we provide numerous examples of identification of classes of time-frequency localizing operators and show that identification is dependent on a wider set of criteria than a single density constraint. Our results are based on decay estimates for time-frequency localizing operators, properties of Gaussian Gabor frames, and localization properties of Gabor molecules. "Compressive sensing and harmonic analysis" Rauhut, HolgerThe talk will give an overview on recent topics at the interplay of compressive sensing, harmonic analysis and random matrix theory. Compressive sensing deals with the recovery of sparse signals from highly undersampled measurements via efficient algorithms, such as convex relaxation (l1-minimization). The full power of the theory can be exploited when using random matrices as measurement maps. It is by now well-understood, that optimal recovery rates can be achieved with Gaussian random measurement matrices. However, Gaussian matrices are of limited use in practice. Often more structure of the measurement matrix is needed. This requirement leads to the study of structured random matrices. The talk will discuss several types of such matrices: 1) Random Fourier type matrices, or more generally, random sampling matrices associated to bounded orthonormal systems. They arise e.g. in the context of recovering sparse trigonometric polynomials from random samples. The underlying mathematics are closely connected to deep problems in harmonic analysis, such as the Lambda_1 problem studied by Bourgain and Talagrand. As a special case, we study recovery of sparse expansions in Legendre polynomials. 2) Partial random circulant matrices. These matrices arise from the problem of recovery from undersampled convolutions. 3) Random Gabor frames. They are connected to time-frequency analysis. "The Loewner and Hadamard variations" Roth, OliverWe give an explicit formula relating the infinitesimal generators of the Loewner differential equation and the Hadamard variation. This is applied to establish an extension of the Hadamard variation to the case of arbitrary simply-connected domains and to prove the existence of Loewner chains with arbitrary smooth initial generator starting at an arbitrary univalent function which is sufficiently smooth up to the boundary. As another application of this method, we show that every subordination chain $f_t$ is differentiable almost everywhere and satisfies a Loewner equation, without assuming that $f_t'(0)$ is continuous. "On well-posedness and blow-up of a hyperbolic fluid model" Saal, JuergenIn various applications a delay of the propagation speed of certain quantities (temperature, fluid velocity, ...) has been observed. Such phenomena cannot be described by standard parabolic models, whose derivation relies on a Fourier type law (Paradoxon of infinite propagation speed). One way to give account to these observations and which was successfully applied to several models, is to replace Fourier's law by the law of Cattaneo. In the case of a fluid, this leads to a hyperbolicly perturbed quasilinear Navier-Stokes system for which well-posedness and blow up investigations will be presented. This is a joint project with Reinhard Racke at the University of Konstanz. "Newton's method as an efficient root finder for complex polynomials" Schleicher, Dierk "Novel phenomena and models of active fluids" Shelley, MichaelFluids with suspended microstructure - complex fluids - are common actors in micro- and biofluidics applications and can have fascinating dynamical behaviors. A new area of complex fluid dynamics concerns "active fluids" which are internally driven by having dynamic microstructure such as swimming bacteria. Such motile suspensions are important to biology, and are candidate systems for tasks such as microfluidic mixing and pumping. To understand these systems, we have developed both first-principles particle and continuum kinetic models for studying the collective dynamics of hydrodynamically interacting microswimmers. The kinetic model couples together the dynamics of a Stokesian fluid with that of an evolving "active" stress field. It has a very interesting analytical and dynamical structure, and predicts critical conditions for the emergence of hydrodynamic instabilities and fluid mixing. These predictions are verified in our detailed particle simulations, and are consistent with current experimental observation. "Global regularity for the three-dimensional primitive equations of atmospheric and oceanic dynamics" Titi, EdrissIn this talk I will show the global existence and uniqueness of strong solutions to the three-dimensional Primitive Equations of atmospheric and oceanic dynamics. Inspired by this result I will also provide a new global regularity criterion for the three-dimensional Navier-Stokes equations involving one component of the pressure gradient. This is a joint work with Chongsheng Cao. "A family of models for rotating shallow water flow" Vasylkevych, SergiyWe consider recently derived family of models for the rotating shallow water flow in the regime known as semi-geostrophic limit, which is typical for mid-latitude large-scale flows in atmosphere and ocean. The family includes, among others, two previously known models (so called $L_1$ and LSG models proposed by R.Salmon) and a completely new model with superior regularity properties. We show that each member of the family is given by a Hamiltonian system on an appropriately chosen diffeomorphism group. Moreover, the Hamiltonian framework turns out to be superior to the original derivation of the equations via asymptotic expansion of the rotating shallow water Lagrangian since it does not place topological restrictions on Coriolis parameter. Furthermore, we prove that all but one model in the family are well-posed in the classical sense for short time as well as global existence of solutions for the distinguished model. "Towards the definition of the Cauchy-Fantappie and Radon transforms" Vidras, AlekosWe define substitutes for the $(n-1)$-Cauchy-Fantappi\'e and Radon transforms in an $n$-complete smooth simplicial toric variety $\mathcal X$ using a definition of $(n-1)$-concavity related to a multi-perspective embedding. Such a construction fits with a notion of bi-concavity for a direct product of two projective spaces. "Existence and stability of solitary water waves with weak surface tension" Wahlen, ErikTwo-dimensional solitary water waves with weak surface tension (0 < \beta < 1/3, where \beta is the Bond number) are constructed by minimising the energy subject to the constraint of fixed momentum. The stability of the set of minimisers follows by a standard principle since the energy and momentum are conserved quantities. 'Stability' must however be understood in a qualified sense because of the lack of a global well-posedness theory for the initial value problem. The variational method relies on the concentration-compactness principle and a penalisation argument. The solitary waves are to leading order periodic wave trains modulated by exponentially decaying envelopes described by the focusing nonlinear Schr?ger equation "Stability of steady states in thin film equations with soluble surfactant" Walker, ChristophThe talk is dedicated to thin film equations with soluble surfactant and gravity. The governing system of degenerate parabolic equations for the film height, bulk and surface surfactant concentration is shown to be locally well-posed. It is also shown that the steady states are asymptotically stable. (This is joint work with J. Escher, M. Hillairet, and Ph. Laurencot.) "Asymptotics in the primitive equations" Wirosoetisno, DjokoWe describe, from physical and mathematical points of view, asymptotic regimes of the primitive equations which are of interest in atmospheric/oceanic dynamics. Some results on the long-time behaviour of the solution in the strongly rotating and/or stratified cases will be discussed. (Joint work with R Temam) "Comparison theorems for number of conjugate points along sub-Riemannian extremals" Zelenko, IgorThe classical Rauch comparison theorem in Riemannian geometry provides the estimation of the number of conjugate points along Riemannian geodesics in terms of bounds for the sectional curvature. We give a generalization of this theorem to extremals of a wide class of optimal control problems including sub-Riemannian extremals. The problem can be reformulated as the problem to estimate the number of conjugate points along a curve in a Lagrangian Grassmannian in terms of the invariants of this curve with respect to the natural action of the Linear Symplectic Group. Our treatment of this problem is based on the construction of the canonical bundle of moving frames and the complete system of symplectic invariants for curves in Lagrangian Grassmannians previously done in the joint works with Chengbo Li. We will explain how appropriately arranged bounds for these symplectic invariants effect the bounds for the number of conjugate points. The application for extremals of natural sub-Riemannian metrics on principal connections of principal bundles with one-dimensional fibers over Riemannian manifolds (i.e. magnetic fields on Riemannian manifolds) will be given.

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