bluuebunny/crossref_metadata_2025 huggingface dataset:
Combinatorics
Authors: Lovasz Laszlo , Prömel Hans Jürgen
Published: December 2009

The workshop
Combinatorics
organised by László Lovász (Redmond) and Hans Jürgen Prömel (Berlin) was held January 1st–January 7th, 2006. This meeting was very well attended with 48 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 plenary lectures provided a very good overview over the current developments in several areas of combinatorics and discrete mathematics. We were very fortunate that some of the speaker reported on essential progress on longstanding open problems. In particular, Ajtai, Komlós, Simonovits, and Szemerédi solved the Erdős–T. Sós conjecture on trees (for large trees) and Tao and Vu greatly improved the asymptotic upper bound on the probability that a Bernoulli matrix is singular. The shorter contributions ranged over many topics. They were a good platform, especially, for younger researchers to present their results. In the following we include the extended abstracts of all talks in the order they were given.

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. The organizers also thank the many participants who traveled on New Year's Eve and New Year's Day to arrive on time for the beginning of the workshop.
Mini-Workshop: Thick Subcategories - Classifications and Applications
Authors: Buchweitz Ragnar-Olaf , Krause Henning , Schwede Stefan
Published: December 2009

Thick subcategories of triangulated categories have been the main topic of this workshop. Triangulated categories arise in many areas of modern mathematics, for instance in algebraic geometry, in representation theory of groups and algebras, or in stable homotopy theory. We give three typical examples of such triangulated categories:

In each case, there is a classification of thick subcategories under some appropriate conditions. Recall that a subcategory of a triangulated category is
thick
, if it is a triangulated subcategory and closed under taking direct factors. Historically, the first classification has been established by Hopkins and Smith for the stable homotopy category, using the nilpotence theorem. A similar idea was then applied by Hopkins and Neeman to categories of perfect complexes over commutative noetherian rings. Later, Thomason extended this classification to schemes. For stable categories of finite group representations, the classification of thick subcategories is due to Benson, Carlson, and Rickard.

The format of the workshop has been a combination of introductory survey lectures and more specialized talks on recent progress and open problems. The mix of participants from different mathematical areas and the relatively small size of the workshop provided an ideal atmosphere for fruitful interaction and exchange of ideas. It is a pleasure to thank the administration and the staff of the Oberwolfach Institute for their efficient support and hospitality.
Differential-Algebraic Equations
Authors: Campbell Stephen L. , März Roswitha , Petzold Linda , Rentrop Peter
Published: December 2009

The topic of Differential Algebraic Equations (DAEs) began to attract significant research interest in applied and numerical mathematics in the early 1980's. Today, a quarter of a century later, DAEs are an independent field of research, which is gaining in importance and becoming of increasing interest for both applications and mathematical theory.

This Oberwolfach workshop brought together 48 experts in applied mathematics, among them, on the one hand, some who have already influenced and formed the developments of the field, and on the other hand, some very young researchers who have shown outstanding creativity and competence in connection with their PhD theses and thus raise great hopes for further advances.

The 16 female and 32 male scientists came from 13 countries to meet and work together in the wonderful, unique Oberwolfach atmosphere, which stimulated a fruitful and pleasant collaboration.

The schedule comprised a total of 34 presentations, 18 of which were arranged into 14 survey lectures (some of them with more than one speaker) offering a broader treatment of a particular subject. 16 shorter contributions supplemented the scientific programmme. The areas can be classified (of course with large overlap) into 4 groups:

The broad range of these areas and the diversity of the participants stimulated fruitful discussions between the different branches and gave rise to new contacts and collaborations. A considerable gain in knowledgde and progress became obvious, which includes the formulation of open questions and challenges for the future.

We are grateful to the Mathematisches Forschungsinstitut Oberwolfach for providing an inspiring setting for this workshop.
Feinstrukturtheorie und Innere Modelle
Authors: Jensen Ronald B. , Magidor Menachem , Schindler Ralf-Dieter
Published: December 2009

The workshop
Fine Structure Theory and Inner Models
, organised by Ronald Jensen (Berlin), Menachem Magidor (Jerusalem) and Ralf Schindler (Münster) was held April 30th - May 6th, 2006. It was attended by most of the leading researchers in the area.

Fine structure theory was initiated by the first organizer, R. Jensen, in the 70ies. It has been exploited ever since for producing a series of spectacular results in set theory. One such is Jensen's Covering Lemma for Gödel's constructible hierarchy,

L

, which says, put informally, that the universe

V

of all sets either resembles

L

to a large extent or else is very different from

L

. Later on, various people (most of which were participants of this workshop) proved versions of the Covering Lemma for larger inner models.

The main goal of “Fine structure theory and inner model theory” is to construct fine structural inner models of set theory, i.e., definable transitive proper class-sized models of the standard axiom system

{\sf ZFC}

of set theory, which reflect the large cardinal structure of the universe, but at the same time admit a fine structure that makes it possible to analyze them in great detail and prove various combinatorial properties in them. Other applications of such inner models are consistency strength investigations, and they can be used as a tool for proving implications which don't mention inner models at all, but for which no “direct” proof is in sight.

A large cardinal concept is one such that

\mathsf{ZFC}

cannot prove that there is an incarnation of it. Our area is the key tool for uncovering the large cardinal structure which is implicit in many (not only set theoretic) hypotheses. In fact, often a given statement which doesn't mention large cardinals at all and a statement about the existence of models with large cardinals turn out to be two sides of the same coin. Breathtaking results by Martin, Steel, Woodin, and others in the 80ies and 90ies have shown that the large cardinal concept of a
Woodin cardinal
is a crucial one here.

The main issues of this area are the following.

The conference had 16 participants. 13 talks were given, and they covered both pure and applied parts of inner model theory. Because this was a gathering of true specialists, there was no need for overview-style talks and we could concentrate on issues which are at the focus of current research. The talks came with intriguing results, but also with promising new perspectives for upcoming research. We had very lively discussions.

It was a fruitful workshop, and many of the ideas which were exchanged are sure to be further elaborated in the near future.
Computational Group Theory
Authors: Hiß Gerhard , Holt Derek F. , Newman Michael F.
Published: December 2009

The workshop
Computational Group Theory
was the fifth of this title held at Oberwolfach. It was attended by

54

participants with broad geographic representation from four continents. Among the participants were

10

young researchers, visiting Oberwolfach for the first time, some of them still graduate students. We are grateful for the special EU funds making their visits possible.

The lecture program was divided into two sections. In the first we had a series of four invited longer lectures, 50 minutes each, given by distinguished, selected participants, surveying recent major developments in their fields. The first of these lectures was held by Eamonn O'Brien (Auckland), who gave a talk on “The latest developments in the matrix groups computation project”. The second was given by Bill Kantor (Eugene) on “A presentation of presentations”, introducing the remarkable result that all finite simple groups (except possibly

^2G_2(3^{2k+1})

) can be presented by a bounded number of generators and relations. In the third talk in this series Michael Vaughan-Lee (Oxford) reported on the recently completed classification of all

p

-groups of orders

p^6

and

p^7

. Finally, Gregor Kemper (München) surveyed new developments in Computational Invariant Theory.

The other section was made up of

37

short twenty-minute lectures on new work or work in progress. We aimed at giving preference to the younger participants to present their results. This program structure prompted positive feedback by many participants.

The short talks reported on various computational aspects – data structures, algorithms, complexity, computer experiments – in a broad range of topics, including matrix groups,

p

-groups, finitely presented groups, permutation groups, representation theory of groups, invariant theory, group cohomology, Lie algebras and combinatorics. The analysis of algorithms requires a thorough theoretical background in the respective fields.

The talks also revealed the close interrelationship between the different topics. Work for finite

p

-groups uses Lie rings and algebraic groups, matrix group algorithms rely on methods for finitely presented groups as well as on representation theory, work on permutation groups uses representation theoretic information. The computational solution of open problems such as the construction of Brauer character tables or the cohomology rings of sporadic groups usually requires the application of almost all the known techniques. Methods from other parts of computer algebra, e.g., Gröbner bases, come into play in computational cohomology theory as well as in the construction of matrix groups which are factor groups of specific finitely presented groups, e.g., Hurwitz groups.

Despite the relatively large number of talks at this workshop, there was plenty of time for discussions. Needless to say that this time was well spent: numerous collaborations were continued and various others were started.
Geometric and Topological Combinatorics
Authors: Björner Anders , Kalai Gil , Ziegler Günter M.
Published: December 2009

The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” was organized by Anders Björner (KTH and Mittag-Leffler Institute, Stockholm), Gil Kalai (Hebrew University, Jerusalem), and Günter M. Ziegler (TU Berlin). It consisted of six one-hour lectures by Isabella Novik, Herbert Edelsbrunner, Carsten Schultz, Igor Pak, Alexander Barvinok and Roy Meshulam, as well as twenty-seven half-hour talks, a problem session (led by Gil Kalai, also documented below), two software demonstrations (on by Michael Joswig, and on by Jesus De Loera), and many more informal sessions, group discussions, and a great variety of small group and pairwise discussions. It was a lively week!

As will become obvious from the sequence of extended abstracts presented below, the conference treated a broad spectrum of topics from Topological Combinatorics (such as poset topology, graph coloring, etc.), Discrete Geometry (polytopes, triangulations, arrangements, Coxeter groups, and polyhedral surfaces), as well as Geometric Topology (triangulated manifolds, persistent homology of geometric data, etc.).

The manifold connections between these themes, with refinements of well-established bridges as well as completely new links between seemingly distant themes, problems, methods, and theories show that the area is very much alive. Even more so this is demonstrated by substantial progress on older problems, which on this conference included Isabella Novik's opening lecture about centrally symmetric polytopes (joint works with Nati Linial and with Alexander Barvinok), or still on the first day Ed Swartz' report about the

f

-vectors of triangulated manifolds.

Of course there is no way to present the richness and depth of the work and presentations of the conference's program on a one page report, or a short collection of abstracts. All the following can offer is an overview of the official program of the conference. It does not cover all the additional smaller presentations, group discussions and blackboard meetings (for example, Tom Braden was coerced into an additional evening presentation of the topology of hypertoric varieties “by popular demand”), nor the lively interactions that occured during the week – for example, a surprising connection was made at the problem session between permutation patterns that appeared in Jonas Sjöstrand's lecture, and a conjecture posed by Alex Postnikov; in subsequent work, the problem was solved by Axel Hultman, Svante Linusson, John Shareshian, and Jonas Sjöstrand (paper in preparation). We are extremely grateful to the Oberwolfach institute, to its director and to all of its staff for providing a perfect setting for an inspiring, intensive week of “Geometric and Topological Combinatorics.”
Mini-Workshop: Positional Games
Authors: Krivelevich Michael , Szabo Tibor
Published: December 2009

The theory of Positional Games is a branch of Combinatorics, whose main aim is to develop systematically an extensive mathematical basis for a variety of two player perfect information games, ranging from such commonly popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. Though a close relative of the classical Game Theory of von Neumann and of Nim-like games popularized by Conway and others, Positional Games are quite different and are of much more combinatorial nature. The first papers on the subject appeared in the 60's and the 70's. J\'ozsef Beck turned it into a well established mathematical discipline through a series of papers spanning the last quarter century. Positional games are strongly related to several other branches of Combinatorics like Ramsey Theory, Extremal Graph and Set Theory, the Probabilistic Method. The subject has proven to be quite instrumental in deriving important results in Theoretical Computer Science, in particular in derandomization and algorithmization of important probabilistic tools. Recently the field of Positional Games has been experiencing an explosive growth with quite a few new and important results in different directions (new versions of game definitions; analysis of biased games; exact results in Ramsey-type games; games of geometric nature; fast winning strategies etc.) appearing. The purpose of this mini-workshop was two-fold: it was aimed to provide an opportunity for leading researchers in the field to present and discuss their recent results on a systematic basis; it was also meant to attract new researchers, including students, to this exciting and rapidly developing field. 17 scientists from different countries participated in the meeting. The organizers and participants would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing an inspiring setting for this event.
Phase Transitions
Authors: Alt Hans Wilhelm , Luckhaus Stephan , Presutti Errico , Salje Ekhard K.H.
Published: December 2009

The present workshop was part of a series with the topic of phase transitions, designed especially to bridge the gap in notation and notions of phase transition in different communities, i.e. physics, statistical mechanics, partial differential equations. Judging from the lively discussions it seem to have achieved its goal at least in part. This workshop is the last organized by H. W. Alt, S. Luckhaus, E. Presutti and E.Salje who are handing over to a new team of organizers. An active area of research discussed in many of the talks was the motion of phase interfaces (Dirr, Dreyer, Eck, Esposito, Freistuehler, Melcher, Planes, Zimmer) also in hyperbolic scaling where many questions are still open. The interaction phase interfaces with crystal structure was another topic (Bismayer, Conti, Janssen) for the physics and pde community alike. From quantum statistical mechanics the Curie Weiss model was discussed (Grimmett, Ioffe). Random potentials or environments were an important topic (Bovier, Dirr, Planes , Orlandi) and models of kinetic type played a bigger role than in the previous workshops (Bellettini, Esposito, Marra, Theil). A strength of the workshop were the discussions between the different groups of participants and the organizers feel that in the field of phase transitions this interdisciplinary discourse is particularly important.
Large Scale Stochastic Dynamics
Authors: Landim Claudio , Olla Stefano , Spohn Herbert
Published: December 2009

“Large Scale Stochastic Dynamics” is at the crossroad of probability theory and statistical physics. One central theme of statistical physics is the emergent behavior resulting from the interaction of many identical components, the paradigm being a fluid or a gas. On the atomistic scale they consist of a huge number of identical molecules. Their motion is governed by Newton's equation of classical mechanics (ignoring quantum effects). The emergent description, valid only for particular initial states and on a sufficiently coarse space-time scale, are the compressible Navier–Stokes equations of fluid dynamics. Roland Dobrushin (1929–1995) and Frank Spitzer (1926–1992) had the vision that in the context of stochastic dynamics with many identical components the issue of emergent behavior is both mathematically challenging and important in modeling applications. The latter judgment turned out to be more than true. Stochastic algorithms, such as kinetic Monte Carlo, importance sampling, Monte Carlo Markov chains, Glauber dynamics, and others, are daily practice. Their mathematical vision has evolved over the past twenty years into a rich, multifaceted research program. Our workshop is like a snap-shot of the current activities, in fact quite distinct from the snap-shot taken exactly three years ago. A partial list of topics reads

We had 44 participants from 11 countries, mostly probabilists, but also experts from partial differential equations and statistical physics. They all enjoyed tremendously the unique and stimulating atmosphere at the Mathematische Forschungsinstitut Oberwolfach and hope to return some day.
Modulformen
Authors: Böcherer Siegfried , Ibukiyama Tomoyoshi , Kohnen Winfried
Published: December 2009

The workshop
Modulformen
, organised by Siegfried Böcherer (Mannheim), Tomoyoshi Ibukiyama (Osaka) and Winfried Kohnen (Heidelberg) was held October 28–November 3, 2007. This meeting was attended by 24 participants with different backgrounds reflecting some of the many aspects of the theory of modular forms. One of the callenging features of this theory is that techniques using Whittaker models do not apply here. On the other hand, the theory is very rich in explicit structures and has immediate connections to arithmetic and geometry.

Some of the main topics (presented in one-hour talks) are

The third and forth topic are not only interesting in their own right but they are also a testing ground for general conjectures. We explictly mention the theory of newforms by Roberts and Schmidt for

GSp(4)

.

Explicit liftings can be viewed as examples for Langlands functoriality. The emphasis is on the explicit description of these liftings in terms of modular forms (not only representations). A prototype is the Duke–Imamoglu–Ikeda-lifting, which appeared in several talks.

Other talks dealt with theta series, Poincare series, generation of spaces of modular forms, converse theorems and spherical functions on p-adic spaces. Beyond the talks, there was much opportunity for scientific interaction among the participants.
Arbeitsgemeinschaft: Julia Sets of Positive Measure
Authors: Buff Xavier , Cheritat Arnaud
Published: December 2009

A polynomial

P:\mathbb C\to :\mathbb C

can be considered as a dynamical system. We are interested in the sequences

(z_n)

defined by induction:

z_0\in :\mathbb C\quad\text{and}\quad z_{n+1}=P(z_n).

The filled-in Julia set

K_P

is the set of points

z_0\in :\mathbb C

for which the sequence

(z_n)

is bounded. This set is compact. The Julia set

J_P

is the boundary of

K_P

. In particular, it has empty interior. There is a small collection of polynomials, for instance

P(z) = z^d\quad ,\quad P(z) = z^2-2,

for which the Julia set can be fairly easily understood, but most exhibit “fractal” geometry and “chaotic” behavior, the analysis of which requires serious tools from complex analysis, dynamical systems, topology, combinatorics,

\ldots

This subject has a fairly long history, with contributions by Koenigs, Schröder, Böttcher in the late 19th century, and the great memoirs of Fatou and Julia around 1920.

There followed a dormant period, with notable contributions by Cremer (1936) and Siegel (1942), and a rebirth in the 1960's (Brolin, Guckenheimer, Jakobson). Since the early 1980's, partly under the impetus of computer graphics, the subject has grown vigorously, with major contributions by Douady, Hubbard, Sullivan, Thurston, and more recently Lyubich, McMullen, Milnor, Shishikura, Yoccoz \ldots

Fatou found sufficient conditions for the boundary of the basin of an attracting fixed point to be a Cantor set with Lebesgue measure equal to

0

. He could not tell whether or not the measure could be positive.

For some time and until the 1990's, the conjecture, reinforced by the analogy with Ahlfors's conjecture on the area of limit sets of Kleinian groups, was that no Julia set of a polynomial could have positive area.

Results in this direction were obtained by Douady and Hubbard in the case of hyperbolic or subhyperbolic maps, by Branner, Hubbard and McMullen in the case of non-renormalizable cubic polynomials with an escaping critical point, by Lyubich and Shishikura in the case of finitely renormalizable quadratic polynomials without indifferent cycles, by Petersen in the case of quadratic polynomials having a Siegel disk with bounded type rotation number.

In the 1990's, Douady began to catch a glimpse of a method for Julia sets of positive area: in the family of degree 2 polynomials with an indifferent Cremer fixed point. Recently, we brought Douady's method to completion.

The Arbeitsgemeinschaft
Julia sets of positive measure
focused on the proof of existence of quadratic polynomials having a Julia set of positive area. It was held March 30th–April 5th, 2008. It was attended by 36 participants.
Atomistic Models of Materials: Mathematical Challenges
Authors: E Weinan , Friesecke Gero , Pettifor David
Published: December 2009

The past decades have witnessed an explosion of interest in the application of mathematical models to science and engineering down at the atomic scale. Atomistic models are large, complex, multiscale, and in particular discrete, and provide a rich source of fascinating challenges for mathematics. In particular, a basic goal is to understand (1) how and at which length- and timescales the behaviour of large atomistic systems becomes well approximated by traditional continuum descriptions (of dislocations, grains, fracture, elasticity or plasticity), and under which circumstances atomistic and continuum “modes” are nontrivially coupled (2) how atomistic models can be accurately and efficiently extracted from quantum mechanical models.

While mathematical research in this area is still in its early stages, interest by mathematicians in atomistic models is fast growing, and the goal of this workshop was to bring together leading mathematicians and materials scientists, in the unique Oberwolfach setting, in order to document recent results, discuss main open challenges, and stimulate an exchange between the two communities.

The workshop focused on the following topics, the basic goals described above being recurrent themes.

The organizers are particularly indebted to the participants from both mathematics and materials science for making such a committed effort to communicate their work to researchers from the other community, both in the actual talks and in this Oberwolfach report. Judging by the success of this effort, we are confident that this report can serve as a starting point and a stimulation for a great deal of future interaction between our communities.
Real Analysis, Harmonic Analysis and Applications
Authors: Müller Detlef , Stein Elias M.
Published: December 2009

This workshop, which continued the triennial series at Oberwolfach on Real and Harmonic Analysis that started in 1986, has brought together experts and young scientists working in harmonic analysis and its applications (such as to dispersive PDE's and ergodic theory) with the objective of furthering the important interactions between these fields.

Three prominent experts, Elon Lindenstrauss (Princeton), Amos Nevo (Technion, Haifa), and Terence Tao (UCLA), gave survey respectively introductory lectures. Their topics included “Effective equidistribution on the torus”, “Non-Euclidean lattice point counting problems, and the ergodic theory of lattice subgroups,” and “The van der Corput lemma, equidistribution in nilmanifolds, and the primes.”

Major further areas and results represented at the workshop are:

The meeting took place in a lively and active atmosphere, and greatly benefited from the ideal environment at Oberwolfach. It was attended by 43 participants. The program consisted of 3 survey lecture series and 25 lectures. The organisers made an effort to include young mathematicians, and greatly appreciate the support through the joint Oberwolfach/NSF program “US Junior Oberwolfach Fellows,” which allowed to invite several outstanding young scientists from the United States.
Geometrie
Authors: Taimanov Iskander A. , Wilking Burkhard , Lott John
Published: December 2009

The official program consisted of 20 lectures and therefore left plenty of space for fruitful informal collaboration for the 44 participants. One emphasis with 9 talks at this meeting was on geometric flows. Exciting progress could be reported on

Another important theme was metric and Finsler geometry with five talks covering the following topics

The other six talks covered other aspects of geometry including

Several connections between the different areas became apparent during the workshop. For example, the initial value problem for the Ricci flow with singular initial data is closely linked to smoothing problems occuring in Alexandrov geometry.
Discrete Geometry
Authors: Henk Martin , Matoušek Jiří , Welzl Emo
Published: December 2009

Discrete Geometry
deals with the structure and complexity of discrete geometric objects ranging from finite point sets in the plane to more complex structures like arrangements of

n

-dimensional convex bodies. Classical problems such as Kepler's conjecture and Hilbert's third problem on decomposing polyhedra, as well as classical works by mathematicians such as Minkowski, Steinitz, Hadwiger and Erdös are part of the heritage of this area. By its nature, this area is interdisciplinary and has relations to many other vital mathematical fields, such as algebraic geometry, topology, combinatorics, computational geometry, convexity, and probability theory. At the same time it is on the cutting edge of applications such as geographic information systems, mathematical programming, coding theory, solid modelling, computational structural biology and crystallography.

The workshop was attended by 40 participants. There was a series of 12 survey talks giving an overview of developments in Discrete Geometry and related fields:

In addition, there were 18 shorter talks and an open problem session chaired by János Pach on Wednesday evening — a collection of open problems resulting from this session can be found in this report. The program left ample time for research and discussions in the stimulating atmosphere of the Oberwolfach Institute.
Discrete Differential Geometry
Authors: Bobenko Alexander I. , Kenyon Richard , Sullivan John M. , Ziegler Günter M.
Published: December 2009

This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete Diﬀerential Geometry.
The Arithmetic of Fields
Authors: Jarden Moshe , Pop Florian , Schneps Leila
Published: December 2009

The workshop “The Arithmetic of Fields” focused on a series of problems concerning the interplay between number theory, arithmetic and algebraic geometry, Galois theory, and model theory, such as: the Galois theory of function ﬁelds / covers of varieties, rational points on varieties, Galois cohomology, local-global principles, lifting/specializing covers of curves, model theory of ﬁnitely generated ﬁelds, etc.
Mini-Workshop: Product Systems and Independence in Quantum Dynamics
Authors: Bhat B.V. Rajarama , Franz Uwe , Skeide Michael
Published: December 2009

Quantum dynamics, both reversible (i.e., closed quantum systems) and irreversible (i.e., open quantum systems), gives rise to product systems of Hilbert spaces or, more generally, of Hilbert modules. When we consider reversible dynamics that dilates an irreversible dynamics, then the product system of the latter is equal to the product system of the former (or is contained in a unique way). Whenever the dynamics is on a proper subalgebra of the algebra of all bounded operators on a Hilbert space, in particular, when the open system is classical (commutative) it is indispensable that we use Hilbert modules. The product system of a reversible dynamics is intimately related to a ﬁltration of subalgebras that are independent in a state or conditionally independent in a conditional expectation of the reversible system. This has been illustrated in many concrete dilations that have been obtained with the help of quantum stochastic calculus. Here the underlying Fock space or module determines the sort of quantum independence underlying the reversible system. The mini-workshop brought together experts from quantum dynamics, product systems and quantum independence who have contributed to the general theory or who have studied intriguing examples. As the implications of the tight relationship between product systems and independence had so far been largely neglected, we expect from our mini-workshop a strong innovative impulse to this ﬁeld.
Control Theory: On the Way to New Application Fields
Authors: Allgöwer Frank , Helmke Uwe , Sontag Eduardo D.
Published: December 2009

Control theory is an interdisciplinary ﬁeld that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently, deep interactions are emerging with new application areas, such as systems biology, quantum control and information technology. In order to address the new challenges posed by the new application disciplines, a special focus of this workshop has been on the interaction between control theory and mathematical systems biology. To complement these more biology oriented focus, a series of lectures in this workshop was devoted to the control of networks of systems, fundamentals of nonlinear control systems, model reduction and identiﬁcation, algorithmic aspects in control, as well as open problems in control.
Mini-Workshop: Numerical Upscaling for Flow Problems: Theory and Applications
Authors: Brandt Achi , Efendiev Yalchin , Iliev Oleg
Published: December 2009

The objective of this workshop was to bring together researchers working in multiscale simulations with emphasis on multigrid methods and multiscale ﬁnite element methods, aiming at chieving of better understanding and synergy between these methods. The goal of multiscale ﬁnite element methods, as upscaling methods, is to compute coarse scale solutions of the underlying equations as accurately as possible. On the other hand, multigrid methods attempt to solve ﬁne-scale equations rapidly using a hierarchy of coarse spaces. Multigrid methods need “good” coarse scale spaces for their eﬃciency. The discussions of this workshop partly focused on approximation properties of coarse scale spaces and multigrid convergence. Some other presentations were on upscaling, domain decomposition methods and nonlinear multiscale methods. Some researchers discussed applications of these methods to reservoir simulations, as well as to simulations of ﬁltration, insulating materials, and turbulence.
Enveloping Algebras and Geometric Representation Theory
Authors: Kumar Shrawan , Littelmann Peter , Soergel Wolfgang
Published: December 2009

The meeting brought together experts investigating Lie theory from the geometric, algebraic and combinatorial points of view to discuss recent progress and bring forward the research in this area by fostering scientiﬁc interaction.
Representations of Finite Groups
Authors: Chuang Joseph , Linckelmann Markus , Malle Gunter , Rickard Jeremy
Published: December 2009

The workshop Representations of Finite Groups was organised by Joseph Chuang (London), Markus Linckelmann (Aberdeen), Gunter Malle (Kaiserslautern) and Jeremy Rickard (Bristol). It covered a wide variety of aspects of the representation theory of ﬁnite groups and related objects like Hecke algebras. A particular focus was placed on the rapidly evolving area of fusion systems.
Hilbert Modules and Complex Geometry
Authors: Douglas Ronald G. , Eschmeier Jörg , Upmeier Harald
Published: December 2009

The major topics discussed in this workshop were Hilbert modules of analytic functions on domains in

ℂ^n

, Toeplitz and Hankel operators, the interplay of commutative algebra, complex analytic geometry and multivariable operator theory, coherent and quasi-coherent sheaves as localizations of Hilbert modules, Hilbert bundles and Jordan varieties on Cartan domains.
Mathematical Biology
Authors: DiBenedetto Emmanuele , Perthame Benoît , Stevens Angela
Published: December 2009

Mathematical biology is a fast growing ﬁeld of research, which on one hand side faces challenges resulting from the enormous amount of data provided by experimentalists in the recent years, on the other hand new mathematical methods may have to be developed to meet the demand for explanation and prediction on how speciﬁc biological systems function.
Algebraische Zahlentheorie
Authors: Kings Guido , Kisin Mark , Venjakob Otmar
Published: December 2009

The workshop brought together researchers from Europe, the US and Japan, who reported on various recent developments in algebraic number theory and related ﬁelds. Dominant themes were
p
-adic methods,
L
-functions and automorphic forms but other topics covered a very wide range of algebraic number theory.
Perfect polynomials over $\mathbb F_4$ with less than five prime factors
Authors: Gallardo Luis H. , Rahavandrainy Olivier
Published: December 2009

A perfect polynomial

A\in\mathbb{F}_4[x]

is a monic polynomial that equals the sum of its monic divisors. There are no perfect polynomials

A\in\mathbb{F}_4[x]

with exactly

3

prime divisors, i.e., of the form

A=P^aQ^bR^c

where

P,Q,R\in\mathbb{F}_4[x]

are irreducible and

a,b,c

are positive integers. We characterize the perfect polynomials

A

with

4

prime divisors such that one of them has degree

1

. Assume that

A

has an arbitrary number of distinct prime divisors, we discuss some simple congruence obstructions that arise and we propose three conjectures.
A version of the Glimm method based on generalized Riemann problems
Authors: Hong John M. , LeFloch Philippe G.
Published: December 2009

We introduce a generalization of Glimm's random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend on the unknown as well as on the time and space variables. The method is based on local approximate solutions of the generalized Riemann problem, which form building blocks in our scheme and allow us to take into account naturally the effects of the flux and source terms. To establish the nonlinear stability of these approximations, we investigate nonlinear interactions between generalized wave patterns. This analysis leads us to a global existence result for quasilinear hyperbolic systems with source-term, and applies, for instance, to the compressible Euler equations in general geometries and to hyperbolic systems posed on a Lorentzian manifold.
A monotone method for fourth order boundary value problems involving a factorizable linear operator
Authors: Habets P. , Ramalho Margarita
Published: December 2009

We consider the nonlinear fourth order beam equation

u^{\textit{iv}}=f(t,u,u''),

with boundary conditions corresponding to the periodic or the hinged beam problem. In presence of upper and lower solutions, we consider a monotone method to obtain solutions. The main idea is to write the equation in the form

u^{\textit{iv}}-cu''+du=g(t,u,u''),

where

c

,

d

are adequate constants, and use maximum principles and a suitable decomposition of the operator appearing in the left-hand side.
Homoclinic solutions of a fourth-order travelling wave ODE
Authors: Moroșanu Gheorghe , Souroujon Diko , Tersian Stepan
Published: December 2009

In this paper we investigate via the shooting method the existence of homoclinic solutions of a fourth-order differential equation arising in the theory of water waves.
Definition and stability of Lorentzian manifolds with distributional curvature
Authors: LeFloch Philippe G. , Mardare Cristinel
Published: December 2009

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the framework of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we investigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curvature tensors and the second fundamental form of the hypersurface and to establish the Gauss–Codazzi equations.
Weak theorems on differential inequalities for two-dimensional functional differential systems
Authors: Sremr Jiri
Published: December 2009

New theorems on differential inequalities for two-dimensional systems of linear functional differential equations are established. Differential systems with argument deviations are considered in more detail, in which case further results are obtained.
On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential
Authors: Carles Rémi
Published: December 2009

We consider the Cauchy problem for nonlinear Schrödinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the H1-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly – no matter how large it is, unless the initial datum decays sufficiently fast at infinity.
An exact multiplicity result for a class of symmetric problems
Authors: Korman Philip
Published: December 2009

We consider positive solutions of a class of semilinear problems

u'' + λa(x)f(u) = 0, \qquad  −1 < x < 1,\qquad u(−1) = u(1) = 0,

with even and positive

a(x)

, depending on a positive parameter

λ

. In case

f(u)

is convex, an exact multiplicity result was given in P. Korman, Y. Li and T. Ouyang [6]; see also P. Korman [4] for the details. It was observed by P. Korman and J. Shi [7] that convexity requirement can be relaxed for large

u

(see also [5]). We show that convexity requirement can also be relaxed for small

u

.
On radicals and polynomial rings
Authors: Tumurbat Sodnomkhorloo , Mendes Deolinda Isabel C. , Mekei Abish
Published: December 2009

For any class

\mathcal M

of rings, it is shown that the class

\mathcal E_ℓ(\mathcal M)

of all rings each non-zero homomorphic image of which contains either a non-zero left ideal in

\mathcal M

or a proper essential left ideal is a radical. Some characterizations and properties of these radicals are presented. It is also shown that, for radicals

γ

under certain constraints, one can obtain a strictly decreasing chain of radicals

γ = γ_{(1)} ⊃ γ_{(2)} ⊃ ··· ⊃ γ_{(n)} ⊃ ···

where, for each positive integer

n

,

γ(n)

is the radical consisting of all rings

A

such that

A[ x_1, … ,x_n ]

is in

γ

, thus giving a negative answer to a question posed by Gardner. Moreover, classes

\mathcal M

of rings are constructed such that there exist several such radicals

γ

in the interval

[ \mathcal E_ℓ(0),\mathcal E _ℓ(\mathcal M) ]

.
(Quasi)periodic solutions in (in)finite dimensional Hamiltonian systems with applications to celestial mechanics and wave equation
Authors: Biasco Luca , Valdinoci Enrico
Published: December 2009

We describe a general method, based on a Lyapunov–Schmidt reduction and perturbative techniques, recently used by the authors to find periodic and quasi-periodic solutions both in finite and in infinite dimensional hamiltonian systems. We also illustrate some concrete applications to celestial mechanics and nonlinear wave equation.
On the solution set of some classes of nonconvex nonclosed differential inclusions
Authors: Cernea Aurelian
Published: December 2009

We consider a nonconvex and nonclosed semilinear differential inclusion and prove the arcwise connectedness of the set of its mild solutions. A similar result is provided for a class of nonconvex, nonclosed second order differential inclusions.
Existence of solutions for a third-order boundary value problem with $p$-Laplacian operator and nonlinear boundary conditions
Authors: Ma De-Xiang , Sun Shu-Zhen
Published: December 2009

In this paper we study the third-order nonlinear boundary value problem

\begin{align*}
&(Φ(u''))'(t) + f(t, u(t), u'(t), u''(t)) = 0\quad \text{a.e. } t ∈ [0,1], \\
&u(0) = 0,\ g(u'(0), u''(0)) = A,\ h(u'(1), u''(1)) = B,
\end{align*}

where

A, B ∈ ℝ

,

f : [0,1] × ℝ^3 → ℝ

is a Carathéodory function,

g, h ∈ C^0(ℝ^2,ℝ)

and

Φ ∈ C^0(ℝ, ℝ)

. Using apriori estimates, the Nagumo condition, upper and lower solutions and the Schauder fixed point theorem, we are able to prove existence of solutions of this problem.
On simple Filippov superalgebras of type $B(0, n)$, II
Authors: Pojidaev Alexander P. , Saraiva Paulo
Published: December 2009

It is proved that there exist no simple finite-dimensional Filippov superalgebras of type

B(0, n)

over an algebraically closed field of characteristic 0.
A note on common range of a class of co-analytic Toeplitz operators
Authors: Meštrović Romeo , Pavićević Žarko
Published: December 2009

We characterize the intersection of the ranges of a class of co-analytic Toeplitz operators by considering this set as the dual space of the Privalov space

N^p

,

1 < p < ∞

, in a certain topology. For a fixed

p

we define the class

H_p

consisting of those de Branges spaces

\mathscr{H}(b)

such that the function

b

is not an extreme point of the unit ball of

H^∞

, and the associated measure

μ_b

for

b

satisfies an additional condition. It is proved that the function

f

analytic on

\mathbf D

is a multiplier of every de Branges space from

H_p

if and only if

f

is in the intersection of the ranges of all Toeplitz operators belonging to the class

H_p

. We show that this is true if and only if the Taylor coefficients

\hat{f}(n)

of

f

decay like

O(\exp (−cn^{1/(p + 1)}))

for a positive constant

c

.
Algebraic connections vs. algebraic $\mathscr D$-modules: inverse and direct images
Authors: Cailotto Maurizio , Fiorot Luisa
Published: December 2009

This paper is devoted to a new proof of the comparison between the derived direct image functor for

\mathscr D

-Modules and the construction of the Gauss–Manin connection for smooth morphisms.
The geometry of 2-calibrated manifolds
Authors: Martínez Torres David
Published: December 2009

We define 2-calibrated structures, which are analogs of symplectic structures in odd dimensions. We show the existence of differential topological constructions compatible with the structure by developing an appropriate approximately holomorphic geometry for 2-calibrated structures.
An oscillation criterion for linear difference equations with general delay argument
Authors: Chatzarakis G. E. , Philos Ch. G. , Stavroulakis I. P.
Published: December 2009

This paper presents a new sufficient condition for the oscillation of all solutions of linear difference equations with general delay argument. The significance of this condition is demonstrated by comparing with known oscillation conditions. An example illustrating the results is also given.
The best constant for an almost critical Sobolev imbedding
Authors: Haraux Alain
Published: December 2009

The inequality

\Vert  u \Vert_∞ ≤ \frac C {√ε} \Vert u \Vert_{H^{N/2 + ε}}\qquad  \text{for all } u \in H^{N/2 + ε}(Ω),

is shown for

Ω

a sufficiently smooth domain in

ℝ^N

or a product of

N

bounded open intervals. This inequality is optimal for any open domain

Ω

.
Degenerations of scrolls to unions of planes
Authors: Calabri Alberto , Ciliberto Ciro , Flamini Flaminio , Miranda Rick
Published: December 2009

In this paper we study degenerations of scrolls to union of planes, a problem already considered by G. Zappa in [23] and [24]. We prove, using techniques different from the ones of Zappa, a degeneration result to union of planes with the mildest possible singularities, for linearly normal scrolls of genus

g

and of degree

d \geq 2g+4

in

\mathbb P^{d-2g+1}

. We also study properties of components of the Hilbert scheme parametrizing scrolls. Finally we review Zappa's original approach.
Schröder Equation in Several Variables and Composition Operator
Authors: Bisi Cinzia , Gentili Graziano
Published: December 2009

Let

\varphi

be a holomorphic self-map of the open unit ball

\mathbb B

of

\mathbb C^n

such that

\varphi(0)=0

and that the differential

d \varphi_{0}

of

\varphi

at

0

is non singular. The study of the Schröder equation in several complex variables

\sigma \circ \varphi = d\varphi _{0} \circ \sigma

is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on

\mathbb B

and to the theory of discrete, complex dynamical systems. An extensive use of the infinite matrix which represents the composition operator associated to the map

\varphi

leads to a simpler approach, and provides new proofs, to results of existence of solutions for the Schröder equation.
Closed curves in $\mathbb R^3$ with prescribed curvature and torsion in perturbative cases — Part 2: Sufficient conditions
Authors: Caldiroli Paolo , Guida Michela
Published: December 2009

We investigate the problem of

(\kappa,\tau)

-loops, namely closed curves in the three-dimensional Euclidean space, with prescribed curvature

\kappa

and torsion

\tau

. In particular we focus on some perturbative cases, taking

\kappa=\kappa_{\varepsilon}(p)

and

\tau=\tau_{\varepsilon}(p)

with

\kappa_{\varepsilon}

and

\tau_{\varepsilon}

converging to the constants 1 and 0, respectively, as

\varepsilon\to 0

. We prove existence of branches of

(\kappa_{\varepsilon},\tau_{\varepsilon})

-loops (for small

|\varepsilon|

) emanating from circles which correspond to stable zeroes of a suitable vector field

M\colon\mathbb T^{2}\times\mathbb R^{3} \to\mathbb R^{5}

.