bluuebunny/crossref_metadata_2025 huggingface dataset:
Mini-Workshop: Multiscale and Variational Methods in Material Science and Quantum Theory of Solids
Authors: Catto Isabelle , Chenchiah Isaac Vikram , Veselić Ivan , Zimmer Johannes
Published: December 2009

This workshop brought together people working in probability theory, multiscale analysis, calculus of variations and spectral theory. The purpose was to stimulate exchanges between the respective communities, and identify open problems at the boundaries and intersections of these areas.

There are several mathematical links between the theory of random Schrödinger operators, quantum mechanics of interacting atoms, and mathematical materials science. For instance, spectral analysis lies at the heart of the study of random Schrödinger operators; it is used in the quantum mechanics of interacting atoms as a natural and important tool for the analysis of Euler–Lagrange equations and their solutions; it was shown in the workshop that spectral analysis plays an important rôle in the study of photonic crystal fibres, e.g., in the study of band gaps.

Other tools of scale-bridging were discussed including homogenisation and the computation of the thermodynamic limits of various classical and quantum systems.

Several expository morning lectures explained the fundaments of these fields to the varied audience, particularly to those in the other communities. These lectures were intended to last 60– 75 minutes but almost always lasted 90 minutes due to lively discussions, both during and after the talks. They were complemented by specialised talks in the afternoon.

The organisers feel that this attempt to bring these different communities together was largely successful, and may have built new bridges between analysis and probability; the credit for this belonging, for the most part, to the speakers. That mini-workshops on “Levy Processes and Related Topics in Modelling” and “Control of Free Boundaries” were being held in parallel afforded many opportunities for further interactions.

There was roughly the same number of analysts and probabilitists among the participants. France, Germany, the UK and the US were about equally represented; there was one participant from Denmark. Almost all participants were relatively early in their careers.

The excellent living and working conditions provided by the institute contributed to a lively scientific atmosphere. The organisers thank the NSF for funding the participation of the three speakers from the US, Patrick Dondl for collecting the extended abstracts (and the “kleine Trommler” for livening up the excursion!).
Algebraic Groups
Authors: Brion Michel , Jantzen Jens Carsten , Rouquier Raphaël
Published: December 2009

The workshop continued a series of Oberwolfach meetings on algebraic groups, started in 1971 by Tonny Springer and Jacques Tits who both attended the present conference. This time, the organizers were Michel Brion, Jens Carsten Jantzen, and Raphaël Rouquier.

During the last years, the subject of algebraic groups (in a broad sense) has seen important developments in several directions, also related to representation theory and algebraic geometry. The workshop aimed at presenting some of these developments in order to make them accessible to a “general audience” of algebraic group-theorists, and to stimulate contacts between participants.

Each of the first four days was dedicated to one area of research that has recently seen decisive progress:

The first three days started with survey talks that will help to make the subject accessible to the next generation. The talks on the last day introduced to several recent advances in different areas: arithmetic groups, eigenvalue problems, counting orbits over finite fields, quivers and reflection functors. In order to leave enough time for fruitful discussions, the number of talks (generally of one hour) was limited to four per day.

Besides the scientific program, the participants enjoyed a piano recital on Thursday evening, by Harry Tamvakis.

The workshop was attended by 53 participants, coming mainly from Europe and North America. This includes 6 PhD students, supported by the Marie Curie program of the European Union. The organizers are grateful to the EU for this support, and to the MFO for providing excellent working conditions.
Arithmetic and Differential Galois Groups
Authors: Harbater David , Matzat B. Heinrich , van der Put Marius , Schneps Leila
Published: December 2009

Galois theory is the study of symmetries in solution spaces of polynomial and differential equations and more generally of the relation between automorphism groups (or group schemes) and the structure of algebraic and differential extensions.

Many of the important problems in this area are connected to the classification of (algebraic or differential) Galois extensions and the study of the respective fundamental groups, e.g. via inverse problems. Other interesting points are the direct problem, i.e., the computation of (ordinary or differential) Galois groups, and related constructive aspects.

This workshop gave an overview on some important new results and developments. A second goal was the discussion of ideas for proving some open questions and conjectures as well as of new directions of research.

The main topics of the workshop were:

Besides this main program we had some reports on related subjects like the Cohen-Lenstra heuristics for number fields by J. Klüners and the behaviour of the Tate–Shafarevich group in anticyclotomic field extensions by M. Çiperiani.

It is always difficult to emphasise highlights or unexpected results, but one of them might be the result of F. Pop that henselian fields always are large, which is of great interest in field arithmetic and inverse Galois theory. Another striking result is the existence of

G_2

-motives over

\mathbb{Q}

shown by M. Dettweiler which gives a positive answer to an old question of J.-P. Serre. A fruit of common effort in research in arithmetic and differential Galois theory is the development of patching methods for differential modules by D. Harbater and J. Hartmann. This method will allow yet unexpected applications in inverse differential Galois theory and other areas, like K-theory. The talk of B. Malgrange gave a vision of Galois theory for nonlinear differential equations. Exploitation of his work for special types of equations, like Painlevé equations, and the generalization of his ideas to non algebraically closed fields of constants as well as to positive characteristic will keep researchers busy for years.

Surely, many other results presented at the workshop should be pointed out here, too, like M. Raynaud's work on fundamental groups in positive characteristic, D. Bertrand's result on Schanuel's conjecture or Ch. Hardouin's generalization of

q

-difference equations to roots of unity by creating an iterative

q

-difference theory.

Altogether, we had a wonderful and inspiring week with lots of interesting lectures and many discussions bearing ideas for future research.

Finally, the organisers want to cordially thank the Oberwolfach administration and its staff for giving us the opportunity to arrange this and earlier workshops on Galois theory as well as for the excellent service.
Partielle Differentialgleichungen
Authors: Ilmanen Thomas , Schätzle Reiner , Trudinger Neil S.
Published: December 2009

The workshop
differential equations
, organised by Tom Ilmanen (ETH Zürich), Reiner Schätzle (Universität Tübingen) and Neil Trudinger (Australian National University Canberra) was held July 22–28, 2007. This meeting was well attended by 50 participants, including 6 females, with broad geographic reprensentation. The program consisted of 19 talks and 3 shorter contributions and left sufficient time for discussions.

One focus was the combination of nonlinear partial differential equations and geometric problems. There were talks on monotonicity formulas in presence of degenerate singularities and on estimates of the singular set for branched minimal immersions. One striking new result was the proof of the diffeomorphic sphere theorem by the use of Ricci flow.

Many leading experts in the regularity of elliptic and variational problems and in fully nonlinear differential equations occuring in optimal transport and in Hessian equations attended the workshop. Several main contributions were given to this subject. Here, we mention the proof of interior estimates for quadratic Hessian equations in three dimensions, which has been an open problem for several years of similar relevance as the gradient estimates for the minimal surface equation.

New results were also presented for the Willmore functional. It was achieved to write the Euler-Lagrange equation in divergence form, and the existence of minimizers in fixed conformal classes with certain energy bounds were obtained.

The organisers and the participants are grateful to the Oberwolfach Institute for presenting the opportunity and the resources to arrange this interesting meeting.
Mini-Workshop: Topology of closed one-forms and Cohomology Jumping Loci
Authors: Farber Michael , Suciu Alexander I. , Yuzvinsky Sergey
Published: December 2009

This Mini-Workshop was organized by M. Farber (Durham), A. Suciu (Boston) and S. Yuzvinsky (Eugene). It brought together researchers working on two distinct, yet related topics:

Even though these two fields share some common roots, so far they have developed in parallel, with not much overlap or interaction. Nevertheless, it is becoming increasingly apparent that there are deep connections between the two theories, with potentially fruitful applications going both ways:

Given the multifaceted nature of these topics, the meeting brought together people with a variety of backgrounds, including topology, algebra, discrete geometry, geometric analysis, and singularity theory. Several participants were recent Ph.D.'s, most of them on their first visit to Oberwolfach. In all, there were 16 people attending the workshop (including the organizers), coming from the United States, Great Britain, France, Romania, Canada, and Germany.

The Mini-Workshop provided a lively forum for discussing a host of questions related to the themes listed above. The day-by-day schedule was kept flexible, and was agreed upon on short notice, making it possible to shape the program on-site, and in response to the interests expressed by the participants. The borderline between problem sessions and formal lectures were often blurred. Spending a concentrated and highly intense week in a relatively small group allowed for in-depth and continuing conversations, in particular with new acquaintances. These opportunities were enhanced by the diversity of backgrounds of the participants.

A basic objective of the Mini-Workshop was to bring together some of the people most actively working in two related fields, and to seek common ground for further advances and collaborations. In the ideally suited research atmosphere at Oberwolfach, participants had the opportunity to explain their respective approaches, and the variety of techniques they use. The lively atmosphere and the free-flow of ideas led to a deeper understanding of the subject, to progress in solving several open problems, and to fruitful insights on how to attack new problems.
Nonlinear Waves and Dispersive Equations
Authors: Kenig Carlos E. , Koch Herbert , Tataru Daniel
Published: December 2009

Dispersive equations occur as asymptotic models for the propagation of linear and nonlinear waves. Mathematically they display an interplay between linear dispersion and nonlinear focusing and defocusing effects. They are linked to diverse areas of mathematics and physics, ranging from nonlinear optics over oscillatory integrals to integrable systems. The workshop focused on a PDE based approach to dispersive equations.

Current activities covered by the workshop include:

There is a large number of promising young mathematicians working in this area. The meeting was attented by 45 participants. The organizers gave a strong preference to talks by young researchers.
Coagulation and Fragmentation Models
Authors: Bertoin Jean , Norris James R. , Wagner Wolfgang
Published: December 2009

Historically deterministic equations were used for modelling the dynamics of the density of clusters of different sizes in a medium that evolves by coagulation and/or fragmentation. An alternative stochastic approach uses the distribution of a typical particle in systems undergoing a random evolution. This approach provides tools for deriving limit equations, it easily extends to more general interactions and, finally, it shares the familiar advantages of particle-based methods for computing distributions in high dimensions. The stochastic theory is undergoing intense development, both at a theoretical level, in the construction of models and analysis of their properties, and at a computational level. Major mathematical challenges in the field of coagulation and fragmentation models are related to phase transitions due to fast coagulation (gelation) or fast fragmentation (formation of dust), and to explicit descriptions of the structure of partitions (e.g. allelic partitions induced by random mutations in models for the evolution of populations). A further direction of intensive study is the detailed analysis of particular coagulation and fragmentation processes, where special features (e.g., scaling properties) allow links to be made to Brownian motion and other Lévy processes, also to some interesting problems in random combinatorics.

Major topics that have been discussed during the workshop include :

1. Spatial models of coagulation and fragmentation

Models incorporating diffusion; scaling limits; derivation from particle dynamics; equilibrium measures for coagulation-fragmentation processes; formation of structured particles; diffusion limited aggregation

2. Phase transitions in coagulation and fragmentation models

Gelation effects; shattering transition (appearance of dust in fragmentation); explosion phenomena; computation of the gelation time; uniqueness issues; convergence of particle systems

3. Aspects of random combinatorics

Links to random graphs and trees; genealogy for certain large populations dynamics; mutation and allelic partitions; stochastic coalescents with multiple collisions (

\Lambda

-coalescents) and fragmentations

4. Computational issues

Approximation and numerics; analysis of algorithms; Monte Carlo issues around sensitivity in initial conditions and kernel parameters
Combinatorics
Authors: Kahn Jeff , Lovász László , Prömel Hans Jürgen
Published: December 2009

The workshop
Combinatorics
organised by Jeff Kahn (Piscataway), László Lovász (Budapest), and Hans Jürgen Prömel (Berlin) was held January 6st–January 12th, 2008. This meeting was very well attended with 46 participants from many different countries. The programme consisted of 15 plenary lectures, accompanied by 18 shorter contributions and a vivid problem session led by Vera T. Sós.

The conference is a workshop on Combinatorics in a very broad sense, and is in part intended to serve as a framework for all other Oberwolfach meetings that focus on particular areas within combinatorics. This meeting over the years has been extremely successful in achieving the sometimes elusive goal of bringing together and fostering interactions among people with a wide range of interests, and we feel that the program this time was close to ideal in its coverage of a large fraction of the most exciting recent developments across the combinatorial spectrum. Quite a few of the talks – for instance those of Chudnovsky, Kühn, Osthus, Ruciński, Sudakov, Taraz and Vu – reported major progress on well-known problems. Some of the plenary speakers were asked to give overviews of areas somewhat distant from the central interests of many of the participants.

The breadth of the conference makes its contents nearly impossible to summarize. One might say that the central foci of the meeting were extremal and probabilistic aspects of combinatorics, and graph theory; but “extremal combinatorics” is an extremely broad term, and a glance at the titles below shows that the list of topics is not much shorter than the list of talks. Thus, even among the talks that could be considered to fall in the above categories, one finds geometry, Fourier analysis, algebra, physics, connections with social sciences, and multiple connections with computer science and related technology.

Again, we consider this breadth to be not a drawback, but a central feature of the meeting. It has promoted an inspiring, interactive atmosphere, and led to fruitful discussions and collaborations, to new awareness of what's happening in different parts of combinatorics, and to the discovery of some unexpected connections.

More than in past meetings, an emphasis was placed on talks (both plenary and shorter) by younger researchers. This too worked very well, and we hope to make it the pattern for future meetings.

On behalf of all participants, the organisers would like to thank the staff and the director of the
Mathematisches Forschungsinstitut Oberwolfach
for providing a stimulating and inspiring atmosphere.
Optimal Control of Coupled Systems of PDE
Authors: Kunisch Karl , Leugering Günter , Sprekels Jürgen , Tröltzsch Fredi
Published: December 2009

The international conference
Optimal Control of Coupled Systems of PDE
, was held March 2nd–March 8th, 2008, organized by K. Kunisch (Karl-Franzens-University Graz), G. Leugering (University of Erlangen-Nürnberg), J. Sprekels (Weierstrass Institute of Applied Analysis and Stochastics Berlin) and Fredi Tröltzsch (Technische Universität Berlin). 44 participants attended this meeting and followed 33 talks on optimal control and related topics.

Mathematically, the control of partial differential equations (PDEs) is concerned with the following type of problems: The solution of a PDE (the state of the system) should be influenced in a desired way by the choice of certain control functions or control parameters (the controls), which may occur in different terms of the differential equation. If the controls are to minimize a certain functional related to the state, then an
optimal control problem
is posed. If the domain underlying the PDE is subject of the control, then a
shape optimization problem
is given. For evolution equations, it can be required to move the solution from a given initial state exactly to a desired final state. This is the question of exact
controllability
.

Optimization and control of partial differential equations continues to be a very active field of research. Scientists working in different fields came together to report on their contributions to the numerical analysis of control problems. It is remarkable that optimal control is a challenge for researchers with backgrounds in related fields such as the theory of systems of nonlinear PDEs, numerical methods for solving them, large scale nonlinear optimization, or the numerical simulation and optimization of complex processes in engineering or medical science.

This diversity was reflected by the conference program. Talks were focused on

All these issues are currently subject of active research. In extensive and lively discussions, the participants of the workshop produced new mathematical ideas and tightened connections of joint cooperation.
Random Schrödinger Operators and Random Matrices
Authors: Götze Friedrich , Kirsch Werner , Klopp Frédéric , Kriecherbauer Thomas
Published: December 2009

It is natural to model the spectral behaviour of disordered quantum systems by the typical spectral properties of random operators. Depending on the underlying physics different kinds of ensembles have been introduced. Two types of ensembles, random Schrödinger operators and random matrices, have proved to be of particular interest because of their wide ranges of applicability and because of their rich mathematical structures. On first sight these two types of random operators may appear to be close relatives. However, as it turns out, their typical spectral properties differ significantly. Moreover, the methods that have been developed for their respective analysis have little in common. One may say that over the years two different cultures have evolved leading to two almost disjoint mathematical communities. It was the goal of this workshop to stimulate exchange between these two communities by highlighting important recent developments in both areas.

The workshop brought together 44 researchers from 9 different countries. This report contains the extended abstracts of the 30 lectures that were delivered during the meeting.

The transition from ordered to disordered system was the main topic of four lectures.

There were 14 talks about random Schrödinger operators. One of the major topics was the theory of Anderson localization/delocalization. The density of states, especially the investigation of Lifshitz tails was the subject of several contributions. The third topic in this field concerns the theory of level statistics which is a very active field at the moment. This area is much inspired by the theory of random matrices.

Twelve of the lectures were related to the theory of random matrices. In these talks a variety of ensembles, results, open problems, and methods of proof were discussed that yield a somewhat representative picture of the current state of the art in this field. The ensembles considered include the two classical cases of Wigner ensembles (independent entries) and of invariant ensembles (invariant under appropriate changes of basis). Some of the lectures dealt with more general classes of ensembles that are motivated by applications in physics and statistics. Most of the results that were presented are concerned with local correlations of eigenvalues, with the distribution of the largest eigenvalue and with (central) limit theorems for spectral quantities. One talk was devoted to the Asymmetric Simple Exclusion Process, generalizing a much celebrated result that provided a connection to random matrix theory.

Finally, it is our happy task to thank all the participants for the lively discussions, the staff of Oberwolfach for providing such perfect and pleasant working conditions and Bernd Metzger for collecting and editing the extended abstracts.
History of Mathematics of the Early 20th Century: The Role of Transition
Authors: Corry Leo , Fenster Della , Schwermer Joachim
Published: December 2009

This conference provided a focused venue to investigate the history of mathematics during a particularly active time in the discipline, that is, roughly between the turn of the 20th century and 1950. Using the lens of transition to explore this vibrant period, the organizers brought together mathematicians, historians of mathematics and historians of science to explore ideas and offer insights from different perspectives. With this wide range of scholars in attendance, speakers had to give careful thought to the presentation of their work. This extra effort not only yielded a sterling set of talks but also inspired scholars to rethink their own work.

The restricted time period revealed an almost unexpected richness in the history of mathematics as the conference participants observed and discussed points of connection between the people, places and ideas from fields as seemingly diverse as class field theory, mathematical physics and algebraic geometry, among others. The extended abstracts below reflect the critical roles of community, politics, and institutions in the history of mathematics at this time. They also call attention to interesting questions related to the collective actions involved in transitions and the “stable” places in between.

The organizers placed an especial emphasis on the presence of and contributions by young scholars. In particular, the conference schedule included almost a full day of presentations by graduate students and recent recipients of the Ph.D. Their fresh perspective fostered a vibrant spirit during the meeting.

One of the first-time participants to Oberwolfach remarked that “this Institute has everything you need.” This “everything you need” includes the intangible components of Oberwolfach that give the most meaning to the experience. The conference participants especially appreciated the interesting mix of people and talks we enjoyed during our week at the MFO. While the idea of transition may have brought us together initially, it is the continued “mathematical talking” that will create further conversations and investigations.
Classical Algebraic Geometry
Authors: Eisenbud David , Harris Joseph , Schreyer Frank-Olaf , Vakil Ravi
Published: December 2009

The Workshop on Classical Algebraic Geometry, organized by David Eisenbud (Berkeley), Joe Harris (Harvard), Frank-Olaf Schreyer (Saarbrücken) and Ravi Vakil (Stanford), was held June 8th to June 14th. It was attended by about 45 participants from USA, Canada, Japan, Norway, Sweden, UK, Italy, France and Germany, among of them a large number of strong young mathematicians. There were 18 talks with a maximum of four talks per day, which left plenty of time for lots of informal discussions among smaller groups.

A widespread variety of themes in classical algebraic geometry were discussed, many of them with a very modern point of view. Although it is tempting to summarize each of the talks, we limit ourselves to four highlights:
Mini-Workshop: Mathematical Approaches to Collective Phenomena in Large Quantum Systems
Authors: Adams Stefan , Biskup Marek , Seiringer Robert
Published: December 2009

The workshop
Mathematical Approaches to Collective Phenomena in Large Quantum Systems
, organised by Stefan Adams (Warwick) and Robert Seiringer (Princeton) was held August 31st–September 6th, 2008. This meeting was well attended with 17 participants with broad geographic representation from all continents and with a good mixture of younger Postdocs/researchers (Boland, Bru, Crawford, Schlein, and Starr) and senior researchers. This workshop was a nice blend of researchers with various backgrounds ranging from analysis, probability theory, and functional integration. The fifteen talks are focused along the three general research directions developing mathematical theories of quantum phase transitions:

(1) Dilute gas limit and nonlinear effective theories

(2) Random walk representations via Feynman–Kac formula

(3) Field theory/semiclassical representations via coherent states

Area (1) is represented with new developments in talks by Lieb and Solovej focused on the Bogoliubov theory to derive second order correction to the ground state energy and in a talk by Yngvason on rotating trapped Bose gases. New developments for quantum spin systems have been reported by Nachtergaele and Starr. A new probabilistic approach has been introduced in a talk by König which shows a modelling of the free energy in the thermodynamic limit, and future will show if this approach goes beyond the existing theory on finite temperature Bose gases. An interesting area has been summarised by Ueltschi on random permutations. This area is recently under much focus in the probability community. The contribution from the youngest participant, the PhD-student Boland, showed an interesting result concerning the Bose condensate density and the density of so-called long cycles. Long cycles appear if probability mass of finite cycles gets lost in the thermodynamic limit, an area which is studied in the combinatorics community via shape measure analysis.

The schedule of the workshop allowed intensive discussion during the afternoons including Friday. The workshop has been closed with an informal discussion/meeting on Friday evening which was attended by the majority of the participants. The workshop showed interesting new developments, and it is the hope of the organisers that there will be more interdisciplinary collaborations, in particular between the analysis and the probability community.
New Perspectives in Stochastic Geometry
Authors: Kendall Wilfried S. , Last Günter , Molchanov Ilya S.
Published: December 2009

The principal aim of stochastic geometry is the mathematical analysis of random geometric structures. Fundamental examples of such structures are point processes of geometric objects, random tessellations of space into convex or non-convex regions, random systems of non-overlapping balls (or more general convex bodies), or excursion and level sets of Gaussian random fields. The workshop was devoted to the discussion and exploration of recent advances in stochastic geometry and its related areas. Among the 23 participants were many leading figures in the field as well as some very promising young scientists.

One main theme of the workshop concerned new results and methodology in classical stochastic geometry. In recent years it has become possible not only to make conclusions about mean values of geometric quantities, but also to derive distributional properties, to prove limit theorems and large deviation results, and to explore higher-order moment properties. Examples include: Hug, Reitzner (approximation of polytopes), Calka (visibility properties of the Boolean model), Schreiber (polygonal Markov fields), Penrose (normal approximation in random geometric graphs), Yukich (limit theorems), Reitzner (random tessellations), Baccelli (point processes and information theory).

A second main theme of the workshop concerned allocation and matching procedures for point processes and random measures.The study of such allocations has resulted in some remarkable progress in the understanding of invariance properties of Palm probability measures and associated transport and coupling questions.There are deep and striking relationships with random tessellations, classical optimal transport theory, potential theory, and complex analysis. Examples include: Peres (fair allocations, optimal matchings, gravitational allocation, ...), Sturm (optimal transportation of measures on metric spaces), Thorisson (transport kernels and mass-stationarity), Last (Cox and Bernoulli transports).

Three series of lectures formed the organizational backbone of the workshop. They were delivered by Matthias Reitzner on "Recent results in stochastic geometry", by Yuval Peres on "Fair allocations" and by Theo Sturm on "New trends in optimal transportation on Riemannian and singular spaces". Many of the contributed talks were related to one or more issues covered by these lectures. Another integral part of the programme was provided by three discussion sessions "Probability models with geometric flavour" organized by Jesper Møller and Sergei Zouev, "Asymptotics in stochastic geometry" organized by Peter Mörters and Mathew Penrose, and "From convex to metric and fractal geometry" organized by Martina Zähle and Wolfgang Weil. These sessions provided space for participants to bring up new ideas and discuss open problems in an informal manner.

Many participants of the workshop co-authored a collection of papers "New Perspectives in Stochastic Geometry" edited by W.S. Kendall and I. Molchanov that will be published by the Oxford University Press in 2009.
Trends and Developments in Complex Dynamics
Authors: Lunde Petersen Carsten , Lyubich Mikhail , Schleicher Dierk , Smillie John
Published: December 2009

Holomorphic dynamics is a thriving field which has experienced tremendous progress over the last 25 years, involving mathematicians such as Douady, Hubbard, McMullen, Milnor, Sullivan, Thurston, and Yoccoz. Holomorphic dynamics is gifted with the tools of conformal and hyperbolic geometry that allow a deep penetration into its nature, with many further applications to real dynamics.

Conformal dynamics is currently in a phase of transition. Previously, most of the successful and deep work has been focused on one-parameter model families: families of quadratic or unimodal polynomials and families of meromorphic transcendental functions with one singular orbit. The theory is particularly advanced in the case of the quadratic family; in fact, a complete measure-theoretic picture of the dynamics was obtained in the real quadratic family (Lyubich, Avila, de Melo, Moreira, and others).

During the past two years, significant breakthroughs have appeared in two important directions: rigidity in polynomial dynamics, and Julia sets of positive measure. These results answer major long-standing open questions and simultaneously open up new perspectives. Both problems have served as landmarks in the field, and both have a long history going back to Fatou (and more recently, to Ahlfors, Smale and Mostow). The recent results have brought us very close to a full understanding of the model families, and in particular of the real unimodal dynamics.

Time is now ripe for holomorphic dynamics to move on from the study of the special model cases to a general theory of holomorphic dynamics, delivering on the promise that the model cases have impact in greater generality. Holomorphic dynamics is a vast area of research giving impetus to this transition from many directions. Given the time and number of participants we can not cover all of them within one workshop. We have thus chosen to focus on 5 such aspects of holomorphic dynamics with recent and substantial progress.

The organising principle of the workshop is to present the state of the art of each aspect, to evaluate the progress achieved and to formulate questions which may guide the transition.
Toric Geometry
Authors: Altmann Klaus , Batyrev Victor V. , Karshon Yael
Published: December 2009

Toric Geometry originated from investigations of torus actions on geometric and algebraic objects. It is addressed through algebraic geometry, symplectic geometry, equivariant topology, as well as the theory of convex polyhedra within discrete mathematics. In spite of using their own language these completely diﬀerent disciplines often observe similar or even identical combinatorial phenomena. Thus toric geometry leads to a fascinating and fruitful interplay between these disciplines.
Low Eigenvalues of Laplace and Schrödinger Operators
Authors: Ashbaugh Mark , Benguria Rafael , Laugesen Richard , Weidl Timo
Published: December 2009

This workshop brought together researchers interested in eigenvalue problems for Laplace and Schrödinger operators. The main topics o of discussions and investigations covered Dirichlet and Neumann eigenvalue problems, inequalities for the spectral gap, isoperimertic problems and sharp Lieb–Thirring type inequalities. The focus included not only the analytic and geometric sides of the problems, but also related probabilistic and computational aspects.
Mini-Workshop: The Pisot Conjecture — From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets
Authors: Berthe Valerie , Damanik David , Lenz Daniel
Published: December 2009

This mini-workshop brought together researchers with diverse backgrounds and a common interest in facets of the Pisot conjecture, which relates certain properties of a substitution to dynamical properties of the associated subshift.
Mini-Workshop: Non-Negativity is a Quantum Phenomenon
Authors: Launois Stéphane , Lenagan Thomas H.
Published: December 2009

In recent publications, the same combinatorial description has arisen for three separate objects of interest: non-negative cells in the real grassmannian (Postnikov, Williams); torus orbits of symplectic leaves in the classical grassmannian (Brown, Goodearl and Yakimov); and, torus invariant prime ideals in the quantum grassmannian (Launois, Lenagan and Rigal). The aim of this meeting was to explore the reasons for this coincidence in matrices and the grassmannian in particular, and to explore similar ideas in more general settings.
Sparse Recovery Problems in High Dimensions: Statistical Inference and Learning Theory
Authors: Bartlett Peter L. , Koltchinskii Vladimir , Tsybakov Alexandre B. , van de Geer Sara
Published: December 2009

The statistical analysis of high dimensional data requires new techniques, extending results from nonparametric statistics, analysis, probability, approximation theory, and theoretical computer science. The main problem is how to unveil, (or to mimic performance of) sparse models for the data. Sparsity is generally meant in terms of the number of variables included, but may also be described in terms of smoothness, entropy, or geometric structures. A key objective is to adapt to unknown sparsity, yet keeping computational feasibility.
Arbeitsgemeinschaft: Optimal Transport and Geometry
Authors: Otto Felix , Sturm Karl-Theodor
Published: December 2009

This Arbeitsgemeinschaft was devoted to recent developments in optimal transport with emphasis on links and applications to geometry. The topics reached from the origin of optimal transport as a variational problem, where one minimizes a transportation cost when transporting one density into another, over the introduction of a metric on the space of probability measures, leading to the Wasserstein-space and convex funtionals on it, to the connection between Ricci-curvature and the optimal mass transport problem.
Homotopy Theory of Function Spaces and Related Topics
Authors: Félix Yves , Lupton Gregory M. , Smith Samuel Bruce
Published: December 2009

This workshop brought together researchers studying a variety of problems related to the homotopy theory of function spaces. Topics covered included: evaluation maps and Gottlieb groups, the classiﬁcation of gauge groups and of other function space components, algebraic models for function spaces both in the rational and in the p-local settings, operads, conﬁguration spaces, free and based loop spaces and inﬁnite-dimensional Lie groups.
Multiplier Ideal Sheaves in Algebraic and Complex Geometry
Authors: Kebekus Stefan , Păun Mihai , Schumacher Georg , Siu Yum-Tong
Published: December 2009

The workshop
Multiplier Ideal Sheaves in Algebraic and Complex Geometry
, organised by Stefan Kebekus (Freiburg), Mihai Paun (Nancy), Georg Schumacher (Marburg) and Yum-Tong Siu (Cambridge MA) was held April 12th – April 18th, 2009. Since the previous Oberwolfach conference in 2004, there have been important new developments and results, both in the analytic and algebraic area, e.g. in the ﬁeld of the extension of _L_2 holomorphic functions, the solution of the ACC conjecture, log-canonical rings, the Kähler–Ricci ﬂow, Seshadri constants and the analogues of mula tiplier ideals in positive characteristic.
Algebraic $K$-Theory and Motivic Cohomology
Authors: Geisser Thomas , Huber-Klawitter Annette , Jannsen Uwe , Levine Marc
Published: December 2009

Algebraic

K

-theory and the related motivic cohomology are a systematic way of producing invariants for algebraic or geometric structures. Its deﬁnition and methods are taken from algebraic topology, but it has also proved particularly fruitful for problems of algebraic geometry, number theory or quadratic forms. 19 one-hour talks presented a wide range of results on

K

-theory itself and applications. We had a lively evening session trading questions and discussing open problems.
Local energy decay for the nonlinear dissipative wave equation in an exterior domain
Authors: Daoulatli Moez
Published: December 2009

In odd dimension space, under a microlocal geometric condition, we give the rate of decay of the local energy for solutions of the wave equation on exterior domain, with localized nonlinear damping.
Totally umbilical semi-invariant submanifolds of a nearly trans-Sasakian manifold
Authors: Khan V. A. , Khan M. A. , Uddin S.
Published: December 2009

We have studied some differential geometric aspects of semi-invariant submanifolds of a nearly trans-Sasakian manifold. That have led to the classification of totally umbilical semi-invariant submanifolds of a nearly trans-Sasakian manifold.
Positive integers divisible by the product of their nonzero digits
Authors: De Koninck Jean-Marie , Luca Florian
Published: December 2009

Let

{\cal N}_0(x)

denote the set of positive integers

n\le x

which are divisible by the product of their nonzero digits. In this note, we show that if

x

is large, then

x^{.495}<\#{\cal N}_0(x)<x^{.654}

.
Congruence and $A^{-1}A^*$
Authors: Furtado Susana , Johnson Charles R.
Published: December 2009

A canonical form under congruence for nonsingular complex matrices may be deduced from the canonical pair form for Hermitian matrices in a straightforward way. This allows the study of

A^{-1}A^*

and what it says about the congruence class of

A

. In turn, alternate congruential forms may be given along with a further generalization of Sylvester's law of inertia and an application to C-S equivalence.
Lax equations, factorization and Riemann–Hilbert problems
Authors: Cámara M. Cristina , dos Santos António F. , dos Santos Pedro F.
Published: December 2009

The paper deals with the problem of existence and calculation of solutions to Lax equations that define finite-dimensional integrable systems. The method presented in the paper is based on Wiener–Hopf factorization and related Riemann–Hilbert problems on Riemann surfaces. The idea behind the method was first proposed by Semenov–Tian–Shansky but, to the authors' knowledge, is here applied, for the first time, in an infinite dimensional setting. The method dealt with in the paper enables one to analyse the global existence of solutions which seems more difficult by other methods. An example of a dynamical system associated with an elliptic curve is completely worked out in the paper.
Steiner's invariants and minimal connections
Authors: Ducret Stephen , Troyanov Marc
Published: December 2009

The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure.
The local structure of nonstandard representatives of distributions
Authors: Vernaeve Hans
Published: December 2009

It is shown that the nonstandard representatives of Schwartz distributions, as introduced by K. D. Stroyan and W. A. J. Luxemburg in their book
Introduction to the theory of infinitesimals
[5], are locally equal to a finite-order derivative of a finite-valued and S-continuous function. By ‘equality’, we mean a pointwise equality, not an equality in a distributional sense. This proves a conjecture by M. Oberguggenberger in
Z. Anal. Anwendungen
10 (1991), 263–264. Moreover, the representatives of the zero-distribution are locally equal to a finite-order derivative of a function assuming only infinitesimal values. These results also unify the nonstandard theory of distributions by K. D. Stroyan and W. A. J. Luxemburg with the theory by R. F. Hoskins and J. Sousa Pinto in
Portugal. Math.
48 (1991), 195–216.
Sharp estimates for periodic solutions to the Euler–Poisson–Darboux equation
Authors: Amorim Paulo , LeFloch Philippe G.
Published: December 2009

We establish sharp estimates for distributional solutions to the Euler–Poisson–Darboux equation posed in a periodic domain. These equations are highly singular, and setting the Cauchy problem requires a precise understanding of the nature of the singularities that may arise in weak solutions. We consider initial data in a space of functions with fractional derivatives such that weak solutions are solely integrable, and we derive sharp continuous dependence estimates for solutions to the initial-value problem. Our results strongly depend on a key parameter arising in the Euler–Poisson–Darboux equation.
Uniqueness at infinity in time for the Maxwell–Schrödinger system with arbitrarily large asymptotic data
Authors: Ginibre Jean , Velo Giorgio
Published: December 2009

We prove the uniqueness of solutions of the Maxwell–Schrödinger system with given asymptotic behaviour at infinity in time. The assumptions include suitable restrictions on the growth of solutions for large time and on the accuracy of their asymptotics, but no restriction on their size. The result applies to the solutions with prescribed asymptotics constructed in a previous paper.
A class of integral domains whose integral closures are small submodules of the quotient field
Authors: Dobbs David E.
Published: December 2009

Let

(V, M)

be a valuation domain which is distinct from its quotient field

K

, and let

π: V → V/M

be the canonical surjection. Let

D

be a subring of

V/M

. It is proved that the pullback

R := π^{− 1}(D)

has the property that

V

(and hence each integral overring of

R

) is a small

R

-submodule of

K

. Applications include all classical

D + M

constructions and locally pseudo-valuation domains.
Exponential stabilization of periodic solutions of a system of KdV equations
Authors: Bisognin Eleni , Bisognin Vanilde , Pereira Jardel Morais
Published: December 2009

We consider a coupled nonlinear dispersive system of Korteweg–de Vries type in the presence of a dissipative mechanism. First we prove that the Cauchy problem is globally well posed in a suitable periodic Sobolev space and our main result says that the

L^2

and

L^∞

norms of the solutions decay exponentially fast as

t → + ∞

.
Inequalities for Riemann’s zeta function
Authors: Alzer Horst
Published: December 2009

Let

ζ

and

Λ

be the Riemann zeta function and the von Mangoldt function, respectively. Further, let

c > 0

. We prove that the double-inequality

\exp\Big(− c \sum\limits^∞_{n = 1} \frac{Λ(n)}{n^{s + α}}\Big) < \frac{ζ(s + c)}{ζ(s)} < \exp\Big(− c \sum\limits^∞_{n = 1} \frac{Λ(n)}{n^{s + β}}\Big)

holds for all

s > 1

with the best possible constants

α = 0\qquad  \text{and}\qquad   β =\frac{1}{\log 2} \log\Big(\frac{c \log 2}{1 − 2^{−c}}\Big).

This extends and refines a recent result of Cerone and Dragomir.
Series of reciprocal products with factors from linear recurrence sequences
Authors: Bundschuh Peter
Published: December 2009

In this article we study the values of power series

∑ z^n/\prod^j_{i = 0} W_{(n + im)k}

at certain points of their domain of meromorphy from the arithmetical point of view.

(W_n)

is a sequence of non-zero integers satisfying a recurrence

W_{n + 1} = pW_n + qW_{n − 1}

with non-zero integers

p, q

such that the discriminant

Δ = p^2 + 4q

is positive but not a square. The main interest is to characterize the situations, where these values lie in the real quadratic number field

ℚ(√Δ)

or even in

ℚ

, but we also include some transcendence problems.
A theory of plastic splashing
Authors: Villaggio Piero
Published: December 2009

The fall on the ground of a highly deformable body is followed by a remarkable spread of the material on the plane of fall. Here we propose a model for determining the extent of the splashed mass and the time of arrest after the first contact with the ground.
Weak convergence, local bifurcations and uniqueness theorems
Authors: Dancer Andrew
Published: December 2009

We show how weak convergence techniques can be used to improve classical theorems on local bifurcation and on uniqueness when a parameter is large.
Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem
Authors: Mahmoudi Fethi , Malchiodi Andrea
Published: December 2009

We consider the equation

- \varepsilon^2 \Delta u + u = u^p

in

\Omega \subseteq \mathbb R^N

, where

\Omega

is open, smooth and bounded, and we prove concentration of solutions along

k

-dimensional minimal submanifolds of

\partial \Omega

, for

N \geq 3

and for

k \in \{1, \dots, N-2\}

. We impose Neumann boundary conditions, assuming

1<p <\frac{N-k+2}{N-k-2}

and

\varepsilon \to 0^+

. This result settles in full generality a phenomenon previously considered only in the particular case

N = 3

and

k = 1

.
Some inequalities of Glaeser-Bronšteĭn type
Authors: Spagnolo Sergio , Taglialatela Giovanni
Published: December 2009

The classical Glaeser estimate is a special case of the Lemma of Bronšteĭn which states the Lipschitz continuity of the roots

\lambda_j(x)

of a hyperbolic polynomial

P(x,X)

with coefficients

a_j(x)

depending on a real parameter. Here we prove a pointwise estimate for the successive derivatives of the

a_j(x)

's in term of certain nonnegative functions which are symmetric polynomials of the roots

\{\lambda_j(x)\}

(hence also of the coefficients

\{a_j(x)\})

. These inequalities are very helpful in the study of the Cauchy problem for homogeneous weakly hyperbolic equations of higher order.