bluuebunny/crossref_metadata_2025 huggingface dataset:
Non-Classical Interacting Random Walks
Authors: Comets Francis M. , Zerner Martin
Published: December 2009

The workshop was organized by Francis Comets (Paris 7) and Martin Zerner (Tübingen). It was attended by fifty participants of about twenty different nationalities. Among the participants was a relatively large number of young researchers, some of which were supported by the European Union within the Marie Curie Conferences Programme and by the National Science Foundation.

The title of the workshop was
Non-Classical Interacting Random Walks
. The study of random walks (RWs) goes back to the beginnings of probability theory in the seventeenth century. Some of the first probabilists like Bernoulli and Pascal investigated the properties of coin tossing sequences and other simple games of chance, which nowadays are modeled by RWs. Such a RW takes steps at fixed time intervals, and at each step a random direction is chosen, which gives a chaotic trajectory. So in their most stringent definition, RWs come up as sums of independent and identically distributed random variables with real or often integer lattice values. A somewhat wider definition includes time-homogeneous Markov chains whose transition probabilities are in some way adapted to a given geometric-algebraic-combinatorial structure of the underlying state space. In this context, many tools from potential theory, graph theory, harmonic analysis and Fourier analysis are available.

The focus of this workshop was on a quite different area of current research on RWs. This is what the term “non-classical” in the title refers to. Some of the RWs, which are currently heavily investigated, are quite “irregular”: Some are not Markovian, but are influenced by their own past. Others are Markovian, but interact with a possibly random environment, which influences the transitions of the walk. Many of the tools used in the more traditional settings of RWs fail in this context.

The main models considered in this workshop were RWs in random environments (RWRE) with talks mainly on Monday, RWs in random potential and random polymers on Tuesday, excited RWs on Wednesday morning, followed by an exciting open problem session in the evening, reinforced RWs on Thursday and various other RWs on Friday. The programme consisted in total of 23 talks of about 50 minutes each and the open problem session. On most days a long lunch break from 12:30 to 16:00 gave plenty of opportunity for interaction between the participants. Following the reinforcement by previous workshops and in spite of the title of the current workshop, on Wednesday afternoon most participants took part in the classical interacting non-random walk to St. Roman.
Adaptive Numerical Methods for PDEs
Authors: Rannacher Rolf , Süli Endre , Verfürth Rüdiger
Published: December 2009

Multidimensional processes in sciences and engineering are usually formulated in terms of partial differential equations. With the incorporation of more and more complete physics these models become increasingly complex. Their accurate numerical simulation requires the use of efficient methods which exploit concepts of adaptivity in setting up the models and their discretization. Over the last two decades these approaches have been developed from simple model situations towards real-life applications, in most cases with striking success. The conference “Adaptive Numerical Methods for PDEs” had its focus on basic questions related to the rigorous mathematical understanding of these methods and to their practical use.

The use of automatic adaptivity has become common in the simulation of all kind of PDE models even in situations where a solid theoretical justification is lacking. Examples are the nonlinear Navier-Stokes equations in fluid mechanics, the nonlinear Lamé–Navier equations in large-strain elasticity, the Maxwell equations, and combinations of these models in nonstationary fluid-structure interaction and magneto-hydrodynamics. Such approaches are usually based on
a posteriori
information in terms of local quantities such as approximate gradients or residuals obtained from the computed solution. The extension of quantitative mathematical analysis of adaptive methods to such complicated models is a challenging problem.

The existing methods of a posteriori error estimation and mesh adaptation are based on several heuristic technical assumptions. These are assumptions about the convergence of higher-order difference quotients, the relative “smallness” of linearization errors, “sharpnes” of stopping criteria of algebraic iterations, the “safe” control of error accumulation in long-time calculations, etc. These aspects are fundamental to adaptive methods and still need rigorous mathematical analysis for a better understanding of the underlying mechanisms.

Recently, several new directions of adaptivity have evolved, e.g., “spectral-adaptivity” in wavelet analysis, “hp-adaptivity” in high-order FE-methods, “anisotropic” refinement in the approximation of layers, adaptivity in the solution of “optimal control” problems, “model adaptivity” for example in spatial dimension reduction or the complexity reduction in diffusion models, and “multi-scale” adaptivity in problems with micro- and macro-scales reaching even towards adaptive turbulence modeling. The latter are the really “hot” topics in current Numerical Analysis research. The mathematical analysis of these approaches has to combine concepts of a posteriori error estimation with techniques from PDE theory. The talks given at this conference were concerned with the following subjects:
Differentialgeometrie im Grossen
Authors: Biquard Olivier , Kleiner Bruce , Leeb Bernhard , Tian Gang
Published: December 2009

The meeting continued the biannual conference series
Differentialgeometrie im Großen
at the MFO which was established in the 60's by Klingenberg and Chern. Traditionally, the conference series covers a wide scope of different aspects of global differential geometry and its connections with topology, geometric group theory and geometric analysis. The Riemannian aspect is emphasized, but the interactions with the developments in complex geometry, symplectic/contact geometry/topology and mathematical physics play also an important role. Within this spectrum each particular conference gives special attention to two or three topics of particular current relevance.

The scientific program consisted of 22 (almost) one hour talks leaving ample time for informal discussions.

This time, a main focus of the workshop were
geometric evolution equations
. 6 talks discussed the Ricci flow and its applications to the geometrization of 3-manifolds, the Ricci flow in arbitrary dimension and applications to Riemannian manifolds of positive curvature, the Kähler–Ricci flow and the (inverse) mean curvature flow.

A second focus was the
geometry of singular spaces
(5 talks), that is, metric spaces with sectional curvature bounds (in the sense of Aleksandrov), Gromov-hyperbolic spaces and Carnot spaces with connections to geometric group theory. One of the talks discussed the theory of collapse with lower curvature bound which is another ingredient (independent of Ricci flow) in the argument for geometrization in dimension

3

.

Other talks covered results about
geometric structures on manifolds
(hyperbolic geometry and representation varieties), from
geometric analysis
(Dirac operators, metrics of positive scalar curvature),
symplectic and contact geometry
(open book decompositions, confoliations, construction of special Lagrangian submanifolds with isolated conical singularities), and
complex geometry
(extremal metrics on Kähler manifolds, Kähler–Ricci flow).

There were 52 participants from 8 countries, more specifically, 23 participants from Germany, 12 from France, 6 from other European countries, and 11 from North-America. 6 were young researchers (less than 10 years after diploma or B.A.), both on doctoral and postdoctoral level. The organizers would like to thank the institute staff for their great hospitality and support before and during the conference.

The financial support of the European Union (in particular for young participants) is gratefully acknowledged.
Mini-Workshop: The Reception of the Work of Leonhard Euler (1707-1783)
Authors: Grattan-Guinness Ivor , Pulte Helmut
Published: December 2009

Quite a number of scholars work on all aspects of Euler, especially in connection with the continuing preparation of the edition of his Opera omnia. But the historical study of the reception of his work during his lifetime and especially after his death has been rather patchy; for example, in the general volume [1] to commemorate the bicentenary of his death and in the recent Euler handbook [2] to note the tercentenary of his birth few articles dealt with aspects of reception in any depth. (This point holds of reception history in general.)

In our workshop the reception of Euler's work was usually considered up to around 1840. After that, it seems that it was ordinarily either used routinely or replaced, or quite forgotten, or was studied historically; a few exceptions are noted.

Sixteen scholars came together from eight countries to present their ideas on Euler's influence in a number of mathematical areas. The selection of topics is indicated in the table of contents; it includes several from applied mathematics, where scholarship is especially limited. We also discussed some examples of Euler's influence in various countries, but this kind of history is even less well developed than the influence by areas! In addition, we aired other neglected historical questions, such as: why was there such a small reception of the many papers and other writings that were posthumously published by the Saint Petersburg Academy between 1783 and 1862?

The sign ‘E

n

’ in the abstracts below indicates the number

n

of an Euler writing according to the list that was prepared in the early 1910s by the historian Gustav Eneström.

The meeting was a success and we are thinking of building up our efforts into a book on Euler's influence.
Komplexe Algebraische Geometrie
Authors: Catanese Fabrizio , Kawamata Yujiro , Tian Gang , Viehweg Eckart
Published: December 2009

The Workshop
Komplexe algebraische Geometrie
, organized by Fabrizio Catanese (Bayreuth), Yujiro Kawamata (Tokyo), Gang Tian (Princeton), and Eckart Viehweg (Essen), drew together 50 participants. There were several young PhD students and other PostDocs in their 20's and early 30's, together with established leaders of the fields related to the thematic title of the workshop. There were 23 talks, each lasting 55 minutes or one hour, and each followed by a lively 10 minutes discussion.

As usual at an Oberwolfach Meeting, the mathematical discussions continued outside the lecture room throughout the day and the night. The Conference fully fulfilled its purported aim, of setting in contact mathematicians with different specializations and non uniform background, of presenting new fashionable topics alongside with new insights on long standing classical open problems, and also cross-fertilizations with other research topics as arithmetic and physics. For the latter, cf. the talk by Bernd Siebert on the new approach to Mirror Symmetry through logarithmic geometry and toric affine Calabi Yau varieties and the one by van Straten on quantization of completely integrable Hamiltonian Systems. For the former, cf. the talks by Winkelmann on integral sets and by McQuillan on the Bloch principle.

A central role was occupied by the new results around the Hodge conjecture presented by Voisin, and the new results by Hacon, McKernan et al which give an essential step towards the final solution of the Minimal Model Program, and were presented here by McKernan with a proposed approach to the Sarkisov program.

There were many expositions dealing with several classical problems and classical and modern theories. It would take too long to dwell on each of the outstanding contributions presented at the Conference. For some topics there were several interrelated talks, for instance we could mention the following classical themes:

There were also expositions on many other beautiful topics:

The variety of striking results and the very interesting and challenging proposals made the participation in the workshop rather strenuous but certainly highly rewarding. We hope that the quality of the expositions in these abstracts will make them quite useful to the mathematical community.
Harmonische Analysis und Darstellungstheorie Topologischer Gruppen
Authors: Kobayashi Toshiyuki , Krötz Bernhard , Lapid Erez , Torossian Charles
Published: December 2009

This international conference was organized by Toshiyuki Kobayashi (University of Tokyo), Bernhard Krötz (MPIM Bonn), Erez Lapid (Hebrew University) and Charles Torossian (Paris VI). The general theme was representation theory of real and p-adic Lie groups, especially in connection to automorphic forms. In addition there were very interesting presentations on infinite dimensional Lie groups and quantization.

It was very pleasing that new results of the highest caliber were presented by the youngest of the participants. There were many discussions at this relaxed and influential meeting. We would like to thank the MFO for having given us this wonderful opportunity. The organizers would like to thank Dmitry Gourevitch for his careful work in compiling and editing this report.
Mini-Workshop: Arithmetik von Gruppenringen
Authors: Jespers Eric , Marciniak Zbigniew , Nebe Gabriele , Kimmerle Wolfgang
Published: December 2009

The mini workshop "Arithmetic of group rings" was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and thesize of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report.

The main highlights of the conference were:

The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio.

Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups.

the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units.

discussion of the present state of art via several survey talks presented and the problem sessions

The group

G

determines its integral group ring

\mathbb Z G

and its group

V(\mathbb Z G)

of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.
Set Theory
Authors: Friedman Sy-David , Magidor Menachem , Woodin W. Hugh
Published: December 2009

This was a highly successful workshop on some of the exciting recent developments in set theory. There were 45 participants. By limiting the number of talks to 25, there was ample time for research collaboration in small groups, which rendered the meeting very relaxed, enjoyable and productive.

Among the highlights of the workshop are the following:

Justin Moore presented some dramatic new work on the existence of a universal Aronszajn line, demonstrating that the existence of a canonical, universal linear ordering is not limited to the countable case. There were several beautiful presentations in the area of descriptive set theory: Christian Rosendal provided a rough classification of Banach spaces, Slawomir Solecki described G-delta ideals of compact sets, Simon Thomas drew some striking connections between set theory and geometric group theory and John Clemens discussed Borel homomorphisms of smooth Sigma-ideals. A major new result was presented by David Schrittesser, who showed that projective measurability does not imply the projective Baire property.

There were two major results in the related areas of large cardinals and determinacy: John Steel calculated the consistency strength of determinacy for games on reals and he together with Ronald Jensen discovered a way of building the core model for a Woodin cardinal in ZFC without additional large cardinal hypotheses. Several talks presented important recent work related to the pcf theory: Todd Eisworth discussed combinatorial properties at the successor of a singular cardinal, Matteo Viale told us about reflection and approachability and Moti Gitik brought us up to date on consistency results regarding the behaviour of the generalised continuum function.

In combinatorial set theory we heard lectures on induced Ramsey theory, MAD families, weak diamonds and simple cardinal invariants. And in the theory of forcing we heard about Laver products, p-points, indestructibility of weak compactness, mixed support iterations and reflection.

Paul Larson's very nice talk on Martin's Maximum and definability rounded out a very successful meeting.
Mini-Workshop: Mathematics of Solvency
Authors: Delbaen Freddy , Eisele Karl-Theodor , Hipp Christian
Published: December 2009

The mini-workshop
Mathematics of Solvency
, organised by Freddy Delbaen (Zurich), Karl-Theodor Eisele (Strasbourg) and Christian Hipp (Karlsruhe) was held during the week February 10th–February 16th, 2008. The 15 participants from academia and practice – most of whom came from supervision authorities – discussed the foundation of mathematical concepts for risk and risk measurement as well as their consequences for various decision or control activities in financial services. Such concepts are of major importance for the definition of a necessary solvency capital – which takes into account diversification effects – for banks, investors, and insurers, from the perspective of supervision, of risk management, of rating, and of capital allocation. They help industry to develop consistent foundations for dealing with insurance and financial risks.

The investigation of risk concepts in the multi period setting leads to new mathematical questions in the theory of stochastic processes. A sensible definition of a risk measure requires the property of super-additivity with respect to different risks as well as a submartingale property with respect to time. Such a definition leads to a theory of non-linear pricing of risks, as well as to a better understanding of the time evolution of risk assessment. These two concepts were not treated in traditional finance.

The consistency of risk measurement over different time intervals leads to monotonicity properties of convex functionals. The representation of these time consistent valuations is related to backward stochastic differential equations and difference equations, respectively. The information available to agents is modeled with filtrations and the difference in information leads to problems known as enlargement of filtrations.

In the talks, methods and concepts from various fields were used: stochastic processes, game theory, functional analysis, and stochastic control. In addition, experience with existing and operating real world risk valuation concepts has been reported and discussed.
Analysis of Boundary Element Methods
Authors: Costabel Martin , Stephan Ernst Peter
Published: December 2009

The workshop
Analysis of Boundary Element Methods
, organized by Martin Costabel (Rennes) and Ernst P. Stephan (Hannover). This meeting brought together 46 experts in numerical analysis.

Boundary element methods (BEM) are well established numerical methods with a wide range of applications. There are still many challenging problems, the aim of this workshop was to coordinate the efforts to tackle these problems.

A central theme of many talks was the error analysis. In the following we focus on some central aspects.

1. BEM for time dependent problems:

Space-time boundary integral equations using integral representations with the fundamental solution of hyperbolic problems have become increasingly important in elastodynamics and electrodynamics. New results were presented concerning error analysis, stability and performance of numerical algorithms. Special attention was given to efficient quadrature of retarded potentials and also to the method of convolution quadrature for the computation of transient acoustic waves.

2.Stability at high frequencies:

The numerical modeling of scattering problems at higher frequencies poses practical challenges due to the need to approximate highly oscillating functions. In addition, the analysis of many of the standard BEM loses usefulness in the high frequency range because the constants in the stability and error estimates either grow rapidly with the frequency of have unspecified dependency on the frequency. New boundary element methods have been presented whose stability deteriorates only slowly with frequency or are even unconditionally stable for all frequencies.

3. Fast algorithms:

Multilevel approaches have produced several families of highly efficient algorithms. Several talks reported on the applications of the fast multipole method and on iterative techniques based on multigrid methods, H-matrices or on Schwarz preconditioners.

4. New concepts for BEM:

Topics have been here: sparse p-version BEM with random loading, meshless methods for pseudodifferential equations, adaptive procedures, least squares FEM-BEM coupling, p-version applied to EFIE, BEM for eigenvalue problems.

5. Specific applications:

Here the following have been addressed: Source identification and source reconstructions in acoustics with the help of inverse boundary elements, BEM for Maxwell's equations (currents and charges boundary integral equations, shape optimization of integrated lens antennas, electromagnetic scattering by a dielectric body), BEM for option pricing.

The list of participants included several of the founders of the discipline of mathematical analysis of BEM, but also some PhD students. This mix of both established and younger researchers created a particularly stimulating research atmosphere.
Nonlinear Evolution Equations
Authors: Ecker Klaus , Shatah Jalal , Struwe Michael
Published: December 2009

Following the successful pattern of the meeting in 2005, this year's workshop on 'Nonlinear Evolution Problems' focussed on a small number of currently very active areas in this field. By far the dominant theme, however, were geometric evolution equations of parabolic type, followed by the topic of wave equations and water waves/Navier-Stokes equations both in a classical and relativistic framework. Among the geometric evolution equations, curvature flows of hypersurfaces were considered with priority. In several talks, applications of these to isoperimetric problems and to formulation of appropriate local mass concepts were presented. One talk dealt with mean curvature flow in certain degenerate spaces. Several talks looked into harmonic and biharmonic map flow as well as Ricci flow. Among the nonlinear hyperbolic equations, the water wave equation, Einstein's equations, semilinear wave equation and, for the first time, the d-brane equation from string theory were featured. For the latter, an

\epsilon

-regularization method was used to establish an existence proof. This is another example of a method which has now been applied to different types of nonlinear equations. It had previously been applied to mean curvature flow and to minimal surfaces. The last theme concerned dispersive equations. This was represented only in a talk by Markus Keel, albeit on seminal work on resonant decompositions applicable to a wide class of example. All together, 21 talks were presented by international specialists from Australia, Canada, Germany, Great Britain, Italy, Sweden, Switzerland and the United States. Many of the speakers were only a few years past their Ph.D., some even still working towards their Ph.D.; 6 out of 42 participants and 4 out of 21 speakers were women. As a rule, three lectures were delivered in the morning session; two lectures were given in the late afternoon, which left ample time for individual discussions.
Arithmetic Algebraic Geometry
Authors: Faltings Gerd , de Jong Johan , Pink Richard
Published: December 2009

The workshop covered developments in the field in the last four years. Roughly speaking arithmetic geometry consider algebraic schemes over rings of integers of numberfields. However an important tool is to first extend the base to a p-adic completion. Although both global and local problems matter this time there was a heavy emphasis on p-adic topics. One of them is the deformation-theory of Galois-representations, leading to a proof of Serre's conjecture. here one starts with a global Galois-representation modulo p, then lifts modulo

p^2

, etc. For the lifts one requires certain local conditions (like being unramified outside a given set of places), and the most important and difficult such conditions arise at primes dividing

p

. Here the most important tool is J.M.Fontaine's theory which relates Galois-representations to filtered Frobenius-crystals. Another spectacular progess is the proof (by Ngo) of the fundamental lemma in the theory of automorphic representations. It postulates identities of p-adicorbital integrals and is reduced to a geometric statement about perverse sheaves on Hitchin-fibrations in positive characteristic. Concerning p-adic cohomology theories we are getting closer to a p-adic theory of D-modules, and of overconvergent crystals, over singular schemes. Also the long awaited etale coverings of p-adic period domains have finally been constructed, after it has been understood that they have "holes" which are visible in the Berkovich-space but not in the conventional rigid space. That they exist is suggested by Fontaine's theory. These period domains classify p-divisible groups. Some of them (Drinfeld, Lubin-Tate) can be covered by explicit affinoid domains, thus giving some type of reduction theory for p-divisible groups. There are attempts to extend this to finite flat group-schemes. Concerning K-theory the classical Borel-regulator from K-theory to Deligne-cohomology has been extended to syntomic cohomology, as well as the computation of its values on Eisenstein-symbols. For the l-adic etale theory general finiteness theorems can now be shown for quasi-excellent schemes. A further topic was the theory of p-adic Banach-representations of p-adic Lie-groups. On a more global level we had talks about (p-adic!) constructions of rational points on elliptic curves, the association of K-classes to abelian varieties, and the theory of tame fundamental groups. Finally the theory of small points has been extended to function fields (over numberfields it leads to equidistribution) using tropical geometry.
Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry
Authors: Knieper Gerhard , Polterovich Leonid , Potyagailo Leonid
Published: December 2009

The workshop “Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry”, organized by Gerhard Knieper, Leonid Polterovich and Leonid Potyagailo took place the second time in Oberwolfach. The general idea to organize a joint meeting of specialists belonging to three different but close and interacting fields turned out to be really fruitful. The aim of our workshop is to give a possibility to people working in the above fields to discuss their results and open problems from different points of view. We are sure that such conferences need to exist not only on the level of World or European Congresses but also in much less official and restrictive atmosphere inherent for Oberwolfach conferences.

Our last meeting brought together world renown senior experts in our fields as well as talented mathematicians in the beginning in their careers. Let us describe shortly the directions of the given talks. The presented talks were devoted to:

Historically, one of the motivating subjects being in the center of our joint interests was the theory of quasi-morphisms and bounded cohomology. In the present time there exist many other subjects which are in our common interests: study of groups of diffeomorphisms using methods of symplectic topology (e.g. the talk of F. Le Roux where he constructed a new candidate for a non-trivial normal subgroup of the group of area-preserving homeomorphisms); dynamical methods describing the geometry of the mapping class group and the group of outer automorphisms of free groups (G. Levitt); probabilistic methods in the geometric group theory (A. Erschler and K. Fujiwara); solving equations in free groups and diophantine geometry (F. Dahmani); strong Tits alternative for linear groups and its consequences: uniform exponential growth and uniform decay of return probability (E. Breuillard); spectral theory for combinatorial Laplacian of Cayley graphs of finite abelian groups (A. Karlsson) etc.

Another important issue of our meetings is that they attract many students working in these subjects. Several students (or recent students) gave 30 minutes talks during our informal evening sessions attended by all participants. The students have been actively participating in the discussions during the conference.

We continued our tradition by organizing the open problem session on Wednesday night successfully chaired by Anatole Katok. We enclose below the problems formulated during this session and hope that they will stimulate further research in our fields. We strongly believe that the tradition of such joint meetings representing several research directions should continue in Oberwolfach.
Von Neumann Algebras and Ergodic Theory of Group Actions
Authors: Bisch Dietmar , Gaboriau Damien , Jones Vaughan F. R. , Popa Sorin
Published: December 2009

The workshop \emph{Von Neumann Algebras and Ergodic Theory of Group Actions} was organized by Dietmar Bisch (Vanderbilt University, Nashville), Damien Gaboriau (ENS Lyon), Vaughan Jones (UC Berkeley) and Sorin Popa (UC Los Angeles). It was held in Oberwolfach from October 26 to November 1, 2008.

This workshop was the first Oberwolfach meeting on von Neumann algebras and orbit equivalence ergodic theory. The organizers took special care to invite many young mathematicians and more than half of the 28 talks were given by them. The meeting was very well attended by over 40 participants, leading senior researchers and junior mathematicians in the field alike. Participants came from about a dozen different countries including Belgium, Canada, Denmark, France, Germany, Great Britain, Japan, Poland, Switzerland and the USA.

The first day of the workshop featured beautiful introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa's deformation/rigidity techniques and applications to rigidity in II

_1

factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa). There were many excellent lectures during the subsequent days of the conference and many new results were presented, some for the first time during this meeting. A few of the highlights of the workshop were Vaes' report on a new cocycle superrigidity result for non-singular actions of lattices in SL(n,

\mathbb R

) on

{\mathbb R}^n

and on other homogeneous spaces (joint with Popa), Ioana's result showing that every sub-equivalence relation of the equivalence relation arising from the standard SL(2,

\mathbb Z

)-action on the

2

-torus

{\mathbb T}^2

is either hyperfinite, or has relative property (T), and Epstein's report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space. Other talks discussed new results on fundamental groups of II

_1

factors, L

^2

-rigidity in von Neumann algebras, II

_1

factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra and new results on topological rigidity and the Atiyah conjecture. Many interactions and stimulating discussions took place at this workshop, which is of course exactly what the organizers had intended.

The organizers would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing the splendid environnment for holding this conference. Special thanks go to the very helpful and competent staff of the institute.
Mini-Workshop: Numerics for Kinetic Equations
Authors: Gamba Irene M. , Rjasanow Sergej , Wagner Wolfgang
Published: December 2009

The workshop
Numerics for Kinetic Equations
, organized by Irene Gamba (Austin, USA), Sergej Rjasanow (Saarbrücken, Germany), and Wolfgang Wagner (Berlin, Germany) was held November 16th–November 22nd, 2008. The meeting was well attended with 15 participants (11 from Europe, 3 from USA, and 1 from Japan). This workshop brought together researchers with various backgrounds: mathematics, physics, chemistry, and engineering science. The interdisciplinarity of the invited participants was essential because the numerical solution of kinetic equations is characterized by a wide area of applications (rarefied gas dynamics, modeling of semiconductors, plasma, or combustion). Furthermore, the strong coupling between different areas of science is needed to understand the complicated and highly nonlinear nature of kinetic equations. The last decades have shown an amazing development in the design of effective numerical methods for kinetic equations describing processes in micro- and nanomachines. A particular example is the Knudsen compressor – an energy efficient micro-scale pump with no moving parts. Furthermore, the inelastic Boltzmann equation is more and more used in applications. Finally, semiconductor transport models are still a challenge for numerics.

In this mini-workshop, we have discussed recent mathematical and computational issues, mainly related to the so called “low Mach number” flows, from the point of view of numerical efficiency and accuracy. When solving those problems by the use of the Direct Simulation Monte Carlo method, one has to deal with a small signal-to-noise ratio. Thus, due to statistical fluctuations, it is difficult to compute the physical flow parameters (density, stream velocity, temperature, and, especially, heat flux) with sufficient accuracy, so variance reduction methods are desperately needed. On the other hand, deterministic algorithms do not exhibit statistical fluctuations, and, therefore, may become competitive for slow flows.

The talks have been devoted to

Furthermore, three extended “round table” discussions were devoted to

These discussions were extremely useful for facilitating a clear understanding of the main difficulties when applying stochastic and/or deterministic methods to kinetic equations close to the hydrodynamic limit.

The participants appreciated the idea of the mini-workshops – to have not too many talks, but instead plenty of time for intensive and detailed discussions. We would strongly support further such workshops. Finally, we would like to thank both the administration and the staff of the MFO for providing excellent working conditions and creating a pleasant atmosphere.
Numerical Techniques for Optimization Problems with PDE Constraints
Authors: Heinkenschloss Matthias , Hoppe Ronald H.W. , Schulz Volker
Published: December 2009

The development, analysis and implementation of eﬃcient and robust numerical techniques for optimization problems associated with partial diﬀerential equations (PDEs) is of utmost importance for the optimal control of processes and the optimal design of structures and systems in modern technology. The successful realization of such techniques invokes a wide variety of challenging mathematical tasks and thus requires the application of adequate methodologies from various mathematical disciplines. During recent years, signiﬁcant progress has been made in PDE constrained optimization both concerning optimization in function space according to the paradigm ’Optimize ﬁrst, then discretize’ and with regard to the fast and reliable solution of the large-scale problems that typically arise from discretizations of the optimality conditions. The contributions at this Oberwolfach workshop impressively reﬂected the progress made in the ﬁeld. In particular, new insights have been gained in the analysis of optimal control problems for PDEs that have led to vastly improved numerical solution methods. Likewise, breakthroughs have been made in the optimal design of structures and systems, for instance, by the socalled ’all-at-once’ approach featuring simultaneous optimization and solution of the underlying PDEs. Finally, new methodologies have been developed for the design of innovative materials and the identiﬁcation of parameters in multi-scale physical and physiological processes.
Wave Motion
Authors: Constantin Adrian , Escher Joachim , Johnson Robin , Strauss Walter A.
Published: December 2009

This workshop was devoted to recent progress in the mathematical study of water waves, with special emphasis on nonlinear phenomena. Both aspects related to the governing equations (free boundary Euler equations) as well as aspects related to various model equations were of interest.
Kommutative Algebra
Authors: Bruns Winfried , Flenner Hubert , Huneke Craig
Published: December 2009

The workshop brought together researchers on various areas of Commutative Algebra. New results in combinatorial commutative algebra, homological methods and invariants, characteristic
p
-methods, and in general commutative algebra and algebraic geometry were presented in the lectures of the workshop.
Combinatorics and Probability
Authors: Alon Noga , Bollobás Béla , Wegener Ingo
Published: December 2009

The effective application of probabilistic reasoning in the study of problems in diverse areas is one of the most exciting recent developments in Mathematics. Probabilistic methods turned out to be very powerful in Discrete Mathematics, Analysis, Number Theory and Theoretical Computer Science. The meeting was dedicated to recent developments in these areas, focusing on the investigation of combinatorial problems for random sets and probabilistic methods, on the study of of questions in percolation, on the design and analysis of randomized algorithms and derandomization techniques, and on applications of probabilistic ideas in the study of questions in Combinatorial Number Theory and in Combinatorial Geometry.
Quadratic Forms and Linear Algebraic Groups
Authors: Hoffmann Detlev , Merkurjev Alexander , Tignol Jean-Pierre
Published: December 2009

Topics discussed at the workshop Quadratic forms and linear algebraic groups included besides the algebraic theory of quadratic and Hermitian forms and their Witt groups several aspects of the theory of linear algebraic groups and homogeneous varieties, as well as some arithmetic aspects pertaining to the theory of quadratic forms over function ﬁelds over local ﬁelds.
Manifold Perspectives
Authors: Hambleton Ian , Pedersen Erik Kjaer , Ranicki Andrew , Reich Holger
Published: December 2009

The study of the global properties of manifolds and their symmetries has had a great impact on both geometric and algebraic topology, and also on other branches of mathematics, such as algebra, diﬀerential geometry and analysis. The purpose of the meeting was to bring together active researchers from diverse areas to discuss these exciting current perspectives on the topology of manifolds.
Strings, Fields and Topology
Authors: Stolz Stephan , Sullivan Dennis , Teichner Peter
Published: December 2009

In recent years, the interplay between traditional geometric topology and theoretical physics, in particular quantum ?eld theory, has played a signi?cant role in the work of many researchers. The idea of this workshop was to bring these people together so that the ?elds will be able to grow together in the future. Most of the talks were related to various ﬂavors of ﬁeld theories and differential cohomology theories.
Spectra in Banach and locally convex algebras
Authors: Gaur A. K.
Published: December 2009

We present a counterexample to the converse of results on approximate point spectra. A norm inequality in terms of a distance function is established for a quotient bounded operator on a locally convex Hausdorff space on the set of complex numbers.
Coincidence situations for absolutely summing non-linear mappings
Authors: Botelho Geraldo , Pellegrino Daniel
Published: December 2009

Situations where every linear operator between certain Banach spaces is absolutely

(p;q)

-summing are very useful in linear functional analysis. In this paper we investigate situations of this type for three classes (two new and one well known) of nonlinear mappings which are generalizations of the notion of absolutely summing linear operators.
Qu'est-ce que l'homogénéisation?
Authors: Tartar Luc
Published: December 2009

This text in French tries to describe what is homogenization and what is not homogenization, but discusses also a few related issues. Of course, homogenization is understood in the general sense that the author had decribed in his Peccot lectures at Collège de France in Paris, in the beginning of 1977, and corresponds to the notion of H-convergence that he had developed with François Murat in the early 1970s, extending the work of Sergio Spagnolo on G-convergence from the late 1960s; for example, it uses no periodicity assumption like in the work where Ivo Babu\v{s}ka first used the word homogenization, or in the earlier pioneering work of Evariste Sanchez-Palencia.
An existence result for a new variant of the nonconvex sweeping process
Authors: Bounkhel Messaoud , Haddad Tahar
Published: December 2009

In this paper we prove the existence of solutions of the following new variant of the nonconvex sweeping process with perturbation

− Au(t) \in N_{C(t)}(\dot{u}(t)) + ü(t) + F(t,\dot{u}(t))

a.e. on

[ 0,T  ]

(

T > 0

).
Existence of solutions for degenerated problems in $L^1$ having lower order terms with natural growth
Authors: Aharouch L. , Azroul Elhoussine , Benkirane A.
Published: December 2009

We prove the existence of a solution for a strongly nonlinear degenerated problem associated to the equation

Au + g(x,u,∇u) = f,

where

A

is a Leray–Lions operator from the weighted Sobolev space

W_0^{1,p}(Ω, w)

into its dual

W^{ −1,p'}(Ω, w^*)

. While

g(x,s,ξ)

is a nonlinear term having natural growth with respect to

ξ

and no growth with respect to

s

, it satisfies a sign condition on

s

, i.e.,

g(x,s,ξ) · s ≥ 0

for every

s∈ℝ

. The right-hand side

f

belongs to

L^1(Ω)

.
Generalized Dieudonné and Hill criteria
Authors: Danchev Peter V.
Published: December 2009

Let

A

be a reduced abelian

p

-group with a nice subgroup

G

. It is proved that if

A/G

is simply presented, then

A

is simply presented precisely when

G

is strongly simply presented in

A

. Moreover, the same type theorem for the class of

ℵ_1

-

Σ

-cyclic

p

-groups is also established without the niceness of

G

in

A

. Some analogous assertions for other exotic sorts of abelian groups are also considered.

The results obtained strengthen previous results due to J. A. Dieudonné (Portugal. Math., 1952), P. D. Hill (Trends in Math., 1999), and some other authors.
Local solutions for a Timoshenko system in noncylindrical domains
Authors: Narciso Vando , Cousin Alfredo T.
Published: December 2009

In this paper we study a Timoshenko system modeling transverse vibrations of thin elastic beams in a moving boundary domain. Existence and uniqueness of a local solution is proved by using Faedo–Galerkin approximation.
Study of the maximal throughput of multiclass queueing systems
Authors: Belarbi Faiza , Bouchentouf Amina A.
Published: December 2009

The stability properties of the bandwidth allocation algorithm First Fit are analyzed for the distributions concentrated on three sizes for the requests and the bin equal to 5. To analyze these processes we introduce the notion of a smooth initial state. Starting from a smooth initial state the fluid limits of these systems are investigated.
A note on the global regularity of steady flows of generalized Newtonian fluids
Authors: Crispo Francesca
Published: December 2009

We establish regularity results for solutions of a generalized Newtonian model in a cubic domain. We prove regularity results in the L2-space for the second derivatives of the velocity and the first derivatives of the pressure. Further, we show that the gradient of weak solutions is integrable up to the boundary with any finite exponent.
Abel’s method on summation by parts and well-poised bilateral ${}_5ψ_5$-series identities
Authors: Chu Wenchang
Published: December 2009

The modified Abel lemma on summation by parts is employed to establish two well-poised bilateral

{}_5ψ_5

-series identities. Several quadratic, cubic and quartic transformation formulae are derived for the truncated partial sums of Bailey’s very well-poised

{}_6ψ_6

-series.
A note on nonlinear elliptic problems with singular potentials
Authors: Badiale Marino , Rolando Sergio
Published: December 2009

We deal with the semi-linear elliptic problem

-\triangle u+V\left( \left| x\right| \right) u=f\left( u\right) ,\quad u\in
D^{1,2}(\mathbb{R}^{N};\mathbb{R})

where the potential

V>0

is measurable,
singular at the origin and may also have a continuous set of singularities.
The nonlinearity is continuous and has a super-linear power-like behaviour;
both sub-critical and super-critical cases are considered. We prove the
existence of positive radial solutions. If

f

is odd, we show
that the problem has infinitely many radial solutions. Nonexistence results for particular potentials and nonlinearities are also given.
Periodic solutions of Birkoff–Lewis type for the nonlinear wave equation
Authors: Biasco Luca , Di Gregorio Laura
Published: December 2009

We prove existence and multiplicity of small amplitude periodic solutions with large period for the wave equation with small ‘mass’. Such solutions bifurcate from resonant finite dimensional invariant tori of the fourth order Birkhoff normal form of the associated hamiltonian system. The number of geometrically distinct solutions and their minimal periods tend to infinity when the ‘mass’ tends to zero.
Nodal solutions of nonlinear elliptic Dirichlet problems on radial domains
Authors: Bartsch Thomas , Degiovanni Marco
Published: December 2009

Let

\Omega\subset\R^N

be a ball or an annulus and

f:\R\to\R

absolutely continuous, superlinear, subcritical, and such that

f(0)=0

. We prove that the least energy nodal solution of

-\Delta u= f(u)

,

u\in H^1_0(\Omega)

, is not radial. We also prove that Fučik eigenfunctions, i.e. solutions

u\in H^1_0(\Omega)

of

-\Delta u=\lambda u^+-\mu u^-

, with eigenvalue

(\lambda,\mu)

on the first nontrivial curve of the Fučik spectrum, are not radial. A related result holds for asymmetric weighted eigenvalue problems. An essential ingredient is a quadratic form generalizing the Hessian of the energy functional

J\in C^1(H^1_0(\Omega))

at a solution. We give new estimates on the Morse index of this form at a radial solution. These estimates are of independent interest.
Homogenization of doubly-nonlinear equations
Authors: Visintin Augusto
Published: December 2009

This note deals with a doubly-nonlinear parabolic-hyperbolic equation, that represents electromagnetic processes in a nonhomogeneous metal surrounded by an insulating environment. Existence of a weak solution is here illustrated. The constitutive relations are then assumed to exhibit periodic oscillations in space. As the period vanishes, the solution converges in the sense of Nguetseng to that of a corresponding two-scale homogenized problem. The homogenization procedure is then completed by eliminating the dependence on the fine-scale variable. An analogous problem issued from phase transitions is also illustrated.