bluuebunny/crossref_metadata_2025 huggingface dataset:
Partielle Differentialgleichungen
Authors: Ilmanen Thomas , Schätzle Reiner , Trudinger Neil S.
Published: December 2009

The workshop
Partial differential equations
, organised by Tom Ilmanen (ETH Zürich), Reiner Schätzle (Universität Tübingen) and Neil Trudinger (Australian National University Canberra) was held July 24-30, 2005. This meeting was well attended by 46 participants, including 6 females, with broad geographic reprensentation. The program consisted of 15 talks and 9 shorter contributions and left sufficient time for discussions.

New results combining partial differential equations and geometric problems were presented in the area of minimal surfaces, free boundaries and singular limits, for example the construction of branched minimal surfaces, the regularity of free boundaries in the wake of the monotonicity formula of Weiss and a proof of a conjecture of De Giorgi.

A major part of the leading experts of partial differential equations with conformal invariance attended the workshop. Here new results were presented in conformal geometry, for the Yamabe problem, the Paneitz operator and the Willmore functional.
Dynamical System Methods in Fluid Dynamics
Authors: Marsden Jerrold E. , Scheurle Jürgen
Published: December 2009

The workshop was organized around the following classical as well new and emerging techniques. Traditional methods in fluid mechanics continue to be invigorated with the infusion of ideas from dynamical systems, geometric methods, multiscale analysis, new developments in scientific computation, and control theory. The specific themes of the workshop were as follows:

Structure of the Meeting.
The meeting had a balance of senior researchers, postdoctoral fellows and graduate students. Consistent with the general approach advocated by Oberwolfach, we selected about 20–25 people to give a lecture at the meeting. They suggested students, postdoctoral fellows and junior faculty all of whom participated through two poster sessions.

Posters.
The poster session was one of the most interesting aspects of the meeting. The organizers decided to award prizes for the best poster and an ad hoc committee (consisting of Friedlander, Haller, Reich, Marsden and Scheurle) was formed to choose the four best posters. This number matched the number of gifts that were available. The winners were (in alphabetical order):

Impressions, Connections, Insights.
The mixture of participants, from those who were very applied and those who came from a more of a dynamical systems background, but all with a strong mathematical dedication, was very fruitful.

Dynamical systems methods have now started to be utilized successfully for the study of complex real fluid systems, from controlled man-made experiments to multi-scale and multivariate natural dynamics, from oceanic motions to flows around jellyfish. Dynamical system approaches are then applied to complex data, models or combinations thereoff. There is a wide range of research opportunities, including the possibilty for new theoretical and methodological progress.

Amongst the specific new connections and insights that were gained, the power of dynamical systems ideas, such as invariant manifolds or Lyapunov exponents, seemed to be quite impressive as well as useful in the context of metereology as well as oceanography. The use of dynamical systems ideas in control of fluids (separation and cavity oscillations, for example) was also quite interesting for a number of the participants.
C*-Algebren
Authors: Anantharaman-Delaroche Claire , Echterhoff Siegfried , Haagerup Uffe , Voiculescu Dan
Published: December 2009

The aim of the workshop

C

^*

-algebras

was to bring together researchers from basically all areas related to operator algebra theory. This gives a unique opportunity to obtain a broader view of the subject and to create new interactions between researchers with different background. The organizers, Claire Anantharaman-Delaroche, Siegfried Echterhoff, Uffe Haagerup, and Dan Voiculescu took special care to invite a good number of young researchers, some of them already being leading experts in their fields. As a result, several contributions in this report are from researchers who, at the time of the workshop, were less then 30 years old.

There were 29 lectures presented at this workshop with topics from Ergodic Theory,

\mathrm{L}^2

-(co-)homology, classification of C

^*

-algebras, Operator Theory, von Neumann algebras,

\mathrm{KK}

-theory and the Baum–Connes conjecture, quantum spaces and quantum groups, mathematical physics, non-commutative probability theory, and the theory of operator spaces. To name some special highlights we can mention the reports on recent developements in the study of “boundary actions” of quantum groups due to S. Vaes and R. Vergnioux, the new results on classification theory of amenable C

^*

-algebras in terms of studying algebras which are stable under tensoring with the Jiang-Su algebra

\mathcal Z

(see the lectures of A. Toms and M. Rørdam), or the report by S. Popa on recent progress in the study of strong rigidity for II

_1

factors associated to equivalence relations. But this is only a very small selection of the interesting lectures on new results presented at this workshop.

It is a pleasure for the organizers of the conference to use this opportunity to thank all participants of the workshop for their contributions – either in lectures held at the workshop or in stimulating discussions following the lectures. We also thank the Mathematisches Forschungsinstitut Oberwolfach for providing a great environment and strong support for organizing this conference. Special thanks go to the very competent and helpful staff of the institute and to the chef de cuisine.
Cohomology of Finite Groups: Interactions and Applications
Authors: Adem Alejandro , Carlson Jon F. , Henn Hans-Werner
Published: December 2009

The aim of this meeting was to bring together people from different fields in which the cohomology of finite groups is an important tool. Several such meetings have taken place in the past (for example at Oberwolfach in 2000) and they have contributed substantially to the development of interactions between such fields, in particular between commutative algebra, homological algebra, homotopy theory, modular representation theory and transformation groups. \smallskip The meeting was attended by 54 participants from about a dozen countries. There were

24

talks of various length. Several of them were directly concerned with the cohomology of finite groups, but beyond that there were talks in which the relation of the cohomology of finite groups with other topics was emphasized: among them talks on the cohomology of infinite groups, of quantum groups and of local rings, on functor cohomology, on transformation groups and

p

-compact groups, on homological algebra and triangulated categories, on endotrivial modules in modular representation theory and on the Alperin conjecture as well as on

K

-theory, on invariant theory, on commutative algebra and relations to stable homotopy theory. \smallskip The schedule allowed for enough time for extensive and lively interactions between the participants. Besides the traditional and popular Wednesday afternoon hike (during a week with an exceptionally nice weather) an excellent concert was organized on Thursday evening by some of the participants of the conference. As always the very pleasant and stimulating atmosphere at the institute contributed to make this a very successful meeting.
Mini-Workshop: Analysis and Computation of Microstructures in Finite Plasticity
Authors: Hackl Klaus , Conti Sergio , Ortiz Michael
Published: December 2009

The miniworkshop was attended by fourteen participants of whom twelve gave presentations. Additionally Kolumban Hutter from the University of Darmstadt, then at the research institute as a research-in-pairs fellow, agreed to give a talk on “Granular flows and avalanches”. There was one discussion session chaired by Stefan Müller, during which the most important problems of the field which have to be addressed in the near future were identified. The individual talks lasted for one and a half hours in the average and were interlaced with lively discussions.

The main theme in the workshop was the variational formulation of crystal plasticity, in a rate-independent setting. For small time-intervals, this leads to a single variational problem, which is often referred to as the “first time step”. One substantial difficulty arises from the fact that the resulting functional is not lower semicontinuous, leading to non-existence of minimizers. The corresponding theory of relaxation has some analogies to the one developed in nonlinear elasticity. A second issue is the formulation of a theory that goes beyond small time intervals. In a framework where time is discretized, the functional at each time step depends on the minimizer at the previous one (assuming it exists). This leads to a sequence of coupled variational problems, and even the precise definition of the time-continuous limit is not yet completely understood.

The subjects treated ranged from the mathematical formulation of the variational problems and existence results for those, to analytical and numerical methods to determine their relaxation, and applications to various problems in mechanics and physics. Research in progress and sometimes of speculative nature was presented rather then established results. Specifically, the main questions treated were the following:

During the miniworkshop progress could be made concerning some of these questions and interesting ideas where developed for the rest of them.
The Arithmetic of Fields
Authors: Geyer Wulf-Dieter , Jarden Moshe , Pop Florian
Published: December 2009

The fifth conference with the title
The Arithmetic of Fields
, organized by Wulf-Dieter Geyer (Erlangen), Moshe Jarden (Tel Aviv), and Florian Pop (Philadelphia), was held February 5–11th, 2006. In contrast to the fourth conference held in February 3–9th, 2002, this conference was a “full” one, namely as many participants were invited as the Institute could host. Due to support from the European Union, more young people were invited in the last few weeks prior to the conference, so that the total number of participants reached 54. The participants came from 13 countries: Germany (20), USA (10), Israel (7), France (7), Denmark (2), Austria (1), Brazil (1), Canada (1), Hungary (1), Japan (1), Romania (1), Russia (1), and South Africa (1). Among the participants there were 9 graduate students and 8 young researchers. Six women attended the conference.

The organisers asked four people before the conference to give surveys of one hour on recent progress made by other colleagues in Field Arithmetic.

Tamás Szamuely (Budapest) reported on the solution by János Kollár of a Problem due to Ax from 1968: Every PAC field of characteristic

0

is

C_1

. The case where the characteristic is positive remains open.

Alexandra Shlapentokh (Greenville) described the progress Bjorn Poonen made on Hilbert's Tenth Problem: There exists a recursive set

T_1

of prime numbers of natural density

0

and a set

T_2

of prime numbers of natural density

1

such that

T_1\subseteq T_2

and for each set

S

with

T_1\subseteq S\subseteq T_2

Hilbert's Tenth Problem for

O_{\mathbb Q,S}

has a negative solution. Here

O_{\mathbb Q,S}

is the ring of all rational numbers whose denominators are divided only by primes in

S

. Whether Hilbert's Tenth Problem for

\mathbb Q

has a negative solution is still open.

Alexander Prestel (Konstanz) presented a theorem of Jochen Koenigsmann: If a

p

-Sylow extension

P

of a field

K

is Henselian and

P

is neither separably closed nor real closed, then

K

itself is Henselian.

Pierre Débes (Lille) surveyed Fried's problem on Modular Towers. He mentioned that the main conjecture is close to completion in the case of

4

branch points (Bayley–Fried). He also reported on a result of Anna Cadoret: The dihedral group has a regular realization over

\mathbb Q_{\rm ab}

with only inertia groups of order

2

.

In addition to these survey talks seventeen participants were invited to report on their own achievements in 45 minutes talks. Altogether, the talks presented the impressive progress made in Field Arithmetic in recent years. The reader may find here extended abstracts of all talks. We hope they will be to the benefit of all of the participants as well as the fans of Field Arithmetic.

The organisers: Wulf-Dieter Geyer, Moshe Jarden, Florian Pop
Numerical Techniques for Optimization Problems with PDE Constraints
Authors: Heinkenschloss Matthias , Hoppe Ronald H.W. , Schulz Volker
Published: December 2009

The numerical solution of optimization problems with partial differential equation (PDE) constraints is vital to a growing number of science and engineering applications. The development of robust and efficient algorithms for the solution of these optimization problems presents many challenges that arise out of, e.g., the intricate mathematical structure of these problems, the complicated interactions between numerical methods for PDE and optimization, the large-scale of the optimization problems, and the increasing complexity of applications. To identify and overcome these challenges an integrated approach is needed that builds on a variety of mathematical sub-disciplines, such as theory of PDEs, distributed parameter systems, numerical solution of PDEs, numerical optimization, and numerical linear algebra. This international workshop has brought together some of the leading experts in the fast developing field of optimization problems with PDE constraints to present recent developments in this area as well as to identify open problems and further research needs.

Among the themes of this workshop were the design and analysis of approaches for the solution of PDE constrained optimization problems with additional point-wise constraints on controls and states (the solution of the governing PDE). State constrained problems are particularly challenging because of the low regularity properties of the Lagrange multipliers associated with point-wise constraints on the states. A second theme was the development of adaptive methods for the solution of PDE constrained optimization problems and, more generally, the development of optimization level model reduction techniques for these problems. The goal here is to develop models (through, e.g., mesh adaptation or proper orthogonal model reduction) of the PDE constrained optimization problems that capture the relevant features of the optimization problems with a specified accuracy, but involve as few degrees of freedom as possible and, hence, are computationally less expensive to work with. A third theme was The efficient solution of linear systems arising in optimization algorithms for discretized PDE constrained optimization problems represented another theme. Finally, a number of talks presented advances and challenges in the solution of PDE constrained optimization problems arising in important science and industrial applications.
Analysis and Topology in Interaction
Authors: Bunke Ulrich , Goette Sebastian , Igusa Kiyoshi , Schick Thomas
Published: December 2009

The MFO workshop “Analysis and Topology in Interaction”, organised by Ulrich Bunke (Göttingen), Sebastian Goette (Regensburg), Kyoshi Igusa (Brandeis) and Thomas Schick (Göttingen), was held from March 12th through March 18th, 2006.

The aim of this meeting was to reflect on the current state of the interaction between mathematical fields of analysis and topology. It brought together representatives of various mathematical communities ranging from homotopy theory, index theory, global analysis up to mathematical physics. The program of the conference has been a mixture of research and overview talks, sometimes with introductory elements. Topics of the latter kind were the connection of Quantum field theory constructions with twisted K-theory, and motivic groups. There were two more informal introductory lectures on stacks in topological and smooth categories, and the insights given by the study of the Brownian motions into Hodge theory.

The research talks reflected recent developments in the corresponding fields and covered a broad area between topology, geometry, and analysis. The intention of the program was to communicate these developments across the borders of mathematical communities. They stimulated discussions and hopefully future collaborations.

More specifically, the subjects covered by research talks included in particular

Altogether, there were 23 “official” talks and a number of informal presentations (partly extended discussions), and ample free interaction between the participants. One of the evenings was devoted to a problem session; the open problems presented are included in this report.

The conference was attended by 52 participants coming mainly from all over Europe, Northern America and East Asia. Among them, a couple of very young pre-doctoral mathematicians, supported by the EU, had the unique opportunity to participate in such a stimulating event at a very early stage of their career. It is a pleasure to thank the institute for providing a pleasant and stimulating atmosphere.
The Rigorous Renormalization Group
Authors: Kopper Christoph , Rivasseau Vincent , Salmhofer Manfred
Published: December 2009

The workshop on \emph{The Rigorous Renormalization Group}, was attended by more than 40 participants coming mainly from Western Europe and from America. The official programme consisted in 19 lectures of 60 minutes each (plus discussion). Four of them were devoted to noncommutative field theory, three of them presented methods used for and results on the construction of a non-gaussian fixed point in a statistical mechanics/quantum field theory model, and two lectures concerned, respectively, nonlinear

\sigma

-models, the functional renormalization group, and quantum electrodynamics. The remaining six lectures were on the Brockett-Wegner version of the renormalization group, on random walks, on Fermi liquids, on anomalies in quantum field theory, on renormalization in curved spaces and on functional integrals for many boson systems. The scientific programme, the atmosphere and the Oberwolfach style of the meeting, leaving much room for informal discussions and joint work, were generally highly appreciated. The abstracts of the lectures are presented in chronological order.
Mini-Workshop: Zeta Functions, Index and Twisted K-Theory; Interactions with Physics
Authors: Paycha Sylvie , Rosenberg Steven , Scott Simon G.
Published: December 2009

This mini-workshop brought together number theorists, analysts, geometers and mathematical physicists to discuss current issues at the common boundary of mathematics and physics. Topics covered included the number theoretic and algebraic structures underlying renormalization, twisted K-theory and higher algebraic structures, modular forms, and arithmetic and spectral zeta functions. A particular theme was around developing interconnections between arithmetic (multiple) zeta functions, spectral zeta functions associated with elliptic operators (and related spectral invariants such as spectral flow) and current issues in physics such as renormalization and mirror symmetry. Multiple zeta functions appear in index theory and

K

-theory via their relation to anomalies, in number theory in their relation to polylogarithms, in renormalization questions in perturbative quantum field theory and Hopf algebras, in duality issues and in twisted

K

-theory for index theorems for projective families of elliptic operators, thereby providing a rich set of overlapping topics with common analytical issues.

This meeting was organized around one hour talks, four each day, with plenty of time between talks for informal discussion and a 45 minute talk in the afternoon for students; three graduate students were among the 16 participants. Some participants lectured for two hours in order to have time to introduce the audience to the subject before entering the technical details.

The organizers and participants would like to thank the
Mathematisches Forschungsinstitut Oberwolfach
for providing a pleasant and stimulating enviroment for this meeting.
Mathematical Biology
Authors: DiBenedetto Emmanuele , Perthame Benoît , Stevens Angela
Published: December 2009

This meeting on Mathematical Biology tied in with the long tradition of these workshops in Oberwolfach and at the same time aimed to account for the fast growing synergy between biology and mathematics of the recent years. The use of new instrumentation and visualization methods at the molecular scale in biological and medical experiments allows for measurements which have not been possible a few years ago. Major questions for theoreticians and experimentalists are how to tackle this vast complexity of biological information and data, and, more important, if underlying principles can be found. Finding these would enable the field to become more predictive. Here is exactly where mathematical modeling, analysis, and simulation can contribute. On the other hand, mathematical biologists and mathematicians are now providing first new models to explain the measurements, and these models are ready for mathematical analysis.

The synergy between mathematics and physics, chemistry, engineering, and material sciences, has already proven to greatly advance the respective sciences and mathematics itself. To further deepen the connections between mathematics and biology, a group of experimental biologists, mathematical biologists, and mathematicians – especially many young scientists – met, joining the lively talks and discussions in this workshop.

The meeting intentionally focussed on some specific biological topics this time. Among those were cell movement, where results on the dynamics of the cellular cytoskeleton were presented, as well as on chemotaxis and cell adhesion. Questions of pattern and structure formation in cell systems were discussed for self-organizing microorganisms and cancer invasion. The analysis of structured population models in this context is new, but has a long tradition in ecology and epidemiology. Further topics of interest with clear mathematical challenges were transport and molecular motors, the organization of cell membranes, and the process of photo-transduction.

Wherever possible, the experimentalists talks were placed in tandem with related presentations on mathematical modeling.

Mathematical topics were: reaction-diffusion equations, parabolic and hyperbolic chemotaxis equations, fluid dynamics, variational principles and methods based on the Wasserstein distance, homogenization, singular perturbations, bifurcation analysis, and numerical simulations.

Besides the lectures, two discussion groups were organized, one on mathematical results for chemotaxis equations and one on cell motility. A round table discussion on ‘mathematical modeling in biology, aims and scopes’ rounded off the meeting, not to forget the nice concert, organized by some of the participants.

We would like to express our sincere thanks to the very dynamic and kind support of the Oberwolfach team before and during the workshop.
Pro-$p$ Extensions of Global Fields and pro-$p$ Groups
Authors: Boston Nigel , Coates John , Grunewald Fritz
Published: December 2009

The meeting

Pro-

p

Extensions of Global Fields and pro-

p

Groups

was organised by Nigel Boston (Madison), John Coates (Cambridge) and Fritz Grunewald (Düsseldorf). As the name of the meeting conveys, a primary aim was to bring together group theorists working in the field of pro-

p

groups and number theorists interested in pro-

p

extensions of global fields. The workshop consisted of over

25

talks, supplemented by informal discussions. Topics included: Galois groups of extensions with restricted ramification, self-similar and automata groups, non-commutative Iwasawa theory, groups acting on rooted trees.

This meeting was well attended with over

50

participants; more than

30

of these came from countries other than Germany. The range of topics and the diverse backgrounds of the participants led to a stimulating exchange of recent results, challenging problems and general ideas. The organisers and participants thank the
Mathematisches Forschungsinstitut Oberwolfach
for providing the setting for this successful workshop. The following extended abstracts appear in the chronological order of the talks; they were collected and edited by Benjamin Klopsch (Düsseldorf).
Applications of Asymptotic Analysis
Authors: Klein Rupert , Sanchez-Palencia Evariste , Sokolowski Jan , Wagner Barbara
Published: December 2009

The workshop
Applications of Asymptotic Analysis
, organised by Rupert Klein (Potsdam), Evariste Sanchez-Palencia (Paris), Jan Sokolowski (Nancy) and Barbara Wagner (Berlin) was held June 18th–June 24th, 2006. This meeting was well attended with 46 participants with a broad geographic representation. This workshop was a nice blend of young and senior researchers with various mathematical backgrounds.

The objective of this workshop was to present the new developments of multiple scale asymptotics as they are developed for problems in various fields of application. It brought together experts working in different areas of asymptotic analysis and application fields and initiated exchange of new ideas and discussions of parallel developments.

On the whole the atmosphere of this workshop was very cheerful and characterized by the mutual interest into each others expertise and approach.

The themes of the workshop included:

Apart from these themes, various other lectures on applications of asymptotic analysis were given, ranging from applications in biology, solid mechanics to fluid mechanics, in particular thin liquid films.

Most of the lectures were given in the morning, with one hour overview lectures, followed by shorter half-hour talks. In the afternoon ample time was left for discussion sessions. Wherever possible the lectures and discussion sessions were grouped according to specific topics, which concerned the interplay of asymptotics and shape optimization (Monday), numerical analysis and asymptotics (Tuesday), mathematical theory and asymptotic methods (Wednesday, Thursday) and some new developments in homogenization theory (Friday).
Quadratic Forms and Linear Algebraic Groups
Authors: Hoffmann Detlev , Merkurjev Alexander , Tignol Jean-Pierre
Published: December 2009

The workshop was organized by Detlev Hoffmann (Nottingham), Alexandr Merkurjev (Los Angeles), and Jean-Pierre Tignol (Louvain-la-Neuve), and was attended by 52 participants. Funding from the Marie Curie Programme of the European Union provided complementary travel support for young researchers, and it also allowed for the invitation of six PhD students in addition to established researchers.

The workshop followed a long and illustrious tradition of Oberwolfach meetings on quadratic forms initiated by M. Knebusch, A. Pfister and W. Scharlau in the 1970's. Initially, the topics ranged from the arithmetic theory of quadratic forms and lattices to real algebraic geometry. In the last decade, however, the algebraic theory of quadratic forms sustained a vigorous development of its own, under the influence of geometric methods, and new connections with linear algebraic groups over arbitrary fields appeared. Recently, striking new results, such as Voevodsky's proof of the Milnor conjecture, were obtained by an infusion of new techniques from motivic cohomology and algebraic topology.

The schedule of the meeting comprised 22 lectures of 45 minutes each, which presented recent progress and interesting new directions in various topics where the algebraic theory of quadratic forms, Galois cohomology, algebraic geometry and the theory of linear algebraic groups mutually stimulate each other, notably Witt groups of triangulated categories, Chow motives of homogeneous varieties, and the essential and canonical dimensions of algebraic groups. Some new connections with representation theory were also discussed.
Combinatorics, Probability and Computing
Authors: Alon Noga , Bollobás Béla , Wegener Ingo
Published: December 2009

The conference was organized by Noga Alon (Tel Aviv), B\'ela Bollob\'as (Cambridge and Memphis) and Ingo Wegener (Dortmund). The programme consisted of 8 main lectures, supplemented by 21 shorter contributions, and covered many areas in Extremal and Probabilistic Combinatorics as well as in Theoretical Computer Science.

In its simplest form, the Probabilistic Method refers to the technique of proving the existence of structures with unexpected properties by showing that a randomly chosen object from an appropriate probability distribution has the properties with positive probability. The probabilistic method has been strikingly successful in Combinatorics, Graph Theory and Combinatorial Number Theory, and the probabilistic point of view has had an enormous influence on theoretical computer science.

The speakers reported on recent developments in Ramsey theory, including variants of these problems for Random Graphs and for Geometric questions, extending classical work in Functional Analysis. Extremal graph and hypergraph theory has also been discussed extensively, combining combinatorial, probabilistic and analytic tools. Additional active topics discussed included Percolation on quasi-random graphs and the properties of random triangulations of point sets in the plane as well as novel results in Complexity Theory including connections between communication complexity and geometric problems, and the study of refutation proofs for random instances of satisfiability.

The aim of the workshop was to focus on the connections and common themes of Combinatorics, Discrete Probability and Theoretical Computer Science. The diversity of the topics and participants stimulated a lot of fruitful discussions between researchers in different and yet related branches and gave rise to interesting discussions and new collaborations between the participants including the younger generation of researchers.

49 scientists, including 36 from countries other than Germany participated in the meeting. The organizers and participants would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing an inspiring setting for this conference.
Mathematical Theory of Water Waves
Authors: Craig Walter , Groves Mark D. , Schneider Guido
Published: December 2009

The 1990s saw vigorous activity in the mathematical theory of water waves by several independent international research groups, and in response we organised a mini-workshop in Oberwolfach in 2001 entitled
Recent Developments in the Mathematical Theory of Water Waves.
The 2001 meeting, which was devoted to the exact equations for water waves as written down by Euler, was a great success. A collection of papers originating at the meeting were published in a special issue of the
Philosophical Transactions of the Royal Society
and lead to significant progress on certain famous and outstanding problems in water waves, particularly local existence and uniqueness, stability, three-dimensional waves, justification of model equations, convexity results for Stokes waves, and effective and accurate numerical schemes.

In view of the interest in water waves generated by the 2001 meeting and our subsequent endeavours, it appeared timely to bring the various research groups together in another Oberwolfach workshop to intensify research in this subject. The following topics were chosen as priority research areas for the workshop since
(i) they represent issues which have been almost fully settled for model equations, but remain almost fully open for the exact water-wave problem; and (ii) pose mathematical challenges whose resolution is likely to prove beneficial in a wide range of situations beyond the water-wave problem:

Significant new results in these areas were reported at the conference and are summarised in the extended abstracts below. The workshop was attended by twenty-three researchers from eight countries; there was a good mix of researchers who had attended the 2001 meeting and those who had not, and a number of younger researchers new to this field attended. Twenty 45-minute talks were held in a friendly and informal atmosphere and, in true Oberwolfach spirit, many collaborative discussions took place.
Mini-workshop: Statistical Methods for Inverse Problems
Authors: Hoffmann Marc , Reiß Markus
Published: December 2009

The mini-workshop on “Statistical methods for inverse problems” gathered seventeen people from the area of statistics and numerical analysis. The goal of the workshop was to stimulate discussions around results and methods that are commonly used by different scientific communities. These methods mainly concern regularization of linear and nonlinear inverse problems in presence of deterministic or stochastic noise.

The week was articulated around three main lectures, divided into two talks each. Yuri Golubev (Université of Marseille) started on Monday–Tuesday mornings to give an extended lecture on inverse problems from a statistical perspective. He was quickly followed by Thorsten Hohage (University of Göttingen) who gave lectures on Monday–Tuesday afternoon on inverse problems from a numerical analysis perspective. A third lecture was given on Wednesday–Thursday morning by Rama Cont (École Polytechnique) who addressed the issue of inverse problems in finance, with both deterministic and probabilistic points of view. The main talks were completed by informal contributed talks of approximately one hour each, at the approximate rate of two talks per day except for the mandatory walk in the Schwarzwald, which was unfortunately canceled due to bad weather conditions that concentrated on the only day of rest!

Round tables were planned to follow the lectures, but the friendly and informal atmosphere soon raised constant questions and discussions among the participants during the lectures so that the workshop progressively moved to informal, yet intense scientific discussions: particular focus was given on optimal tuning parameter choice, stochastic or deterministic error modeling and the complexity of numerical schemes. These discussions took place during several extended talks that skipped the initially planned schedule, fortunately tightened by the strict meal hours of the Oberwolfach center.

The excellent atmosphere of the mini-workshop was made possible thanks to the exceptional working conditions at the MFO to which we would like to express our deep gratitude.
Mini-workshop: Dirac Operators in Differential and Noncommutative Geometry
Authors: Bär Christian , Sitarz Andrzej
Published: December 2009

The mini-workshop, organized by Christian Bär and Andrzej Sitarz, had a very special character. The participating scientists came from two different mathematical communities: differential geometry (working mainly on problems related to the Dirac operator on spin manifolds) and noncommutative geometry (working mainly on concepts of Dirac operators in the framework of spectral geometry as postulated by Alain Connes).

Spin geometry has become an established and very active subfield of Differential Geometry, after Lichnerowicz observed that the Index Theorem yields a topological obstruction against the existence of metrics with positive scalar curvature. The Dirac operator plays a key role in the deep work of Gromov, Lawson, Rosenberg, Stolz and others on manifolds admitting metrics with positive scalar curvature.

The birth of noncommutative geometry offered completely new possibilities for extending some notions of differential geometry into the realm of operator algebras. In Connes' notion of spectral triples the Dirac operator was used to define a (possibly noncommutative) geometry itself rather than being an object derived from a geometry. Since then many interesting examples of noncommutative spaces and Dirac operators were studied. The equivalence theorem, allowing reconstruction of a spin manifold from a spectral geometry of a commutative algebra was proved only recently and the proof was presented at the workshop.

The aim of the workshop was twofold: to show current interests, methods and results within each group and open the possibility for interaction between two groups. Due to the character of the meeting, first three days were devoted to the expository presentations, when we tried to cover the possibly broadest scope of topics from one subject presented for the participants from the other group. The remaining two days were devoted to talks on advanced current research problems and results, which had closer links to the topics of both groups. During problem sessions in the evenings various open questions were discussed some of which were solved during the week.
Classical and Quantum Mechanical Models of Many-Particle Systems
Authors: Arnold Anton , Cercignani Carlo , Desvillettes Laurent
Published: December 2009

The Oberwolfach meeting described here aimed at presenting the latest mathematical results in the field of kinetic theory (both classical and quantum).

There were 50 participants, among which 15 young participants (PhD students, post-docs or young assistant professors). Two of them (M.-P. Gualdani and R. Strain) were invited within the program “US Junior Oberwolfach Fellows”: they are promising young researchers working in the US.

The program of the meeting was made in such a way that a lot of time remained for people to meet informally and discuss about scientific issues. It also ensured that almost everybody attended all (or most of) the scheduled talks .

The program was structured in the following way: three subtopics were defined (relationships between micro/meso/macroscopic models; kinetic theory for complex particles: granular media, coagulation/fragmentation, chemotaxis and sprays; quantum mechanical kinetic theory). For each subtopic, there were a few (2 or 3) longer talks (about 40 minutes) by senior participants. For those talks, a specific effort of clarity was asked to the speakers, and the subject had to be rather broad.

Then, shorter talks of about 20 minutes were planned , on more specialized issues. Finally, a special (much longer) talk in two parts was presented by one of the participants (C. Villani), in order to describe in a very didactical way an emerging link between kinetic theory, optimal transport, and Riemannian geometry.

Some participants did not give a talk within this program, but organized an informal discussion around a poster, with an audience composed of specially interested people.

Here is a brief description of each of the subtopics.

We finally would like to emphasize the ideas and trends which have emerged within the two or three last years and which have been present in this Oberwolfach meeting: a new approach of the relations between the hydrodynamic and non hydrodynamic part of the Boltzmann equation; explicit estimates for the spectrum of the linearized Boltzmann equation when the kernel is singular; the treatment of coagulation/fragmentation equations as infinite-dimensional reaction-diffusion equations with the help of entropy methods; the use of the ideas of kinetic theory in fields coming out from the industry rather than from theoretical physics (supply chains, industrial sprays, etc.).
Mini-Workshop: Lévy Processes and Related Topics in Modelling
Authors: Barndorff-Nielsen Ole E. , Jacod Jean , Klüppelberg Claudia
Published: December 2009

The theory of infinite divisibility, Lévy processes and Lévy bases and its applications is an area of very active current interest. This trend is of relatively new origin, dating back around 10 years. In this period larger research conferences on “Lévy Processes and their Applications” have been held on a regular basis, about every 2 years and at different locations around Europe. In addition intermingled between these conferecnes there has been a number of more specialized workshops within the area. One of these was organized at the Isaac Newton Institute, Cambridge, 2 1/2 years ago, the organizers being Professor Neil Shephard and Ole E. Barndorff-Nielsen, on invitation by the Newton Institute. The format of that workshop where the focus was on new ideas and ongoing research, rather than on finished work, and where the presentations were rather short, in some cases even down to about 15 minutes, was found to be very stimulating and productive. The same format was largely followed at the Mini-Workshop under reporting here and was found again to be very productive. Although the workshop was somewhat specialized a rather wide range of topics were treated in the presentations and in informal discussion groups.

Half of the participants were junior researchers, visiting the MFO for the first, or in some cases the second, time. Like the more senior people they were delighted by the warm atmosphere and excellent conditions for research provided by the Institute. And by lucky coincidence, due to the weather the traditional midweek afternoon excursion was shifted to the Thursday afternoon which was the first day of the Fastnacht celebrations in Oberwolfach with the parade of the “coffee aunts”. This made for a rather unique and refreshing break in the otherwise intense programme.

The first talk of the workshop, by Philip Protter, presented an overview of issues in mathematical finance. The talks by Fred Espen Benth, Jean Jacod, Thilo Meyer-Brandis, Mark Podolskij, Robert Stelzer, and Viktor Todorov discussed modelling and inference on volatility and jumps for financial dynamics. A variety of theoretical results on Lévy processes were presented in talks by Søren Asmussen, Friedrich Hubalek, Andreas Kyprianou, Alexander Lindner and Tina Marquardt, while other talks, by Makoto Maejima, Víctor Pérez-Abreu, Jan Rosiński and Steen Thorbjørnsen concerned work on new aspects of infinite divisibility. Finally, the talk by Anna Amirdjanova treated problems in the theory of Gaussian processes.
Analysis and Numerics for Rate-Independent Processes
Authors: Dal Maso Gianni , Francfort Gilles A. , Mielke Alexander , Roubíček Tomáš
Published: December 2009

This conference brought together the mathematicians and engineers concerned with the study of rate-independent processes. The topics involved the physical and mechanical modeling, the analytical treatment as well as the development of numerical algorithms and the computational aspects.

Rate-independent processes occur when a system undergoes loadings varying on a time scale much slower than the internal relaxation times of the system. However, the system still stays in metastable states and does not find its thermodynamical equilibrium because of internal frictional processes. There are many well-established macroscopic theories, which show good agreement with experiments, for instance the classical theory of linearized elastoplasticity. This theory has been established a century ago, but its mathematical foundations were only understood 30 years ago using the theory of convex analysis and variational inequalities. Another classical driving theme in this area is the theory of Coulomb friction between rigid or elastic bodies.

Within the last decade similar models have been developed in several areas of mechanics of solids, some of them on a phenomenological basis and others relying on an improved understanding of the microstructures. Such areas include multifunctional or active materials (e.g., shape-memory alloys, ferromagnetism, magneto- and electro-strictivity) and inelastic processes in solids (e.g., elastoplasticity, damage, delamination, fatigue, brittle fracture, crack growth).

Although many of these models are used in engineering, their mathematical understanding is insufficient and efficient numerical algorithms for the computation of approximate solutions are few. The main problem is the inherent nonsmoothness of the modeling because of rate independence.

We discussed a new approach based on an energetic formulation, which provides a broad basis for treating quite general models using topological, geometrical and measure theoretical methods that do not require a Banach space structure. It was shown in various presentations that this approach allows one to go beyond linearized elastoplasticity and to treat fully nonlinear mechanical problems, like crack formation and growth, or elastoplasticity with softening.

Finally, the modeling of the formation and evolution of microstructures was debated in several talks and gave rise to a better understanding of the physical and mathematical underlying principles.

The specific methods discussed were:

The specific applications discussed were:

The continuous downpour of Black Forest Rain created a cosy atmosphere that fostered the interaction between the different mathematical and engineering communities.
Mini-Workshop: Shape Analysis for Eigenvalues
Authors: Bucur Dorin , Buttazzo Giuseppe , Henrot Antoine
Published: December 2009

The mini workshop
Shape analysis for eigenvalues
, organised by Dorin Bucur (Chambéry), Giuseppe Buttazzo (Pisa) and Antoine Henrot (Nancy) was held April 8th–April 14th, 2007. This meeting was attended by 18 participants.

The question of localizing or optimizing the eigenvalues differential operators has applications in several domains like acoustics, quantum mechanics, visualization, solid or fluid mechanics and bio-mathematics. Those questions have also a specific mathematical interest since they melt geometrical questions with analysis of partial differential equations and provide model problems for general and applied shape optimization problems.

Despite (or because) their false simplicity, several problems are still open, although formulated hundreds year ago (see for instance the recent books by Bucur & Buttazzo
Variational Methods in Shape Optimization Problems
, Birkhäuser-Verlag, 2005 and Henrot
Extremum problems for eigenvalues of elliptic operators
, Birkhäuser, 2006) and give rise to important debates within the international scientific community. Among them, we can cite

Each of these problems is associated to one or several constraints for the geometries, which may be local or not local (convexity, connectedness).

One of the goals of the proposed mini-workshop was to bring together two mathematical communities working on the topics above by quite different methods. On the one hand explicit solutions to optimal shape problems for eigenvalues are searched by means of direct estimations, symmetrizations, rearrangements; on the other hand recent techniques of variational type have been developed to prove the existence of an optimal shape and intensive research is carried out to prove the regularity of the free boundary and to analyze the optimality conditions.

A second main issue was to present the state of the art on some famous open problems and conjectures, and try to make a step forward in the direction of solving them.

The organization of the mini-workshop was the following: mornings were devoted to lectures by participants while each afternoon began by an open problems session and smaller working groups (4-5 persons) were organized to work on these problems.
Poisson Geometry and Applications
Authors: Alekseev Anton , Loja Fernandes Rui , Meinrenken Eckhard , Pflaum Markus
Published: December 2009

The half-size workshop was organized by Anton Alekseev (Geneve), Rui Loja Fernandes (Lisboa), Eckhard Meinrenken (Toronto) and Markus Pflaum (Frankfurt). Marius Crainic (Utrecht) also acted as an unofficial organizer. The programme consisted of 17 lectures and covered a range of areas in Poisson Geometry and its applications where significant progress has been achieved recently. The aim of the workshop was to emphasize the main themes in Poisson Geometry that play the role of driving forces and organizing principles of the field. A significant number of young researchers, who have made already important contributions to the field, participated in this meeting. During the workshop all participants were involved in a great number of informal discussions, some of which gave rise to new collaborations. In total, 27 researchers have participated in this meeting from institutions in 9 different countries in Europe, USA and Canada, including 4 researchers from German institutions. The organizers and participants thank the Mathematisches Forschungsinstitut Oberwolfach for providing a truly inspiring atmosphere for this conference. In the following we include the abstracts in alphabetical order.