new trends in harmonic and complex analysis - jacobs...

New Trends in Harmonic and Complex Analysis

Jacobs University, Bremen, Germany

June 29 – July 3, 2010

Monday, June 28

18:00–20:00 Welcome receptionJacobs University Club

Tuesday, June 29

08:50–09:00 Welcome address (Research II Lecture Hall)

09:00–10:00 Keynote talk (Research II Lecture Hall)Pete Casazza (Missouri)The Kadison-Singer problem in harmonic analysis

10:00–10:30 Coffee break

10:30–12:30 Parallel Sessions

Research II New developments in deterministic Loewner theory ISession organizers: Manuel D. Contreras (Sevilla),Christian Pommerenke (Berlin)

10:30 Oliver Roth (Wurzburg)The Loewner and Hadamard variations

11:00 Santiago Dıaz-Madrigal (Sevilla)Geometry behind chordal Loewner chains

11:30 Gabriela Kohr (Babes-Bolyai University)Parametric representation and Loewner chainsin several complex variables

Research III Spaces of analytic functions and applications ISession Organizer: Yurii Lyubarskii (NTNU Trondheim)

10:30 Luıs Daniel Abreu (Coimbra)Beyond analyticity: time-frequency analysis of Bergmanand Fock-type spaces

11:00 Yury Belov (NTNU Trondheim)Discrete Hilbert transform on sparse sequences

11:30 Alexander Borichev (Marseille)A uniqueness theorem for harmonic functions

12:00 Monika Dorfler (University of Vienna)Time-frequency localization in phase-space

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12:30–13:30 Lunch


Research II Complex dynamical systemsSession Organizers: Walter Bergweiler (Kiel),Dierk Schleicher (Jacobs University)

13:30 Helena Mihaljevic-Brandt (Kiel)Linearisers of entire maps

14:00 Dzmitry Dudko (Gottingen and Jacobs University)The decoration theorem and symmetries of the Mandelbrot set

14:30 Yauhen Mikulich (Jacobs)Classification of postcritically finite Newton maps

15:00 Dierk Schleicher (Jacobs)Newton’s method as an efficient root finder for complex polynomials

Research III Time-frequency analysisSession Organizer: Gotz Pfander (Jacobs University)

13:30 Stefan Kunis (Chemnitz and HelmholtzZentrum Munchen)Accuracy and stability of the butterfly fast Fourier transform

14:00 Peter Rashkov (Jacobs)The identification problem for time-frequency localizing operators

14:30 Kasso A. Okoudjou (Maryland)Beurling-Helson theorem for modulation spaces

15:00 Gotz Pfander (Jacobs)A few remarks on Gaussian Gabor frames


16:00–17:00 Keynote talk (Research II Lecture Hall)Alexandre Eremenko (Purdue University)Singular perturbation of polynomial potentialswith application to PT-symmetric families

17:00–18:00 Keynote talk (Research II Lecture Hall)Vladimir Protasov (Moscow State University)Wavelets and random power series: what is in common?

18:00 Dinner

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Wednesday, June 30

09:00–10:00 Keynote talk (Research II Lecture Hall)Holger Rauhut (Bonn)Compressive sensing and harmonic analysis



Research II Geometric Complex Analysis ISession Organizers: Mark Agranovsky (Bar-Ilan),David Shoikhet (ORT Braude College)

10:30 Manuel Contreras (Sevilla)Holomorphic self-maps of the disk intertwining two linear fractional maps

11:00 Lev Aizenberg (Bar-Ilan)Remarks on the Bohr and Rogosinski phenomena for power series

11:30 Alekos Vidras (University of Cyprus)Towards the definition of the Cauchy-Fantappie and Radon transforms

Research III Geometric Multiscale Analysis ISession Organizer: Gitta Kutyniok (Osnabruck)

10:30 Demetrio Labate (University of Houston)Characterization of singularities in multidimensionsusing the continuous shearlet transform

11:00 Wang-Q Lim (Osnabruck)Construction and applications of compactly supported shearlets

11:30 Jakob Lemvig (Osnabruck)Optimally sparse approximations of 3D data using shearlets

12:00 Stephan Dahlke (Marburg)The continuous shearlet transform in arbitrary space dimensions

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12:30–13:30 Lunch


Research II Patterns in Fluid Dynamics ISession Organizers: Joachim Escher, Mats Ehrnstrom (Hannover)

13:30 Erik Wahlen (Universitat des Saarlandes and Lund University)Existence and stability of solitary water waveswith weak surface tension

14:00 Martin Kohlmann (Hannover)The Cauchy problem for the weakly dissipative µ-DP equation

14:30 Jurgen Saal (Konstanz)On well-posedness and blow-up of a hyperbolic fluid model


15:30 Anca-Voichita Matioc (Hannover)On particle trajectories in linear water waves

16:00 Mirela Kohr (Babes-Bolyai University)Boundary value problems for Brinkman operatorson Lipschitz domains. Applications

Research III Geometric Multiscale Analysis IISession Organizer: Gitta Kutyniok (Osnabruck)

13:30 Gerlind Plonka (Duisburg-Essen)Curvelet-wavelet regularized split Bregman iterationfor compressed sensing

14:00 Gabriel Peyre (Paris-Dauphine)Sparse geometric processing of images

14:30 Jalal Fadili (CNRS France)3D geometrical sparse representations with applicationto inverse problems and video processing


15:30 Laurent Demaret (HelmholtzZentrum Munchen)Anisotropic Delaunay triangulations and denoising

15:30 Fernando Gomez-Cubillo (Valladolid)Spectral models for orthonormal wavelets and multiresolution analysis

16:30 Martin Ehler (Maryland and National Institutes of Health)The probabilistic frame potential and random tight fusion frames

18:00 Dinner

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Thursday, July 1

09:00–10:00 Keynote talk (Research II Lecture Hall)Jean-Yves Chemin (Jussieu)Large global regular solutions of 3-D incompressible Navier-Stokes



Research II Mathematical Fluid DynamicsSession Organizer: Marcel Oliver (Jacobs University)

10:30 Sergiy Vasylkevych (Jacobs)A family of models for rotating shallow water flow

11:00 Djoko Wirosoetisno (Durham)Asymptotics in the primitive equations

11:30 Rafaela Guberovic (ETH Zurich)Localization of analytic regularity criteria on the vorticity and balancebetween the vorticity magnitude and coherence of the vorticitydirection in the 3D NSE

12:00 Zoran Grujic (Virginia)A regularity criterion for the 3D NSE vorticity in a local version of BMO

Research III Spaces of analytic functions and applications IISession Organizer: Yurii Lyubarskii (NTNU Trondheim)

10:30 Victor Katsnelson (Weizmann Institute)Spectral theory of the Fourier operator truncated on the positive half-axis

11:00 Elijah Liflyand (Bar-Ilan)Representation of a function via the absolutely convergent Fourier integral

11:30 Yurii Lyubarskii (NTNU Trondheim)Radial behavior of functions in the Korenblum class

12:30–13:30 Lunch

13:30–14:30 Keynote talk (Research II Lecture Hall)Emmanuel Candes (Stanford)Recovery of data matrices from incomplete and corrupted entries:Theory and algorithms

14:30–18:00 Explore the city of Bremen

18:00–21:00 Conference Dinner on a BoatDeparture: “Martinianleger” in the city of Bremen at 18:00Arrival: near Jacobs University at 21:00

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Friday, July 2

09:00–10:00 Keynote talk (Research II Lecture Hall)Mike Shelley (Courant Institute)Novel phenomena and models of active fluids



Research II Patterns in Fluid Dynamics IISession Organizers: Joachim Escher, Mats Ehrnstrom (Hannover)

10:30 Christoph Walker (Hannover)Stability of steady states in thin film equations with soluble surfactant

11:00 Georg Prokert (Eindhoven)Travelling bubbles in Hele-Shaw flows with kinetic undercooling

11:30 Bogdan-Vasile Matioc (Hannover)On a three phase Muskat-like problem

12:00 Jonatan Lenells (Hannover)A disk rotating around a black hole

Research III New developments in deterministic Loewner theory IISession organizers: Manuel D. Contreras (Sevilla),Christian Pommerenke (Berlin)

10:30 Pavel Gumenyuk (Bergen)Loewner evolution in doubly connected domains

11:00 Dmitry Prokhorov (Saratov State University)Integrability of the Loewner equation

11:30 Filippo Bracci (Universita di Roma “Tor Vergata”)Loewner’s theory in the abstract

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12:30–13:30 Lunch


Research II Geometric Complex Analysis IISession Organizers: Mark Agranovsky (Bar-Ilan),David Shoikhet (ORT Braude College)

13:30 Anatoly Golberg (Holon Institute of Technology)Weak conformality at a point

14:00 Eric Lundberg (University of South Florida)Laplacian growth: toward exact solutionsin dimensions higher than two

14:30 Filippo Bracci (Universita di Roma “Tor Vergata”)Parabolic Attitude

Research III Analysis on Lie groups and Mathematical Physics ISession Organizers: Alexander Vasiliev,Irina Markina (University of Bergen, Norway)

13:30 Helene Airault (Picardie)Extension of vector fields in the disk and conformal welding

14:00 Ugo Boscain (Ecole Polytechnique)The intrinsic hypoelliptic Laplacian and its heat kernelon unimodular Lie groups

14:30 Yuri Neretin (University of Vienna)Multi-operator colligations and multi-variate characteristic functions


16:00–17:00 Keynote talk (Research II Lecture Hall)Karlheinz Grochenig (Uni Wien)Gabor frames and complex analysis

17:00–18:00 Keynote talk (Research II Lecture Hall)Edriss Titi (UC Irvine & Weizmann Institute)Global regularity for the three-dimensional primitive equationsof atmospheric and oceanic dynamics

18:00 Dinner

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Saturday, July 3

09:00–10:00 Keynote talk (Research II Lecture Hall)Nikolai Makarov (Caltech)Radial explorer



Research II Geometric Complex Analysis IIISession Organizers: Mark Agranovsky (Bar-Ilan),David Shoikhet (ORT Braude College)

10:30 Mark Elin (ORT Braude College)Boundary behavior and rigidity of semigroups of holomorphic mappings

11:00 Santiago Dıaz-Madrigal (Sevilla)A non-autonomous version of the Denjoy-Wolff theorem

11:30 Jerzy Kozicky (Uniwersytet Marii Curie-Sklodowskiej)Bogoliubov functionals and the dynamics of particles in continuum

Research III Analysis on Lie groups and Mathematical Physics IISession Organizers: Alexander Vasiliev,Irina Markina (University of Bergen, Norway)

10:30 Roland Friedrich (MPI Bonn)TBA

11:00 Irina Markina and Alexander Vasiliev (Bergen)Virasoro algebra and distribution in the space of univalent functions

11:30 Igor Zelenko (Texas A&M University)Comparison theorems for number of conjugate pointsalong sub-Riemannian extremals

12:30–13:30 Keynote talk (Research II Lecture Hall)Darren Crowdy (Imperial College)A new calculus for two dimensional ideal fluid dynamics

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Abstracts

Abreu, Luıs Daniel

Beyond analyticity: time-frequency analysis of Bergman and Fock-type spaces

Using Gabor and wavelet frames, we will study Bergman and Fock-type spaces whenthe elements of the space are not necessarily analytic. The study is based on the ob-servation that Bergman-type spaces (of non-analytic functions) have an orthogonal de-composition in several subspaces (of polyanalytic functions), where the first subspacesare the classical Bergman and Fock space. In particular, this approach leads to newsampling and interpolation results. As in the collaboration of Grochenig with Borichevand Lyubarskii, it shows how a blend of frame and complex analysis methods can leadto new and unexpected results.

Airault, Helene

Extension of vector fields in the disk and conformal welding

References:

1. “Modulus of continuity of the canonic Brownian motion on the group of diffeomor-phisms of the circle” by Airault-Ren, J. Funct. Anal. 196 (2002), 395–426.

2. “Canonical Brownian motion on the space of univalent functions and resolutionof Beltrami equations by a continuity method along stochastic flows” by Airault-Malliavin-Thalmaier, J. Math. Pures Appl. 83 (2004), 955–1018.

Aizenberg, Lev

Remarks on the Bohr and Rogosinski phenomena for power series

The following problems are discussed in this work.

1. The Bohr and Rogosinski radii for Hardy classes of functions holomorphic in thedisk.

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 Cn.

4. The Bohr and Rogosinski radii for the mappings of Reinhardt Domains into Rein-hardt domains.

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Belov, Yurii

Discrete Hilbert transform on sparse sequences

Let H((an)) =∑

n an ∗vn/(z− tn) be a discrete Hilbert transform. We are looking fornecessary and sufficient conditions for H being bounded operator from l2(vn) to L2(µ,C)where µ is some measure. The special interest for us is the case when µ is a discretemeasure. For fast increasing sequences |tn| we are able to find such conditions. They aresimilar to classical Muckenhoupt condition.

Discrete Hilbert transform naturally appears when we study Hilbert spaces of en-tire functions with Riesz basis of reproducing kernels (Paley-Wiener spaces, de Brangesspaces, some Fock-type spaces).

From our results we can obtain the description of all Carleson measures (and , inparticular, Bessel sequences) and complete interpolating sequences for such spaces.

As an application we verify Feichtinger conjecture for such spaces (and reproducingkernels) and give a counterexample for Baranov’s conjecture about Bessel sequences inde Branges spaces.

Borichev, Alexander

A uniqueness theorem for harmonic functions

Boscain, Ugo

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannianstructures with constant growth vector, using the Popp’s volume form introduced byMontgomery. This definition generalizes the one of the Laplace-Beltrami operator inRiemannian geometry. In the case of left-invariant problems on unimodular Lie groupswe prove that it coincides with the usual sum of squares.

We then extend a method (first used by Hulanicki on the Heisenberg group) to com-pute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie groupof type I. The main tool is the noncommutative Fourier transform. We then study somerelevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), andthe group SE(2) of rototranslations of the plane.

Our study is motivated by some recent results about the cut and conjugate loci onthese sub-Riemannian manifolds. The perspective is to understand how singularities ofthe sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heatequation.

Bracci, Filippo

Loewner’s theory in the abstract

In this talk I am going to describe an abstract approach to Loewner’s theory whichwork also on complex hyperbolic manifolds. In particular I will show how to relate theevolution families, Herglotz vector fields and Loewner’s chains in a abstract way andwhich consequences this implies.

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Bracci, Filippo

Parabolic Attitude

In this talk I will explain how apparently non-parabolic germs of holomorphic diffeo-morphisms might have parabolic-like dynamics and will describe some invariants whichdetermine the existence of parabolic basins of attraction in presence of resonances.

Candes, Emmanuel

Recovery of data matrices from incomplete and corrupted entries: Theory and algorithms

A problem of considerable interest concerns the recovery of a data matrix from asampling of its entries. In partially filled out surveys, for instance, we would like to inferthe many missing entries. In the area of recommender systems, users submit ratings ona subset of entries in a database, and the vendor provides recommendations based onthe user’s preferences. Because users only rate a few items, we would like to infer theirpreference for unrated items (this is the famous Netflix problem). Formally, suppose thatwe observe a few entries selected uniformly at random from a matrix. Can we completethe 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 bysolving a simple convex optimization program, namely, a convenient semidefinite program.The second is that exact recovery via convex programming is further possible even insituations where a positive fraction of the observed entries are corrupted in an almostarbitrary fashion. In passing, this suggests the possibility of a principled approach toprincipal component analysis that is robust vis a vis outliers and corrupted data. Wediscuss algorithms for solving these optimization problems emphasizing the suitabilityof our methods for large scale problems. Finally, we present applications in the areaof video surveillance, where our methodology allows for the detection of objects in acluttered background, and in the area of face recognition, where it offers a principled wayof removing shadows and specularities in images of faces.

Casazza, Peter

The Kadison-Singer problem in harmonic analysis

We will see that the 1959 Kadison-Singer Problem in C∗-Algebras generates severalfundamental open problems in Harmonic Analysis including the Feichtinger Conjectureand Paving Conjecture for Laurant Operators. We will look at these open problems andtheir relationship to the Kadison-Singer Problem and recent advances on these problems.

Chemin, Jean-Yves

Large global regular solutions of 3-D incompressible Navier-Stokes

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The purpose of this talk is to construct a class of large initial data which generatesglobal smooth solutions of the incompressible Navier-Stokes system in three space dimen-sion without boundary. We shall first recall what means large initial data, using the Kochand Tataru theorem. The initial data we consider are slowly varying vector fields. Afterrescaling in one direction, we studied a problem on profiles (i.e. rescaled vector fields)which seems ill-posed. We prove a global Cauchy-Kovalevska theorem on this systemwhich requires smallness in a norm which measure the radius of analyticity of the initialdata for this problem. Going back to the original Navier-Stokes problem, the initial dataturns out to be very large.

Contreras, Manuel D.

Holomorphic self-maps of the disk intertwining two linear fractional maps

In this talk we characterize the holomorphic self-maps of the unit disk that intertwinetwo given linear fractional self-maps of the disk. The proofs are based on iteration and adetailed analysis of the solutions of Schroeder’s and Abel’s equations. In particular, wecharacterize the maps that commute with a given linear fractional map. This is a jointwork with S. Dıaz-Madrigal, D. Vukotic, and M.J. Martın.

Crowdy, Darren

A new calculus for two dimensional ideal fluid dynamics

In classical fluid dynamics, an important problem arising in a variety of applicationsis to understand how vorticity interacts with solid objects (e.g. aerofoils, obstacles orstirrers). For planar flows, a variety of powerful mathematical results exist (complexvariable methods, conformal mapping, Kirchhoff-Routh theory) that have been used tostudy such problems but the constructions are usually restricted to problems with justone, or perhaps two, objects. Expressed another way, most studies deal only with fluidregions that are simply or doubly connected. There has been a general and longstandingperception that problems involving fluid regions of higher connectivity are too challengingto be tackled analytically.

The talk will show that there is a way to formulate the theory so that the relevant fluiddynamical 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 ofmultiple objects/aerofoils/obstacles/stirrers in ideal flow and their interaction with freevortices. The approach rests on use of a special transcendental function associated withmultiply connected planar domains called the Schottky-Klein prime function.

While the presentation will be in the context of ideal fluid dynamics, the same methodsprovide a constructive calculus for solving general potential theory problems in planarmultiply connected domains.

Dahlke, Stephan

The continuous shearlet transform in arbitrary space dimensions

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In this talk, we shall be concerned with the generalization of the continuous shearlettransform to higher dimensions. Similar to the two-dimensional case, our approach isbased on translations, anisotropic dilations and specific shear matrices. It turns out thatthe associated integral transform again originates from a square-integrable representationof a specific group, the full n-variate shearlet group. Moreover, one can verify that byapplying the coorbit theory, canonical scales of smoothness spaces and associated Banachframes can be derived. We also indicate how our transform can be used to characterizesingularities in signals.

This is joint work with G. Steidl (Mannheim) and G. Teschke (Neubrandenburg).

Demaret, Laurent

Anisotropic Delaunay triangulations and denoising

Analyzing the nature of boundaries between objects in a scene is one of the main tasksin image processing. Isotropic approximation methods -like wavelets- do not provide op-timal approximation rates when applied to functions with singularities along curves. Thesolution to this problem is a challenging mathematical issue. Anisotropic triangulationshave proved to be a fruitful ansatz to solve this problem: they provide sparse and flexibleimage representations, as well as a good reconstruction quality.

Adaptive thinning is a greedy refinement technique which proceeds to the removalof the points leading to minimal reconstruction error: the output is a content-adaptedDelaunay triangulation, with elongated triangles in areas where gradients are highly di-rected, in particular along singularities. These methods have already been successfullyused in the context of terrain modelling, image compression and more recently of com-pression of video sequences.

In this talk we present a new application field of these methods: image denoising. Thekey observation is the following: the approximation produced by Delaunay based adaptivethinning acts as a highly nonlinear filtering of images, but are simultaneously edge- andshape-preserving. After a short general introduction to the topic of anisotropic Delaunaytriangulations for image approximation, the focus is on the practical and theoreticalquestions related to the problem of denoising based on these schemes.

Dıaz-Madrigal, Santiago

A non-autonomous version of the Denjoy-Wolff theorem

The aim of this work is to establish the celebrated Denjoy-Wolff Theorem in thecontext of generalized Loewner chains. In contrast with the classical situation whereessentially convergence to a certain point in the closed unit disk is the unique possibil-ity, several new dynamical phenomena appear in this framework. Indeed, omega-limitsformed by suitable closed arcs of circumferences appear now as natural possibilities ofasymptotic dynamical behavior.

Dıaz-Madrigal, Santiago

Geometry behind chordal Loewner chains

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Loewner Theory is a deep technique in Complex Analysis affording a basis for manyfurther important developments such as the proof of the famous Bieberbach’s conjectureand the well-celebrated Schramm’s Stochastic Loewner Evolution. Two cases have beendeveloped in the classical theory, namely the radial and the chordal Loewner evolutions,referring to the associated families of holomorphic self-mappings being normalized at aninternal or boundary point of the reference domain, respectively. Recently there hasbeen 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 weaddress the question what kind of systems of simply connected domains can be describedby means of Loewner chains of chordal type.

Dorfler, Monika

Time-frequency localization in phase-space

Inspired by the classical work of Slepian/Pollak/Landau we aim at giving similarresults for eigenfunctions and eigenvalues of localization operators defined directly inphase space. We will give basic definitions, some existing results on localization operatorsand some first results concerning localization in phase space. Possible consequences andconnections to other problems in harmonic analysis will be discussed.

Dudko, Dzmitry

The decoration theorem and symmetries of the Mandelbrot set

Ehler, Martin

The probabilistic frame potential and random tight fusion frames

A unit norm frame for Rd is a finite sampling of the unit sphere that spans Rd. Wegeneralize this concept to probability distributions on the sphere. In fact, we introduceprobabilistic p-frames and a probabilistic p-th frame potential. Its minimizers then arecharacterized by means of probabilistic p-frames that are “tight”. A fusion frame canbe considered as a finite sampling of the Grassmannian manifold that “spans” Rd. Wegeneralize this concept to probability distributions. Finally, we prove that the randomsampling of the Grassmannian manifold converges (with growing sample size) towards atight fusion frame.

Elin, Mark

Boundary behavior and rigidity of semigroups of holomorphic mappings

The asymptotic behavior of semigroups of holomorphic mappings and their boundarybehavior have attracted considerable attention of many mathematicians during a longperiod, but especially in the last decade.

In this talk we discuss some recent quantitative characteristics of the boundary asymp-totic behavior of such semigroups acting on the open unit disk of the complex plane.

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In particular, we present new results on the limit curvature of semigroup trajectoriesat the boundary Denjoy-Wolff point. These results enable us to establish an asymptoticrigidity property for semigroups of parabolic type. This is joint work with D. Shoikhet.

Eremenko, Alexandre

Singular perturbation of polynomial potentials with application to PT-symmetric families

In the first part, eigenvalue problems of the form −w′′ + Pw = Ew with complexpotential P and zero boundary conditions at infinity on two rays in the complex planeare discussed. Sufficient conditions for continuity of the spectrum when the leadingcoefficient of P tends to 0 are given. In the second part, these results are applied to thestudy of topology and geometry of the real spectral loci of PT-symmetric families withP of degree 3 and 4, and prove several related results on the location of zeros of theireigenfunctions.

Fadili, Jalal

3D geometrical sparse representations with application to inverse problems and videoprocessing

This work describes several 3D multiscale transforms among which two new 3Dcurvelet transforms.

The latter are built as 3D extensions of the first generation 2D curvelets and thenextend their sparsifying properties depending on the geometrical content of the data. Thefirst one, called BeamCurvelet transform, is well designed for representing 1D filamentarystructures in a 3D space, while the second, the RidCurvelet transform, is designed toefficiently capture 2D surfaces. Thus, the first one will sparsify smooth content away fromsingularities along lines, and the second will provide near optimal sparse representationsof piecewise volumes away from smooth 2D boundaries. We finally demonstrate thepotential applicability of these transforms to a variety of tasks such as object detection,denoising, inpainting and video processing.

Golberg, Anatoly

Weak conformality at a point

The condition of conformality in a domain is satisfied by a rich but rather narrow classof mappings in the plane. In higher dimensions the class is in a sense even narrower as thewell-known theorem of Liouville states that the only conformal mappings, even in R3, areof Mobius transformations. On the other hand a class that naturally extends conformalityin the plane and higher dimensions - the class of quasiconformal mappings guaranteesreal differentiability almost everywhere. A notion closely related to real differentiabilityis “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,

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to be Holder continuous, to preserve angles, etc. Such properties have applications inMechanics and Physics.

A joint work with Melkana Brakalova, University of Fordham, US.

Gomez-Cubillo, Fernando

Spectral models for orthonormal wavelets and multiresolution analysis

Spectral representations of the dilation and translation operators on L2(R) are builtthrough appropriate bases. Orthonormal wavelets and multiresolution analysis are thendescribed in terms of rigid operator-valued functions defined on the functional spectralspaces. The approach is useful for computational purposes.

Grochenig, Karlheinz

Gabor frames and complex analysis

Gabor analysis offers many problems at the intersection of harmonic analysis andcomplex analysis. One of the goals of Gabor analysis is to characterize those lattices inRd that generate a Gabor frame with a given basis function. At this time the answeris known only for the Gaussian (Lyubarskii, 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 withGaussians and Hermite functions.

(a) For the characterization of Gaussian Gabor frames in dimension 1 one needs pre-cise 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 ofinterpolation problem for entire functions that is not at all understood.

(c) In higher dimensions the study of Gabor frames with the Gaussian leads to aninterpolation problem for entire functions of several complex variables. We explainsome preliminary results and some challenges to complex analysts.

The talk will cover joint work with Ya. Lyubarskii and A. Borichev.

Grujic, Zoran

A regularity criterion for the 3D NSE vorticity in a local version of BMO

A spatio-temporal localization of a BKM-type Kozono-Taniuchi regularity criteria forthe solutions to the 3D Navier-Stokes equations; namely, the time-integrability of theBMO-norm of the vorticity, will be presented. The localization is based on an explicitlocalization of the vortex-stretching term and a version of local nonhomogeneous Div-CurlLemma. This is a joint work with Rafaela Guberovic.

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Guberovic, Rafaela

Localization of analytic regularity criteria on the vorticity and balance between thevorticity magnitude and coherence of the vorticity direction in the 3D NSE

In ’95 daVaiga has shown that ‖Du‖2q

2q−3q ∈ L1(0, T ) is a regularity class for the

Navier-Stokes Equations for any 3 ≤ q < ∞ . Previously, Beale-Kato-Majda provedthe regularity when the time-integrability of the L∞-norm of the vorticity holds. In thistalk we will show the localized versions of the aforementioned conditions imply localenstrophy remains bounded. The geometric conditions on the vorticity direction field areimportant in studying the regularity and the localized version has been recently obtainedby Grujic. The special scaling invariant regularity class of weighted LpLq type for thevorticity magnitude with the coherence factor as a weight will be demonstrated in alocalized setting.

Gumenyuk, Pavel

Loewner evolution in doubly connected domains

Loewner Theory constitutes an essential part of Geometric Function Theory and isknown to have applications both in Complex Analysis and Mathematical Physics. Fromthe modern point of view Loewner Theory can be described by the relations and interplayof three concepts: Loewner chains, Herglotz vector fields and evolution families.

In the talk we will extend this general approach for doubly connected case. In contrastto the simply connected case, evolution families do not in general consist of self-mappingsof any fixed reference domain and one has to consider families of holomorphic mappingsbetween an expanding system of annuli. Accordingly, Herglotz vector fields in doublyconnected case cannot be described as mappings from [0; +1) to the set Gen(D) of allinfinitesimal generators of one-parametric semigroups.

In the talk we will give definitions of evolution families and Herglotz vector fieldsin doubly connected setting and establish one-to-one correspondence between these twoclasses of objects.

Katsnelson, Victor

Spectral theory of the Fourier operator truncated on the positive half-axis

The spectral theory of the Fourier operator truncated on the positive half-axis isdeveloped.

Kohlmann, Martin

The Cauchy problem for the weakly dissipative µ-DP equation

We discuss the Cauchy problem for the periodic µ-DP equation with weak dissipationand prove local well-posedness in the Sobolev spaces Hs for s > 3/2. For s = 2 weestablish the precise blow-up setting and consider solutions with finite existence time andglobal solutions.

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Kohr, Gabriela

Parametric representation and Loewner chains in several complex variables

In this talk we shall present recent results related to the theory of Loewner chainsin several complex variables. Applications to extreme and support points for mappingswhich have parametric representation on the unit ball will be also given.

Kohr, Mirela

Boundary value problems for Brinkman operators on Lipschitz domains. Applications

In this talk we present recent results in the area interfacing potential theory andboundary value problems for Brinkman operators on Lipschitz domains in Rn. Extensionto boundary value problems for Brinkman operators on Lipschitz domains in Riemannianmanifolds are also considered.

Kozicki, Jurij

Bogoliubov functionals and the dynamics of particles in continuum

Markov evolution of infinite systems of particles in Rd is described by means of gen-erating functionals. It is shown that such functionals can be continued to holomorphicfunctions on Banach spaces. As a result, the evolution is described in spaces of such func-tions. A number of theorems are proven which establish the properties of such dynamicalsystems.

Kunis, Stefan

Accuracy and stability of the butterfly fast Fourier transform

A straightforward discretization of high dimensional problems often leads to a seriousgrowth in the number of degrees of freedom and thus even efficient algorithms like the fastFourier transform have high computational costs. Utilizing sparsity allows for a severedecrease of the problem size but asks for the customization of efficient algorithms to thesethinner discretization. We discuss recent generalizations of the FFT based on a multilevelapproximation scheme and present bounds on the approximation error as well as on itsnumerical stability.

Labate, Demetrio

Characterization of singularities in multidimensions using the continuous shearlettransform

Directional multiscale systems such as the Shearlet Transform were recently intro-duced to overcome the limitations of the traditional wavelet transform in dealing withmultidimensional data. In particular, while the continuous wavelet transform is able toidentify the location of singularities of functions and distributions through its asymptotic

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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 character-ize 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 andfeature extraction.

Lemvig, Jakob

Optimally sparse approximations of 3D data using shearlets

Many important problem classes are governed by anisotropic features such as singu-larities concentrated on lower dimensional embedded manifolds. To analyze the abilityof representation systems to reliably capture and sparsely represent anisotropic struc-tures, Donoho introduced the model situation of so-called cartoon-like images, i.e., two-dimensional functions which are C2 smooth apart from a C2 discontinuity curve. In thepast years, it was shown that curvelets, contourlets, and shearlets all have the ability toessentially optimal sparsely approximate cartoon-like images measured by the L2-errorof the (best) n-term approximation. Traditionally, this type of results has only beenavailable for band-limited generators, but recently Kutyniok and Lim showed that opti-mal sparsity also holds for spatial compactly supported shearlet generators under weakmoment conditions.

In this talk, we introduce three-dimensional cartoon-like images, i.e., three-dimen-sional functions which are C2 except for discontinuities along C2 surfaces, and considersparse approximations of such. We first derive the optimal rate of approximation whichis achievable by exploiting information theoretic arguments. Then we introduce three-dimensional pyramid-adapted shearlet systems with compactly supported generators andprove that such shearlet systems indeed deliver essentially optimal sparse approximationsof three-dimensional cartoon-like images. Finally, we even extend this result to the sit-uation of surfaces which are C2 except for zero- and one-dimensional singularities, andagain derive essential optimal sparsity of the constructed shearlet frames.

This is joint with G. Kutyniok and W.-Q Lim (University of Osnabruck).

Lenells, Jonatan

A disk rotating around a black hole

Two of the most well-studied solutions of the axisymmetric Einstein equations arethe Kerr black hole and the Neugebauer-Meinel disk. In this talk I will present exactsolutions of a class of boundary value problems for the Einstein equations which combinethe 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 hyperellipticRiemann surfaces of genus four.

Liflyand, Elijah

Representation of a function via the absolutely convergent Fourier integral

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New sufficient conditions for representation of a function via the absolutely convergentFourier integral are obtained. In one dimension, a new feature is that this is controlledby the behavior near infinity of both the function and its derivative. Most of such andrelated results are extended to any dimension d ≥ 2. The sharpness of the obtainedtheorems is checked on the known multipliers (joint work with R.M. Trigub).

Lim, Wang-Q

Construction and applications of compactly supported shearlets

It is now widely acknowledged that analyzing the intrinsic geometrical features ofthe underlying object is essential in many applications. In order to achieve this, severaldirectional representation schemes have been proposed in the past. Of those, shearlettight frames have been extensively studied during the last years due to their optimal ap-proximation properties of 2D data governed by curvilinear singularities and their unifiedtreatment of the continuum and digital setting. However, these studies only concernedshearlet tight frames generated by a band-limited shearlet, whereas for practical purposescompact support in spatial domain is crucial.

In this talk, we will discuss a novel approach to construct shearlets which not only havecompact support in spatial domain but can also provide optimally sparse approximationof curvilinear singularities. We will provide sufficient conditions for compactly supportedshearlet 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.

Lundberg, Erik

Laplacian growth: toward exact solutions in dimensions higher than two

The Laplacian Growth (or Hele-Shaw) Problem is a nonlinear moving boundary prob-lem 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 con-trast, ellipsoids are the only exact solutions known in higher dimensions. It is temptingto blame this on the lack of conformal maps in space. However, exact solutions can beunderstood without the time-dependent conformal map by studying dynamics of singu-larities of the Schwarz function. We will lift this point of view to higher dimensions usingthe “Schwarz potential” introduced by D. Khavinson and H.S. Shapiro. We give a newexact solution in R4, and we discuss possible future directions to take in this program.

Lyubarskii, Yurii

Radial behavior of functions in the Korenblum class

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Makarov, Nikolai

Radial explorer

I will define two random curve models which could be thought of as radial versionsof 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 underlyingfield theories are different (Ramond and Neveu-Schwatz boundary conditions in physicalterminology). The talk is based on the joint work with Nam-Gyu Kang and DapengZhan.

Markina, Irina and Alexander Vasiliev

Virasoro algebra and distribution in the space of univalent functions

Matioc, Anca-Voichita

On particle trajectories in linear water waves

We determine the phase portrait of a Hamiltonian system of equations describing themotion of the particles in linear water waves.

The particles experience in each period a forward drift which is minimal on the flatbed.

Matioc, Bogdan-Vasile

On a three phase Muskat-like problem

In this talk we present a model for the evolution of two fluid phases in a porousmedium.

The fluids are separated from each other and the wetting phases from air by interfaceswhich evolve in time. Within parabolic theory we establish existence and uniqueness ofclassical solutions for the problem and study the stability properties of the finger-shapedequilibria.

Mihaljevic-Brandt, Helena

Linearisers of entire maps

Mikulich, Yauhen

Classification of postcritically finite Newton maps

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Neretin, Yury

Multi-operator colligations and multi-variate characteristic functions

In the spectral theory of non-self-adjoint operators there is a well-known operationof product of operator colligations. Many similar operations appear in the theory ofinfinite-dimensional groups as multiplications of double cosets. We construct character-istic functions for such double cosets and get semigroups of matrix-valued functions inmatrix balls.

Okoudjou, Kasso

Beurling-Helson theorem for modulation spaces

The main problem I would like to address in this talk is the following. Given a functionf with certain properties, what are the operations we can perform on the function, thatwill result in new functions possessing the same properties as f? In particular, I willfocus on what change of variables φ will guarantee that f and f(φ) have exactly the sameproperties.

I will first discuss this problem in the context of Fourier transforms and show how atheorem of Beurling and Helson in this setting generalizes to functions defined by time-frequency methods.

Peyre, Gabriel

Sparse geometric processing of images

In this talk, I will review recent works on the sparse representations of natural images.I will in particular focus on the application of these emerging models for the resolutionof various imaging problems, which include compression, denoising and super-resolutionof images, as well as compressive sensing and compressive wave computations. Natu-ral images exhibit a wide range of geometric regularities, such as curvilinear edges andoscillating textures.

Adaptive image representations select bases from a dictionary of orthogonal or re-dundant frames that are parametrized by the geometry of the image. If the geometry iswell 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 usingsparsity constraints in these adapted representations. I will also discuss the application ofthese ideas for the acceleration of computing problems, such as the reverse time migrationin seismic imaging.

Pfander, Gotz

A few remarks on Gaussian Gabor frames

In his original work, Dennis Gabor considered representing functions as linear combi-nations of time-frequency shifted copies of Gaussians. The The Gaussian is consequently

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the best studied Gabor window when it comes to analyzing functions in one variable.Gaussians in multiple variables have not been fully explored yet, we will present a fewsimple but surprising observations.

Plonka, Gerlind

Curvelet-wavelet regularized split Bregman iteration for compressed sensing

Compressed sensing is a new concept in signal processing. Assuming that a signalcan be represented or approximated by only a few suitably chosen terms in a frameexpansion, compressed sensing allows to recover this signal from much fewer samplesthan the Shannon-Nyquist theory requires. Many images can be sparsely approximatedby 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 sparsityconstraints in two different frames as e.g. wavelets and curvelets. We present an efficientiteration 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 thatit can be understood as a special cases of the Douglas-Rachford Split algorithm. Nu-merical experiments for compressed sensing based Fourier-domain random imaging showgood performances of the proposed curvelet-wavelet regularized split Bregman (CWSpB)method, where we particularly use a combination of wavelet and curvelet coefficients assparsity constraints.

Prokert, Georg

Travelling bubbles in Hele-Shaw flows with kinetic undercooling

We discuss a two-dimensional free boundary problem for the Laplacian with Robinboundary conditions in an exterior domain. This problem describes so-called Hele-Shawflows with kinetic undercooling and is also discussed in the context of models for gasionization processes. The formulation of the problem as evolution equation leads to anonlinear, degenerate transport equation on the circle line with a nonlocal lower orderterm. We identify the solutions that correspond to uniformly translating bubbles neartrivial (circular) ones. It is shown that locally there is precisely one such nontrivialbubble for any translation velocity. In particular, the trivial solutions are unstable in aco-moving frame of reference.

The degenerate character of the problem is reflected in a loss of regularity for thesolutions of the linearized problem. Moreover, there is an upper bound on the regularityof the nontrivial solutions which depends smoothly on the regularization parameter. Theproof of the results uses quasilinearization by differentiation, index theorems for degen-erate ordinary differential operators on the circle line, and perturbation arguments forunbounded Fredholm operators. (Joint work with M. Guenther, Leipzig)

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Prokhorov, Dmitry

Integrability of the Loewner equation

We consider a special integrable case for the chordal version of the Loewner differentialequation and describe its singular trajectories.

Protasov, Vladimir

Wavelets and random power series: what is in common?

The problem of distributions of power random series has been known since the fa-mous works of P. Erdos of 1939–40. It has found many applications in the study offunctional equations, PDE, random walks, Bernoulli convolutions, etc. Suppose we havea power series, whose coefficients are independent identically distributed random vari-ables. 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 appliedin the study of compactly-supported wavelets. This phenomenon explains many commonproperties of wavelets and distributions of random series. We will also discuss applicationsof joint spectral characteristics of linear operators (joint spectral radius, the p-radius, theLyapunov exponent, etc.) to this problem.

Rashkov, Peter

The identification problem for time-frequency localizing operators

The talk will discuss the study of general classes of Hilbert-Schmidt operators withtime-frequency localizing properties. Identification of incompletely known linear opera-tors based on the observation of restricted input and output signals is quite importantin communications engineering. We develop a general setup for identification based on aGabor series discretization of the operator. A study by Kozek and Pfander has suggestedthat the identification problem might be dual to the statement of density theorem forGabor frames for a particular class of operators. In the general case we provide numerousexamples of identification of classes of time-frequency localizing operators and show thatidentification is dependent on a wider set of criteria than a single density constraint. Ourresults are based on decay estimates for time-frequency localizing operators, propertiesof Gaussian Gabor frames, and localization properties of Gabor molecules.

Rauhut, Holger

Compressive sensing and harmonic analysis

The 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 undersam-pled measurements via efficient algorithms, such as convex relaxation (l1-minimization).

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The full power of the theory can be exploited when using random matrices as measure-ment maps. It is by now well-understood, that optimal recovery rates can be achievedwith Gaussian random measurement matrices. However, Gaussian matrices are of lim-ited use in practice. Often more structure of the measurement matrix is needed. Thisrequirement 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 associ-ated to bounded orthonormal systems. They arise e.g. in the context of recoveringsparse trigonometric polynomials from random samples. The underlying mathe-matics are closely connected to deep problems in harmonic analysis, such as the Λ1

problem studied by Bourgain and Talagrand. As a special case, we study recoveryof sparse expansions in Legendre polynomials.

2. Partial random circulant matrices. These matrices arise from the problem of recov-ery from undersampled convolutions.

3. Random Gabor frames. They are connected to time-frequency analysis.

Roth, Oliver

The Loewner and Hadamard variations

We give an explicit formula relating the infinitesimal generators of the Loewner dif-ferential equation and the Hadamard variation. This is applied to establish an extensionof the Hadamard variation to the case of arbitrary simply-connected domains and toprove the existence of Loewner chains with arbitrary smooth initial generator startingat an arbitrary univalent function which is sufficiently smooth up to the boundary. Asanother application of this method, we show that every subordination chain ft is differen-tiable almost everywhere and satisfies a Loewner equation, without assuming that f ′t(0)is continuous.

Saal, Jurgen

On well-posedness and blow-up of a hyperbolic fluid model

In various applications a delay of the propagation speed of certain quantities (tem-perature, fluid velocity, . . . ) has been observed. Such phenomena cannot be describedby standard parabolic models, whose derivation relies on a Fourier type law (Paradoxonof infinite propagation speed).

One way to give account to these observations and which was successfully appliedto several models, is to replace Fourier’s law by the law of Cattaneo. In the case of afluid, this leads to a hyperbolicly perturbed quasilinear Navier-Stokes system for whichwell-posedness and blow up investigations will be presented. This is a joint project withReinhard Racke at the University of Konstanz.

Schleicher, Dierk

Newton’s method as an efficient root finder for complex polynomials

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Shelley, Michael

Novel phenomena and models of active fluids

Fluids with suspended microstructure – complex fluids – are common actors in micro-and biofluidics applications and can have fascinating dynamical behaviors. A new areaof complex fluid dynamics concerns “active fluids” which are internally driven by havingdynamic microstructure such as swimming bacteria. Such motile suspensions are im-portant to biology, and are candidate systems for tasks such as microfluidic mixing andpumping. To understand these systems, we have developed both first-principles particleand continuum kinetic models for studying the collective dynamics of hydrodynamicallyinteracting microswimmers. The kinetic model couples together the dynamics of a Stoke-sian fluid with that of an evolving “active” stress field. It has a very interesting analyticaland dynamical structure, and predicts critical conditions for the emergence of hydrody-namic instabilities and fluid mixing. These predictions are verified in our detailed particlesimulations, and are consistent with current experimental observation.

Titi, Edriss

Global regularity for the three-dimensional primitive equations of atmospheric andoceanic dynamics

In this talk I will show the global existence and uniqueness of strong solutions to thethree-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.

Vasylkevych, Sergiy

A family of models for rotating shallow water flow

We consider recently derived family of models for the rotating shallow water flowin 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 previouslyknown models (so called L1 and LSG models proposed by R. Salmon) and a completelynew model with superior regularity properties.

We show that each member of the family is given by a Hamiltonian system on anappropriately chosen diffeomorphism group. Moreover, the Hamiltonian framework turnsout to be superior to the original derivation of the equations via asymptotic expansion ofthe rotating shallow water Lagrangian since it does not place topological restrictions onCoriolis parameter.

Furthermore, we prove that all but one model in the family are well-posed in theclassical sense for short time as well as global existence of solutions for the distinguishedmodel.

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Vidras, Alekos

Towards the definition of the Cauchy-Fantappie and Radon transforms

We define substitutes for the (n− 1)-Cauchy-Fantappie and Radon transforms in ann-complete smooth simplicial toric variety X using a definition of (n−1)-concavity relatedto a multi-perspective embedding. Such a construction fits with a notion of bi-concavityfor a direct product of two projective spaces.

Wahlen, Erik

Existence and stability of solitary water waves with weak surface tension

Two-dimensional solitary water waves with weak surface tension (0 < β < 1/3, whereβ is the Bond number) are constructed by minimizing the energy subject to the constraintof fixed momentum. The stability of the set of minimizers follows by a standard principlesince the energy and momentum are conserved quantities. “Stability” must however beunderstood in a qualified sense because of the lack of a global well-posedness theory for theinitial value problem. The variational method relies on the concentration-compactnessprinciple and a penalization argument. The solitary waves are to leading order periodicwave trains modulated by exponentially decaying envelopes described by the focusingnonlinear Schrodinger equation

Walker, Christoph

Stability of steady states in thin film equations with soluble surfactant

The talk is dedicated to thin film equations with soluble surfactant and gravity. Thegoverning system of degenerate parabolic equations for the film height, bulk and surfacesurfactant concentration is shown to be locally well-posed. It is also shown that thesteady states are asymptotically stable. (This is joint work with J. Escher, M. Hillairet,and Ph. Laurencot.)

Wirosoetisno, Djoko

Asymptotics in the primitive equations

We describe, from physical and mathematical points of view, asymptotic regimes of theprimitive equations which are of interest in atmospheric/oceanic dynamics. Some resultson the long-time behaviour of the solution in the strongly rotating and/or stratified caseswill be discussed. (Joint work with R. Temam.)

Zelenko, Igor

Comparison theorems for number of conjugate points along sub-Riemannian extremals

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The classical Rauch comparison theorem in Riemannian geometry provides the esti-mation of the number of conjugate points along Riemannian geodesics in terms of boundsfor the sectional curvature. We give a generalization of this theorem to extremals of awide class of optimal control problems including sub-Riemannian extremals. The problemcan be reformulated as the problem to estimate the number of conjugate points along acurve in a Lagrangian Grassmannian in terms of the invariants of this curve with respectto the natural action of the Linear Symplectic Group. Our treatment of this problem isbased on the construction of the canonical bundle of moving frames and the completesystem of symplectic invariants for curves in Lagrangian Grassmannians previously donein the joint works with Chengbo Li. We will explain how appropriately arranged boundsfor these symplectic invariants effect the bounds for the number of conjugate points. Theapplication for extremals of natural sub-Riemannian metrics on principal connections ofprincipal bundles with one-dimensional fibers over Riemannian manifolds (i.e. magneticfields on Riemannian manifolds) will be given.

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