Degenerate Structures in Complex Analysis From the Past to the Future

Transcription

1 An International Conference in Complex Analysis organized by the Complex Analysis group at the University of Wuppertal Degenerate Structures in Complex Analysis From the Past to the Future (in honor of Professor Klas Diederich) in Bergisch Gladbach, Germany from May 22 to May 26, 2006 Scientific Committee: B. Berndtsson (Gothenburg) J.E. Fornæss (Ann Arbor) J. Leiterer (Berlin) N.V. Shcherbina (Wuppertal) sponsored by Deutsche Forschungsgemeinschaft and the University of Wuppertal

2 Schedule May 23 May 24 May 25 May J. E. Fornæss J. Duval F. Forstnerič T. C. Dinh 9.50 (Ann Arbor) (Toulouse) (Ljubljana) (Paris) S. Pinchuk S. Ivashkovich J. D. McNeal D. Popovici (Bloomington) (Lille) (Ohio) (Warwick) Coffee Coffee Coffee Coffee D. H. Phong B. Jöricke J. Brinkschulte E. Chirka (Columbia) (Uppsala) (Leipzig) (Moscow) Lunch Lunch Lunch Lunch E. J. Straube J. Merker A. Sukhov (Texas) (Marseille) (Lille) S. Nemirovski E. Mazzilli G. Tomassini (Moscow) (Lille) (Pisa) Coffee Coffee Coffee M. C. Shaw Ch. Laurent-Thiebaut S. Fu (Notre Dame) (Grenoble) (Rutgers) Parallel Sessions Dinner Dinner Conference Dinner

3 Parallel Sessions Room E37 Room E45 May X. Zhou M. Kosek (Beijing) (Cracow) A. Brudnyi J. Wiegerinck (Calgary) (Amsterdam) May H. Samuelsson F. Haslinger (Wuppertal) (Wien) T. Akahori P. Liczberski (Hyogo) (Lodz) May I. Majcen J. Michel (Ljubljana) (Calais) J. Ruppenthal (Bonn)

4 Abstracts Takao Akahori The Rumin complex and Hamiltonian mechanism Let (V, o) be an isolated singularity in a comlpex euclidean space. Then, from this complex euclidean space, we have the induced Kaehler metric on V o and we can discuss Hamiltonian mechanics with respect to this Kaehler metric. While let M be the intersection of this V and the real hypersphere, centered at o. Then over M, we have an induced CR structure, which is strongly pseudo convex. This CR structure is of importance. It determines the isolated singularity V, uniquely. With this, the deformation theory of CR structures has been developed ([A],[AGL1]). In this work, we study Hamitonian mechanics for isolated singularity (V, o), and we see how it relates with CR structures through the Rumin complex. Rumin complex (CR-version) plays an important role in the CR geometry (see [AGL1]). In this work, we consider Hamilton mechanics with respect to the induced Kaehler metric. By using this, we introduce a new differential complex. It is shown that; this complex is a complex manifold version for the Rumin complex. And we see that: Theorem 1. Assume that W is a Calabi-Yau manifold. Then, our new complex on W recovers H 1 (W, Θ) (the Kodaira-Spencer group). Here, if our W satisfies W admits the trivial canonical line bundle, (1) H 1 (W, O) = 0, (2) then we call W a Calabi-Yau manifold (in this definition, W may be open, and if (V, o) is a hypersurface isolated singularity with dim C V 3, then W = V o is an open Calabi-Yau manifold). Furthermore we discuss how our new complex relates to lemma on V o. 4

5 References [A] T. Akahori, The new estimate for the subbundles E j and its application to the deformation of the boundaries of strongly pseudo convex domains, 63 (1981), Inventiones mathematicae. [AGL1] T. Akahori, P. M. Garfield and J. M. Lee, Deformation theory of fivedimensional CR structures and the Rumin complex, 50 (2002), , Michigan Mathematical Journal. Judith Brinkschulte The -equation on q-convex domains The regularity of the Cauchy-Riemann equation on strictly pseudoconvex domains is by now very well understood. It is also very well known that even though various pathologies might occur on pseudoconvex domains, the -equation can be solved with regularity up to the boundary. If the Levi-form of the domain is allowed to have some negative eigenvalues, however, the Cauchy- Riemann equation is not yet well understood. The aim of this talk is to present some new results concerning the -equation with regularity and support conditions on domains with weakly q-convex boundary by means of L 2 -estimates. Alex Brudnyi Holomorphic functions of slow growth on coverings of strongly pseudoconvex manifolds In the talk I will answer some questions posed in the paper of Gromov-Henkin- Shubin on holomorphic L 2 functions on coverings of strongly pseudoconvex manifolds. I also present some connection of the obtained results with the Shafarevich conjecture on holomorphic convexity of universal coverings of complex projective manifolds. Evgeni Chirka On the holomorphicity of foliations 5

6 Let M = S τ be a foliation of a complex manifold M by complex hypersurfaces S τ such that almost all of leaves are Liouville manifolds (H = C). Then the foliation is holomorphic, i.e., for any a M there is a nbh U a and a holomorphic map f : U D C such that any component of S τ U (if non-empty) is equal to f 1 (z) for some z D. Tien-Cuong Dinh Complex dynamics in higher dimension We discuss some results on dynamics of holomorphic and meromorphic maps of several variables obtained in collaboration with N. Sibony. In particular, we consider some questions on the entropy, invariant currents, invariant measures and their properties. The main tools are apropriate spaces of test functions and estimate on solutions of the equation. Julien Duval On Brody s lemma We give a quantitative version of Brody s lemma, a basic tool in complex hyperbolicity. John Erik Fornæss Foliations in P 2 I ll discuss some recent joint work with Nessim Sibony on foliations by Riemann surfaces. Foliated sets carry harmonic currents, and we show their uniqueness under natural conditions. Franc Forstnerič Gluing holomorphic sprays and applications 6

7 A holomorphic spray is a family of holomorphic sections of a holomorphic fibration, depending holomorphically on a Euclidean parameter. I shall describe a result (joint with B. Drinovec-Drnovsek) on gluing pairs of sprays over strongly pseudoconvex Stein manifolds, with control up to the boundary. Applications include a construction of proper holomorphic maps of finite bordered Riemann sufaces to (n 1)-convex n-dimensional complex spaces, the linearization of a tubular neighborhood of a section, an up-to-the-boundary Oka-Grauert principle for sections of fiber bundles with flexible fibers over strongly pseudoconvex Stein manifolds, and approximation theorems for holomorphic maps to arbitrary complex manifolds. Siqi Fu Hearing the type of a domain in C 2 with the -Neumann Laplacian We should discuss interplays between spectrum of the -Neumann Laplacian and geometry of a domain in several complex variables. In particular, we show how one can determine the type of a domain in two complex variables via the spectrum of the -Neumann Laplacian. Friedrich Haslinger Compactness in the Neumann Problem and Schrödinger Operators with Magnetic Fields (common work with Bernard Helffer, Orsay) We discuss compactness of the canonical solution operator to on weigthed L 2 spaces on C n. For this purpose we apply ideas which were used for the Witten Laplacian in the real case and various methods of spectral theory of Schrödinger operators with magnetic fields. Sergej Ivashkovich On a topological version of the Levi-type extension theorem Let R be some compact Riemann surface with boundary R consisting from a finite number of closed simple curves {γ i } b i=1 and let {f i : S 1 X} b i=1 be a couple of smooth maps from the unit circle S 1 = {z C : z = 1} to a complex manifold X. We shall say that the couple f = {f i } b i=1 extends to R if there 7

8 exist diffeomerphisms h i : γ i S 1 and a holomorphic map ˆf : R X such that ˆf γi = f i h i for all i = 1,..., b. Let s call this a soft extension condition, or extension after a reparameterization. We shall give a Levi-type theorem for soft extensions. Namely, if a holomorphic family of boundary data is given such that it is soft-extendable for sufficiently many values of parameter, then there is a global reparameterisation that gives a meromorphic extension. Burglind Juhl-Jöricke Pluripolar hulls and fine analytic continuation We relate the description of the pluripolar hull of a piece of a complex curve in C 2 to fine analytic continuation. Fine analytic continuation of an analytic function implies that its graph is not complete pluripolar in C 2. This allows to give simple examples, e.g., of smooth nowhere analytically extendible (even univalent) functions with non-complete pluripolar graphs and of graphs with infinitely-sheeted pluripolar hull. On the other hand, for the pluripolar hull E of the graph of an analytic function f the following result holds. The projection of E to the first coordinate plane is finely open. If the pluripolar hull E is non-trivial and is itself the graph of a smooth function then f has fine analytic continuation. Results are joint with Tomas Edlund. Marta Kosek Julia type sets in C N associated with infinite arrays of polynomial mappings We generalize the notion of a filled-in Julia set associated with one polynomial mapping and obtain two types of sets: partly filled-in composite Julia sets and their polynomially convex hulls: composite (filled-in) Julia sets. We show that the first ones are strongly analytic and the other weakly analytic when regarded as multifunctions of the generating mappings. We study also some pluripotential properties of these sets. (The lecture is a report on a joint work with Maciej Klimek). 8

9 Christine Laurent-Thiebaut On the Hartogs-Bochner-Weinstock extension phenomenon in CR manifolds After recalling the geometrical situation when the Hartogs-Bochner-Weinstock extension theorem holds for CR manifolds, we shall give some regularity results for the extension in the case of 1-concave manifolds. Piotr Liczberski and Victor V. Starkov On locally biholomorphic mappings from multi-connected domains onto simply connected canonical domains E. Ligocka [Li] has found a class of multi-connected domains D of the complex plane C that can be mapped onto the open unit disc locally biholomorphically and m-valently, with m 24. She has also proved that for every finitely connected domain D C, not biholomorphic to C \ {0}, there exist m N and a locally biholomorphic m-valent mapping from D onto C. The first result was inspired by a Fornaess - Stout theorem [FS] and the second one by a Gunning - Narasimhan theorem [GN]. During the lecture we will decide two questions refering to the cited facts: 1. Is it possible to decrease essentialy the constant m in the case f(d) =, for a wide class of multi-connected domains D? 2. Is it possible to estimate uniformly the constant m in the case f(d) = C, for a given class of domains D including all domains finitely connected? We will base on the results of our paper [LS] and on some our new results. References [FS] Fornaess J.E., Stout E.L., Spreading polydisc on complex manifolds, Amer. J. Math. 99 (1977), [GN] Gunning R.C., Narasimhan R., Immersion of open Riemann surfaces, Math. Ann. 174 (1967), [LS] Liczberski P., Starkov V.V., On locally biholomorphic mappings from multi-connected onto simply connected domains, Ann. Polon. Math. 85 (2005),

10 [Li] Ligocka E., On locally biholomorphic surjective mappings, Ann. Polon. Math. 82 (2003), Irena Majcen Closed holomorphic 1-forms without zeros on Stein manifolds It is well known that every class of the de Rham cohomology group is represented by a closed holomorphic 1-form (H. Cartan and K. Stein and generalised by J.- P. Serre in 1953). We show that in each class of the de Rham group there is a closed holomorphic 1-form without zeros. Emmanuel Mazzilli Residue currents and applications In this work, we define currents with same properties of residue currents by an elementary way (without the resolution s singularities theorem)in some complete intersections. As an application, we obtain a new decomposition integral formula for functions in an holomorphic ideal. To illustrate, we consider the case of ideals generated by two holomorphic functions in C 2 with an isolated zero at the origin. Jeffery D. McNeal Weighted Bergman projections and extension of biholomorphic maps I will discuss how certain weighted Bergman projections are connected to biholomorphic mappings. The connection is very similar to the one observed by Bell and Ligocka for the ordinary Bergman projection; however the weight functions we discuss are not smooth up to the boundary, so there are some additional features. Combined with regularity results for a weighted Laplacian, this modified Bell-Ligocka program gives new extension results on biholomorphic mappings. Joël Merker Analytic reflection principle for CR mappings between highly degenerate hypersurfaces 10

11 We associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat hat. We show that every C -smooth CR diffeomorphism h : M M between two globally minimal real analytic hypersurfaces in C n (n 2) is real analytic at every point of M if M is holomorphically nondegenerate. More generally, we establish that the reflection function R h associated to such a C -smooth CR diffeomorphism between two globally minimal hypersurfaces in C n (n 1) always extends holomorphically to a neighborhood of the graph of h in M M, without any nondegeneracy condition on M. This gives a new version of the Schwarz symmetry principle to several complex variables. Finally, we show that every C -smooth CR mapping h : M M of maximal generic rank between two essentially finite real analytic hypersurfaces is real analytic at every point of M. Joachim Michel Holomorphic support functions on K-convex domains K-convex domains generalise the notion of convex domains with smooth boundary in the space of n complex variables. K-convex bounded domains of finite type share many properties with convex domains of finite type. In particular, for any point of the boundary there is a biholomorphic transformation depending on this point such that the type and the multitype coincide with their linear counterparts. In this talk we show how to construct on K-convex domains of finite type a holomorphic support function admitting optimal lower estimates in view of possible Hölder estimates for the Cauchy-Riemann equations. This construction generalises the corresponding construction of Diederich and Fornæss on convex domains. Finally, the notion of McNeal polydics can also be generalised to K-convex domains in a natural way. Stefan Nemirovski Coverings and weak pseudoconvexity 11

12 The uniformization theorem for strictly pseudoconvex Stein domains with real analytic boundary states that such a domain is covered by the unit ball if and only if its boundary is locally biholomorphic to the sphere. The purpose of this talk is to show what can happen to this result if the domain is only weakly pseudoconvex. Duong H. Phong The degenerate Monge-Ampere equation and geodesics of Kaehler potentials We consider the explicit construction of solutions of the Dirichlet problem for certain completely degenerate complex Monge-Ampere equations. In the process, we also derive suitable extensions of the Bedford-Taylor theory for weak solutions. The underlying motivation is the construction of geodesics in the space of Kaehler potentials. These geodesics are central to Donaldson s program for constant scalar curvature metrics, the existence of which should be, by a well-known conjecture of Yau, equivalent to stability in the sense of geometric invariant theory. We also provide a brief discussion of these surrounding problems. The work prsented is joint work with Jacob Sturm. Sergey Pinchuk Reflection Principle in C n I will discuss the reflection principle with respect to different problems of analytic continuation of CR and proper holomorphic mappings in C n. Dan Popovici Singular Morse Inequalities We prove singular Morse inequalities estimating the asymptotic growth of the cohomology groups of high tensor powers of singular Hermitian holomorphic line bundles twisted by the corresponding multiplier ideal sheaves over a compact complex manifold. The main step in the proof is the construction of a new regularisation of closed almost positive (1, 1)-currents with controlled Monge- Ampère masses. To this end, we prove two results describing the asymptotic growth of multiplier ideal sheaves associated with increasingly singular metrics: 12

13 almost linear growth and an effective version of their coherence property. The asymptotics of Bergman kernels associated with singular metrics will play an important part. Jean Ruppenthal Hölder regularity of the Cauchy-Riemann equations on certain singular analytic sets We establish a new class of weighted Hölder estimates for the well-known basic -homotopy formula on the ball. As an application, we construct a solution operator for the Cauchy-Riemann equations on certain strictly pseudoconvex domains in analytic sets and give Hölder estimates for this operator. Håkan Samuelsson Various approaches to the residue of a complete intersection Consider a holomorphic mapping f = (f 1,..., f m ) : X C m from a complex manifold X and assume that f defines a complete intersection, i.e., that the codimension of f 1 (0) is m. Using Hironaka s desingularization theorem, Coleff and Herrera proved that the limit of the residue integral, ϕ, ϕ D n,n m (X), T ǫ = { f 1 2 = ǫ 1,..., f m 2 = ǫ m }, f 1 f m T ǫ as ǫ = (ǫ 1,..., ǫ m ) tends to zero along certain admissible paths, exists and defines the action of a (0, m)-current, R f CH, the Coleff-Herrera residue current. In many respects, R f CH has proved to be the correct notion of the residue associated to f. The restriction to limits along admissible paths is not merely for technical reasons; Passare and Tsikh found an example where the limit does not exist unrestrictedly. Various ways to get a more rigid approach to R f CH, based on averages of the residue integral, have therefore been proposed. I will explain some of these approaches and present recent progress. Mei-Chi Shaw Bounded plurisubharmonic functions and the -Cauchy problem in the complex projective spaces 13

14 In this talk we will discuss bounded plurisubharmonic exhaustion functions on pseudoconvex domains in the complex projective spaces. Such functions are used to study the function theory via the -Cauchy problem. We also discuss the application on the nonexistence of Lipschitz Levi-flat hypersurfaces in the complex projective space of dimension greater or equal to 3 (Joint work with Jianguo Cao) Emil J.Straube A sufficient condition for global regularity of the Neumann problem A theory of global regularity of the -Neumann problem is developed which unifies the two principal approaches to date, namely the one via compactness due to Kohn-Nirenberg and Catlin and the one via plurisubharmonic defining functions and/or families of vector fields that commute approximately with due to Boas and the author. Alexandre Sukhov Filling real hypersurfaces by pseudoholomorphic discs We discuss several recent results concerning the geometry of Bishop discs attached to a real submanifold of an almost complex manifold. Giuseppe Tomassini Holomorphic C-fibrations and pseudoconvexity Convexity properties of open subsets D of C n on which there exist Stein holomorphic fibrations are discussed. In particular, if D admits a Stein fibration E D then it is pseudoconvex of order n 2 (joint paper with V. Vajaitu). Jan Wiegerinck Fine aspects of pluripotential theory 14

15 Finely holomorphic functions of one variable were introduced in the seventies by Bent Fuglede. Only recently it was observed by Edlund and Jöricke that fine holomorphic extension of classical holomorphic functions explains at least partly the phenomenon, discoverd by Edigarian and myself, that the graph of a complete holomorphic function may have non-trivial pluripolar hull. We will discuss the role of the fine topology in pluripotential theory. In particular we discuss recent work joint work with El Marzguioui stating that the pluri-fine topology is locally connected and ongoing work with Edigarian and El Marguioui to the effect that graphs of fine holomorphic function are complete pluripolar. Xiangyu Zhou Group actions in several complex variables We present some results in several complex variables related to group actions, displaying the role of group actions in several complex variables. 15

16 Addresses Takao Akahori, Department of Mathematics, University of Hyogo, Himeji , Hyoga, JAPAN, William Alexandre, Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral Côte d Opale, 50 rue F. Buisson, B.P. 699, Calais Cedex, FRANCE, william-alexandre@wanadoo.fr Éric Amar, UFR de Mathématiques et Informatique, Université de Bordeaux I, Talence, FRANCE, Eric.Amar@math.u-bordeaux1.fr Léa Blanc-Centi, Centre de Mathématiques et d Informatique (CMI), Université d Aix-Marseille I (Université de Provence), Marseille, FRANCE, Lea.Blanc-Centi@cmi.univ-mrs.fr Judith Brinkschulte, Institut für Mathematik, Universität Leipzig, D Leipzig, GERMANY, brinkschulte@math.uni-leipzig.de Alex Brudnyi, Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, CANADA, albru@math.ucalgary.ca Josep M. Burgués, Department of Mathematics, Autonomous University of Barcelona, Bellaterra (Barcelona), SPAIN, josep@mat.uab.es Philippe Charpentier, Mathématiques Pures, Université de Bordeaux I, Talence, FRANCE, Philippe.Charpentier@math.u-bordeaux1.fr Evgeni Chirka, V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow, RUSSIA, chirka@mi.ras.ru Giuseppe Della Sala, Istituto Matematico, Scuola Normale Superiore, Pisa, ITALY, g.dellasala@sns.it Klas Diederich, Fachbereich C Mathematik, Bergische Universität Wuppertal, D Wuppertal, GERMANY, Klas.Diederich@math.uni-wuppertal.de Tien-Cuong Dinh, Institut de Mathématiques, Université de Paris VI (Pierre et Marie Curie), Paris, FRANCE, dinh@math.jussieu.fr Julien Duval, Laboratoire de Mathématiques Emile Picard, Université de Toulouse III (Paul Sabatier), Toulouse, FRANCE, duval@picard.ups-tlse.fr Thomas Eckl, Mathematisches Institut, Universität zu Köln, D Cologne, GERMANY, teckl@math.uni-koeln.de Bert Fischer, Fachbereich C Mathematik, Bergische Universität Wuppertal, D Wuppertal, GERMANY, fischer@math.uni-wuppertal.de John Erik Fornæss, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, fornaess@umich.edu 16

17 Franc Forstnerič, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, 1000 Ljubljana, SLOVENIA, Klaus Fritzsche, Fachbereich C Mathematik, Bergische Universität Wuppertal, D Wuppertal, GERMANY, Klaus.Fritzsche@math.uni-wuppertal.de Siqi Fu, Department of Mathematics, Rutgers University, Camden, NJ 08102, USA, sfu@camden.rutgers.edu Friedrich Haslinger, Institut für Mathematik, Universität Wien, 1090 Vienna, AUSTRIA, friedrich.haslinger@univie.ac.at Anne-Katrin Herbig, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA, herbig@umich.edu Gregor Herbort, Fachbereich C Mathematik, Bergische Universität Wuppertal, D Wuppertal, GERMANY, Gregor.Herbort@math.uni-wuppertal.de C. Denson Hill, Department of Mathematics, SUNY Stony Brook University, Stony Brook, NY 11790, USA, Dhill@math.sunysb.edu Sergej Ivashkovich, UFR de Mathématiques Pures et Appliquées, Université de Lille I (Sciences et Techniques de Lille Flandres Artois), Villeneuve d Ascq, FRANCE, Serguei.Ivachkovitch@math.univ-lille1.fr Michal Jasiczak, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, POLAND, mjk@amu.edu.pl Burglind Jöricke, Department of Mathematics, Uppsala University, Uppsala, SWEDEN, joericke@math.uu.se Marta Kosek, Mathematics Institute, Jagiellonian University, Kraków, POLAND, Marta.Kosek@im.uj.edu.pl Christine Laurent-Thiebaut, Institut Fourier, Université de Grenoble I (Joseph Fourier), Saint-Martin-d Hères, FRANCE, Christine.Laurent@ujf-grenoble.fr Jürgen Leiterer, Institut für Mathematik, Humboldt-Universität, D Berlin, GERMANY, leiterer@mathematik.hu-berlin.de Piotr Liczberski, Institute of Mathematics, Technical University of Lódź, Lódź, POLAND, piliczb@p.lodz.pl Ingo Lieb, Universität Bonn, Mathematisches Institut, Beringstraße 1, D Bonn, GERMANY, ilieb@math.uni-bonn.de Irena Majcen, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, 1000 Ljubljana, SLOVENIA, irena.majcen@fmf.uni-lj.si George Marinescu, Johann Wolfgang Goethe-Universität, Institute für Mathematik, Robert-Mayer-Str. 6-10, Frankfurt am Main, GERMANY, marinesc@math.uni-frankfurt.de 17

18 Emmanuel Mazzilli, UFR de Mathématiques Pures et Appliquées, Université de Lille I (Sciences et Techniques de Lille Flandres Artois), Villeneuve d Ascq, FRANCE, Emmanuel.Mazzilli@math.univ-lille1.fr Jeffery D. McNeal, Department of Mathematics, Ohio State University, Columbus, OH 43210, USA, mcneal@math.ohio-state.edu Joël Merker, Laboratoire d Analyse, Topologie et Probabilités (LATP), Université d Aix-Marseille I (Université de Provence), Marseille, FRANCE, merker@latp.univ-mrs.fr Joachim Michel, Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral Côte d Opale, 50 rue F. Buisson, B.P. 699, Calais Cedex, FRANCE, Joachim.Michel@lmpa.univ-littoral.fr Stefan Nemirovski, V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, Moscow, RUSSIA, stefan@mi.ras.ru Giorgio Patrizio, Dipartimento di Matematica, Università di Firenze, Florence, ITALY, patrizio@math.unifi.it Peter Pflug, Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, D Oldenburg, GERMANY, pflug@mathematik.uni-oldenburg.de Duong H. Phong, Department of Mathematics, Columbia University, New York, NY 10027, USA, phong@cpw.math.columbia.edu Sergey Pinchuk, Department of Mathematics, Indiana University, Bloomington, IN 47401, USA, pinchuk@indiana.edu John C. Polking, Department of Mathematical Sciences, Rice University, Houston, TX 77251, USA, polking@rice.edu Anca Popa-Fischer, Institut für Mathematik, Universität Flensburg, Auf dem Campus 1, Flensburg, GERMANY, popa-fischer@uni-flensburg.de Dan Popovici, Department of Mathematics, University of Warwick, Coventry CV4 7AL, ENGLAND, popovici@maths.warwick.ac.uk R. Michael Range, Department of Mathematics and Statistics, University at Albany (SUNY), Albany, NY 12222, USA, range@albany.edu Jean Ruppenthal, Universität Bonn, Mathematisches Institut, Beringstraße 1, D Bonn, jean@math.uni-bonn.de Fatiha Sahraoui, Laboratoire de Mathématiques, Université Djilali Liabès, B.P. 89, Sidi Bel Abbès, ALGERIA, douhy fati@yahoo.fr Håkan Samuelsson, Fachbereich C Mathematik, Bergische Universität Wuppertal, D Wuppertal, GERMANY, hasam@math.chalmers.se Alberto Saracco, Istituto Matematico, Scuola Normale Superiore, Pisa, ITALY, alberto.saracco@gmail.com 18

19 Alla Sargsyan, Institut für Mathematik, Universität Leipzig, D Leipzig, GERMANY, Wolfgang Schwarz, Fachbereich C Mathematik, Bergische Universität Wuppertal, D Wuppertal, GERMANY, Wolfgang.Schwarz@math.uni-wuppertal.de Rasul Shafikov, Department of Mathematics, University of Western Ontario, London, ON N6A 3K7, CANADA, shafikov@uwo.ca Mei-Chi Shaw, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA, shaw.1@nd.edu Nikolay Shcherbina, Fachbereich C Mathematik, Bergische Universität Wuppertal, D Wuppertal, GERMANY, Nikolay.Shcherbina@math.uni-wuppertal.de Alexandru Simioniuc, Istituto Matematico, Scuola Normale Superiore, Pisa, ITALY, a.simioniuc@sns.it Tadej Starcic, Institute of Mathematics, Physics and Mechanics, University of Ljubljana, 1000 Ljubljana, SLOVENIA, tadej.starcic@fmf.uni-lj.si Victor V. Starkov, Institute of Mathematics, Technical University of Lódź, Lódź, POLAND, vstar@p.lodz.pl Emil J. Straube, Department of Mathematics, Texas A&M University, College Station, TX 77843, USA, straube@math.tamu.edu Alexandre Sukhov, UFR de Mathématiques Pures et Appliquées, Université de Lille I (Sciences et Techniques de Lille Flandres Artois), Villeneuve d Ascq, FRANCE, Alexandre.Sukhov@math.univ-lille1.fr Giuseppe Tomassini, Istituto Matematico, Scuola Normale Superiore, Pisa, ITALY, giuseppe.tomassini@sns.it Jernej Tonejc, Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA, tonejc@math.wisc.edu Jan Wiegerinck, Korteweg-de Vries Instituut, Universiteit van Amsterdam, 1018 TV Amsterdam, THE NETHERLANDS, janwieg@science.uva.nl Xiangyu Zhou, Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing, PEOPLES REPUBLIC OF CHINA, xyzhou@math.ac.cn 19