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European Mathematical Society Publishing House
2024-03-28 10:17:00
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=OWR&vol=3&iss=2&update_since=2024-03-28
Oberwolfach Reports
Oberwolfach Rep.
OWR
1660-8933
1660-8941
General
10.4171/OWR
http://www.ems-ph.org/doi/10.4171/OWR
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© Mathematisches Forschungsinstitut Oberwolfach
3
2006
2
Representations of Finite Groups
Alexander
Kleshchev
University of Oregon, EUGENE, UNITED STATES
Markus
Linckelmann
City University, LONDON, UNITED KINGDOM
Gunter
Malle
Universität Kaiserslautern, KAISERSLAUTERN, GERMANY
Jeremy
Rickard
University of Bristol, BRISTOL, UNITED KINGDOM
The meeting was organized by A. Kleshchev (Eugene), M. Linckelmann (Aberdeen), G. Malle (Kaiserslautern) and J. Rickard (Bristol). This meeting was attended by over 50 participants with broad geographic representation. It covered a wide variety of aspects of the representation theory of finite groups and related objects like Hecke algebras. This workshop was sponsored by a project of the European Union which allowed us to invite in addition to established researchers also a couple of young people working on a PhD in representation theory. In eleven longer lectures of 40 minutes each and twentytwo shorter contributions of 30 minutes each, recent progress in representation theory was presented and interesting new research directions were proposed. Besides the lectures, there was plenty of time for informal discussion between the participants, either continuing ongoing research cooperation or starting new projects. The topics of the talks came roughly from two major areas: on the one hand side, the investigation of representation theoretic properties of general finite groups and related objects, on the other hand the determination and detailed analysis of representations of special classes of finite groups and related objects like Hecke algebras. \par One of the main topics touched upon in several talks was the investigation of the various open conjectures on characters and blocks of finite groups, like Alperin's, Brou\'e's and Dade's conjecture. A major breakthrough presented at this meeting by G. Navarro was the reduction (jointly with M. Isaacs and G. Malle) of the McKay conjecture on character degrees to a statement purely about simple groups, and the verification of this condition for certain families of finite simple groups. In the same direction, Puig announced some reduction statements for Alperin's weight conjecture.
Group theory and generalizations
General
895
978
10.4171/OWR/2006/15
http://www.ems-ph.org/doi/10.4171/OWR/2006/15
Higher Torsion Invariants in Differential Topology and Algebraic K-Theory
Sebastian
Goette
Universität Freiburg, FREIBURG, GERMANY
Kiyoshi
Igusa
Brandeis University, WALTHAM, UNITED STATES
E. Bruce
Williams
University of Notre Dame, NOTRE DAME, UNITED STATES
The classical Franz-Reidemeister torsion and its cousins, the Whitehead torsion and Ray-Singer analytic torsion, are topological invariants of manifolds with local coefficient systems (or flat vector bundles) that can distinguish homotopy equivalent spaces that are not homeomorphic. The purpose of this Arbeitsgemeinschaft was to learn about several natural generalisations of these classical invariants to families of manifolds. Regard a family~$p\colon E\to B$ of compact manifolds~$M$, equipped with a flat vector bundle~$F\to M$. Then the fibrewise cohomology groups~$H^\bullet(E/B;F)$ form flat vector bundles over the base~$B$. The starting point for our investigations are analogues of the Atiyah-Singer family index theorem that relate~$F$ to~$H^\bullet(E/B;F)$. To a flat vector bundle~$F\to M$, one associates Kamber-Tondeur characteristic classes $c_\bullet(F)$ in~$H^{\text{odd}}(M;\mathbb R)$, which vanish if~$F$ carries a parallel metric. By Bismut-Lott~\cite{BLin}, one has $$\sum_i(-1)^ic_\bullet\bigl(H^i(E/B;F)\bigr) =\int_{E/B}e(TM)\,c_\bullet(F) \quad\in H^*(B;\mathbb R)\;,$$ where~$e(TM)$ is the Euler class of the vertical tangent bundle, and the right hand side is the Becker-Gottlieb transfer in de Rham cohomology. If one specifies some additional geometric data, then all classes above are naturally represented by specific differential forms. On the level of differential forms, the equation above only holds up a correction term~$d\mathcal T$. Here~$\mathcal T$ is the higher analytic torsion, which depends naturally on the fibration and the geometric data. If both~$H^\bullet(E/B;F)$ and~$F$ admit parallel metrics, then~$\mathcal T$ gives rise to a secondary characteristic class~$\mathcal T(E/B;F)\in H^{\text{even},\ge2}(B;\mathbb R)$. Dwyer-Weiss-Williams \cite{DWWin} construct Reidemeister torsion for a smooth fiber bundle $p\colon E \to B$ as a byproduct of a family index theory. If $p$ is any fiber bundle with fibers compact topological manifolds and base a CW complex, then the family index theory states that $\chi(p)$, the A-theory Euler characteristic of $p$ is determined by the A-theory Euler class of $\tau_{fib}(p),$ the tangent bundle along the fiber. Here A-theory is algebraic K-theory of spaces in the sense of Waldhausen. More precisely, by applying fiberwise Poincare duality, and then an assembly map to the A-theory Euler class, one gets the A-theory Euler characteristic. If $p$ is a smooth bundle, then one gets a stronger smooth index theorem where the A-theory Euler class is replaced by the Becker-Euler class, which lives in the (twisted) stable cohomotopy of $E$. When $B$ is a point this result is equivalent to the classical Poincare-Hopf theorem. The third approach is due to Igusa-Klein~\cite{Ibookin}, and is somewhat different in nature. Here, one regards a generalised fibrewise Morse function on~$M\to B$. Together with a flat vector bundle~$F\to M$, this gives rise to a classifying map from~$B$ to a Whitehead space, and the higher Franz-Reidemeister torsion is the pullback of a universal class on the Whitehead space. There are conjectural relations between all three definitions of higher torsion. In a special case, Igusa has characterized higher Franz-Reidemeister torsion axiomatically; checking these axioms for either of the other higher torsions would prove equality. For some bundles, equality of higher Franz-Reidemeister torsion and higher analytic torsion can be shown analytically using the Witten deformation. Finally, one expects that higher Franz-Reidemeister torsion can be recovered from Dwyer-Weiss-Williams torsion. It turns out that higher torsion invariants are somewhat finer than classical FR torsion, since they detect higher homotopy classes of the diffeomorphism group of high-dimensional manifolds that vanish under the forgetful map to the homeomorphism group. In particular, these invariants distinguish differentiable structures on a given topological fibre bundle~$M\to B$, where one may even fix differentiable structures on~$M$, $B$ and the typical fibre. There are also applications of higher torsions to problems in graph theory and moduli spaces of compact surfaces. Some of these were sketched throughout this Arbeitsgemeinschaft. The talks were grouped as follows. \begin{enumerate} \item The first talk gave a short introduction to classical torsion invariants. \item In talks 2--7, we discussed the Dwyer-Weiss-Williams homotopy theoretical approach. \item Parametrized Morse theory, Kamber-Tondeur classes and Igusa-Klein torsion were discussed in talks 8--16, and some applications were given. \item Finally, based on talks 10 and 11, we introduced analytic torsion in the talks 17--19. \end{enumerate} The meeting took place from April 2nd till April 8th 2006 and was organized by Sebastian Goette (Regensburg), Kiyoshi Igusa (Brandeis) and Bruce Williams (Notre Dame). It was attended by 43 participants, coming mainly from Europa and the USA. \begin{thebibliography}{99} \bibitem{BLin} J.-M. Bismut, J. Lott, \textit{Flat vector bundles, direct images and higher real analytic torsion}, J. Am. Math. Soc. \textbf{8} (1995), 291--363. \bibitem{DWWin} W.~Dwyer, M.~Weiss, B.~Williams, \textit{A parametrized index theorem for the algebraic $K$-Theory Euler class}, Acta Mathematica \textbf{190} (2003), 1--104. \bibitem{Ibookin} K.~Igusa, \textit{Higher Franz-Reidemeister Torsion}, AMS/IP Studies in Advanced Mathematics 31, International Press, 2002. \end{thebibliography}
Algebraic geometry
Algebraic topology
General
979
1026
10.4171/OWR/2006/16
http://www.ems-ph.org/doi/10.4171/OWR/2006/16
The Rigorous Renormalization Group
Christoph
Kopper
École Polytechnique, PALAISEAU CEDEX, FRANCE
Vincent
Rivasseau
École Polytechnique, PALAISEAU CEDEX, FRANCE
Manfred
Salmhofer
Universität Heidelberg, HEIDELBERG, GERMANY
The workshop on \emph{The Rigorous Renormalization Group}, was attended by more than 40 participants coming mainly from Western Europe and from America. The official programme consisted in 19 lectures of 60 minutes each (plus discussion). Four of them were devoted to noncommutative field theory, three of them presented methods used for and results on the construction of a non-gaussian fixed point in a statistical mechanics/quantum field theory model, and two lectures concerned, respectively, nonlinear $\sigma$-models, the functional renormalization group, and quantum electrodynamics. The remaining six lectures were on the Brockett-Wegner version of the renormalization group, on random walks, on Fermi liquids, on anomalies in quantum field theory, on renormalization in curved spaces and on functional integrals for many boson systems. The scientific programme, the atmosphere and the Oberwolfach style of the meeting, leaving much room for informal discussions and joint work, were generally highly appreciated. The abstracts of the lectures are presented in chronological order.
Quantum theory
General
1027
1076
10.4171/OWR/2006/17
http://www.ems-ph.org/doi/10.4171/OWR/2006/17
Differential-Algebraic Equations
Stephen
Campbell
North Carolina State University, RALEIGH, UNITED STATES
Roswitha
März
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Linda
Petzold
University of California, SANTA BARBARA, UNITED STATES
Peter
Rentrop
TU München, GARCHING BEI MÜNCHEN, GERMANY
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: \begin{itemize} \item abstract differential algebraic systems, coupled systems, partial differential algebraic systems; \item analysis of (ordinary) differential algebraic equations and application of numerical methods to problems having new mathematical complexity; \item innovative and improved numerical integration methods to solve highly complex application problems; \item optimization with constraints described by DAEs and control problems concerning DAEs. \end{itemize} 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.
Numerical analysis
General
1077
1168
10.4171/OWR/2006/18
http://www.ems-ph.org/doi/10.4171/OWR/2006/18
Topological and Geometric Methods in Group Theory
Martin
Bridson
University of Oxford, OXFORD, UNITED KINGDOM
Peter
Kropholler
University of Glasgow, GLASGOW, UNITED KINGDOM
Karen
Vogtmann
Cornell University, ITHACA, UNITED STATES
The focus of this meeting was the use of topological and geometric methods to study infinite discrete groups. These methods are increasingly being supplemented by powerful new techniques from analysis. Key topics included group actions on CAT(0) and tree-like spaces, filling invariants, cohomology, K-theory and $\ell^2$-cohomology, amenability and Kazhdan's property~(T), and deformation spaces, curve complexes and Teichm\"uller space. More specific information is contained in the abstracts which follow in this volume. The meeting was organized around a series of 23 lectures each of 50~minutes' duration representing the major recent advances in the area. The first day's lectures were selected from the abstracts and talk proposals which had been submitted in advance of the meeting, and the remainder were decided on Monday and Tuesday morning; in this way were able to take advantage of a full range of abstracts and also to incorporate last-minute information about exciting developments in the field. We had interesting proposals from virtually every participant but lecture slots for fewer than half to speak. We posted all abstracts on the wall of the lecture building and drew attention to them by running a poster event on Tuesday evening, at which every participant took the opportunity to introduce their research and poster. This worked very effectively and was valuable especially to younger people or those visiting the Forschungsinstitut for the first time. It also led to small groups getting together for unofficial lecture sessions in the evening and afternoon breaks on subsequent days. There were 51 participants from a wide range of countries, including Germany, France, the United States, the United Kingdom, Greece, Russia, Poland, Switzerland and Australia. We are grateful to the European Union, which provided funds to support 6 advanced graduate students, a number of recent Ph.~Ds and a few senior researchers. We feel that the meeting was exciting and highly successful. The quality of the lectures was outstanding, and outside of lectures there was a constant buzz of intense mathematical conversations. One indication of the high degree of current activity and interest in the subject is the fact that four of the participants (including two of the organizers) have been invited to speak at the upcoming International Congress of Mathematicians in Madrid.
Group theory and generalizations
General
1169
1214
10.4171/OWR/2006/19
http://www.ems-ph.org/doi/10.4171/OWR/2006/19
Feinstrukturtheorie und Innere Modelle
Ronald
Jensen
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Menachem
Magidor
The Hebrew University, JERUSALEM, ISRAEL
Ralf-Dieter
Schindler
Universität Münster, MÜNSTER, GERMANY
The workshop \emph{Fine Structure Theory and Inner Models}, organised by Ronald Jensen (Berlin), Menachem Magidor (Jerusalem) and Ralf Schindler (M\"unster) 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 G\"odel'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 ${\sf ZFC}$ 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 {\em Woodin cardinal} is a crucial one here. The main issues of this area are the following. \begin{itemize} \item {\em Fine structure theory.} This is a general theory of the definability over the structures that form the building blocks of the inner models one wants to construct. In most cases, these structures are {\emph premice}, that is, models constructed from sequences of extenders which code fragments of elementary embeddings. The existence of such embeddings is the essence of the crucial large cardinal concepts. \item {\em Iterability.} That a (well-founded) structure be iterable means that we can keep taking ultrapowers of it (i.e., decoding the elementary embedding coded by some extender on the sequence of the structure, along with the target model, which, by elementarity, is again a model constructed relative to a sequence of extenders) without ever producing non-well-founded structures. In fact, what one needs for iterability is an iteration strategy for the given structure. The iterability of a premouse is the key property one needs in order to choose the next building block in the construction of an inner model in a canonical way. This eventually makes the resulting inner model {\em definable} in some reasonable way. Also, without iterability we wouldn't know how to prove key (fine structural) first order properties which we require of our premice and which are then inherited by the inner model we are about to construct. For instance, the fact that the (generalized) continuum hypothesis holds in the inner models we construct relies on iterability. It is important here to isolate criteria for the iterability of a premouse which are not too strong so that sufficiently many iterable premice can be shown to exist. \item {\em The model construction.} The construction of an inner model is done by recursion on its ``building blocks''. In order to verify that the construction doesn't trivialize one has to prove that sufficiently many premice meet the iterability criterion one works with. Also, one wants to show that a Covering Lemma holds for the inner model which was built. \item {\em Applications of inner models.} Woodin's core model induction makes use of ``locally defined'' inner models which are used for verifying inductively that (sufficiently iterable) models of ${\sf ZFC}$ plus there are such-and-such many Woodin cardinals exist. This induction can therefore be used for showing that a given (say, combinatorial) statement implies that definable sets of reals are determined. It turns out that in order to organize such an induction properly, one has to construct a new kind of ``hybrid'' premice which are constructed not only relative to a sequence of extenders, but also relative to iteration strategies for certain structures. \end{itemize} 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.
Mathematical logic and foundations
General
1215
1244
10.4171/OWR/2006/20
http://www.ems-ph.org/doi/10.4171/OWR/2006/20
Mini-Workshop: Zeta Functions, Index and Twisted K-Theory; Interactions with Physics
Sylvie
Paycha
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Steven
Rosenberg
Boston University, BOSTON, UNITED STATES
Simon
Scott
King's College London, LONDON, UNITED KINGDOM
This mini-workshop brought together number theorists, analysts, geometers and mathematical physicists to discuss current issues at the common boundary of mathematics and physics. Topics covered included the number theoretic and algebraic structures underlying renormalization, twisted K-theory and higher algebraic structures, modular forms, and arithmetic and spectral zeta functions. A particular theme was around developing interconnections between arithmetic (multiple) zeta functions, spectral zeta functions associated with elliptic operators (and related spectral invariants such as spectral flow) and current issues in physics such as renormalization and mirror symmetry. Multiple zeta functions appear in index theory and $K$-theory via their relation to anomalies, in number theory in their relation to polylogarithms, in renormalization questions in perturbative quantum field theory and Hopf algebras, in duality issues and in twisted $K$-theory for index theorems for projective families of elliptic operators, thereby providing a rich set of overlapping topics with common analytical issues. This meeting was organized around one hour talks, four each day, with plenty of time between talks for informal discussion and a 45 minute talk in the afternoon for students; three graduate students were among the 16 participants. Some participants lectured for two hours in order to have time to introduce the audience to the subject before entering the technical details. The organizers and participants would like to thank the {\em Mathematisches For\-schungs\-institut Oberwolfach} for providing a pleasant and stimulating enviroment for this meeting.
Number theory
Quantum theory
General
1245
1284
10.4171/OWR/2006/21
http://www.ems-ph.org/doi/10.4171/OWR/2006/21
Mini-Workshop: Studying Original Sources in Mathematics Education
Fulvia
Furinghetti
Università di Genova, GENOVA, ITALY
Hans Niels
Jahnke
Universität Duisburg-Essen, ESSEN, GERMANY
Jan
van Maanen
Universiteit Utrecht, UTRECHT, NETHERLANDS
In the last thirty years quite some initiatives evolved and much material was developed for using the history of mathematics in the teaching of mathematics at all levels. There is a growing consensus that historical work of pupils and students may contribute to further through: \begin{itemize} \item providing insights into the development of mathematical concepts; \item developing a deeper understanding of the role of mathematics in our surrounding world and its relation to applications, culture and philosophy; and \item encouraging the perception of the subjective dimensions of mathematics: of aims and intentions in the building of mathematical concepts and algorithms, of alternative methods and of personal and creative aspects. \end{itemize} \noindent Among the various possible activities by which historical aspects might be integrated into the teaching of mathematics, the study of an original source is the most demanding and the most time consuming. It requires a detailed and deep understanding of the mathematics in question, of the time when it was written and of the general context of ideas. The aspect of language becomes important in ways which are completely new compared with usual practices of mathematics teaching. Thus, reading a source is an especially ambitious enterprise, but rewarding and capable of substantially deepening the mathematical understanding. In principle, the aims and effects which might be pursued through the study of an original source will not be different from those attained by other types of historical activities. However, there are some ideas which are specifically supported by reading mathematical sources. \begin{enumerate} \item Studying an original source replaces the usual with something different: it allows student and teacher to see mathematics as an intellectual activity, rather than as just a corpus of knowledge or a set of techniques. For example, Newton's letter to Leibniz of 1676 in which he described how as a young man of 22 years he arrived at the general binomial formula (a cornerstone in his fluxional calculus) is a unique document for a process of mathematical invention progressing by bold generalisations and analogies. Through the reading of the letter, the student more or less feels the presence of the inventor. \item Integrating sources in mathematics challenges the learner's perceptions through making the familiar unfamiliar. Coming to grips with a historical text can cause a reorientation of the learner's views and thus deepen his or her mathematical understanding. All too often in teaching, concepts appear as if already existing and they are manipulated with no thought for their construction. Sources remind students that these concepts were invented and that such invention did not happen all by itself. As an example, we might refer to Leibniz' version of the calculus. There are many experiences which show that students are motivated to reflect about the limit approach to calculus when they study Leibniz' way of dealing with infinitely small quantities. Also the teacher may gain insight by concentrating on the unfamiliar. It is often difficult enough to cope with unexpected solutions by students; however, studying sources enables to the teacher and students to keep an open mind. \item Integrating the study of sources in mathematics education invites the learner to place the development of mathematics in the scientific and technological context of a particular time and in the history of ideas and societies. One of many examples from antiquity to the present is provided by Heron's textbook (1st century A.D.) on land surveying called \emph{The Dioptra}. Reading parts of it connects the topic of similarity to the context of ancient surveying techniques and shows the astonishingly high achievements of ancient engineers in this and other areas. Such sources may as well provoke students to engage in practical activities (simulations, measurements, theatre), which otherwise would not come to their mind or to the mind of their teacher. \item Reading a source is a type of activity which is oriented more to processes of understanding than to final results. The complete meaning of a historical text may remain open, and it occurs quite often that the same text leads to different readings. Of course, this does not entail arbitrariness. The reader has to give reasons in support of his or her interpretation. As an example we refer to the highly interesting story of negative and/or complex numbers. Reading sources about this topic poses in every case the question whether, and if yes, in which sense these creations were understood as legitimate numbers in different historical times. Doubts that students themselves have from time to time are reflected by the doubts that existed through the ages. \item When working with original sources at least three different languages interact in the classroom: the language of the source, the modern terminology of the mathematical topic in question and the everyday language which has evolved in the classroom. This requires of the learner competencies of translation and switching between these languages which are highly desirable from an educational point of view since communication between expert mathematicians and people who want a problem solved mathematically is one of the main problems of applying mathematics. \end{enumerate} \section{Theoretical and practical orientations} \noindent The mini-workshop comprised sessions of different types. Most of the meetings were devoted to traditional presentations of papers. On the other hand, in some sessions the participants discussed the needs and aims of the future development of the field. As a result, research questions were identified which evolved from work in the past and might be helpful in orienting future work. They reflect central issues related to the integration of original sources from the history of mathematics into mathematics education. Each of the questions addresses both the learning of mathematics (by secondary school and university students and by prospective or in-service teachers) and the teaching of mathematics (at the secondary and university levels). In both cases, each of the questions retains its general formulation; however, each is approached differently by the authors according to the target population and their intended educational goals. Thus each question may have more than one answer. \begin{enumerate} \item What are the possible epistemological/theoretical basis and frameworks for research and development towards the integration of original sources into the teaching and learning of mathematics? \item What are the characteristics of viable models for implementing the integration of original sources in the teaching and learning of mathematics? \item What is the actual impact of these models on students' and teachers' learning and understanding of mathematics, and on teachers' teaching practices? \item How can historical research and practice inspire, impact, support or supply explanatory frameworks and working tools for research on learning and teaching mathematics? \item How can research and practice in mathematics education inspire, support and broaden the research in the history of mathematics in general, and on original sources in particular? \end{enumerate} Another issue that came up in the workshop was the problem of upscaling. There is no reason to believe that teaching which is done by an enthusiast with good results can easily and successfully be repeated by the average teacher. It is a welcome development that new materials are being published and that research is being done on projects where "average teachers" are doing the teaching. In many countries mathematics education standards are in the process of being elaborated. These standards often appear as collections of mathematical problems. The approach of reading sources can be successful in the future only if the community will produce problems in a format adequate to be included among these standard problems. Some contributions during the meeting showed that this is in fact possible. A related issue is the important role history of mathematics, particularly the reading of sources, might play in the training of teachers for all levels. Studying sources can provide awareness for subtleties in the meaning of mathematical concepts which cannot be afforded otherwise. Thus, the sensitivity of teachers in regard to content-related difficulties of their students might be considerably enhanced. The workshop showed that many activities are in place worldwide which try to take advantage of an approach which includes the study of sources. \section{The Workshop} \noindent Most of the contributions during the workshop were related to and inspired by one or more of the research questions outlined in the previous section. \begin{itemize} \item An important development in recent years is that more empirical research studies on the integration of original sources are being done, many of which include a large number of students. A few of them were presented here (Glaubitz; Clark; Peters; van Maanen, reporting about his student Iris van Gulik-Gulikers). \item Other talks were focusing on theoretical issues based on examples from practice (Arcavi; Bardini, Radford). \item Some papers gave examples from the presenters' own practice with comments on their theoretical background (Barbin; Dematt\'{e}; Wann-Sheng Horng; van Maanen; Rasfeld; Reich). \item In two sessions the audience was invited to take part in working on historical sources (Pengelley; Jahnke). \item One talk gave an overview on curricula, textbooks and teachers and their roles in making history of mathematics part of mathematics education (Smestad). \end{itemize} The organizers thank the Institute staff for providing a comfortable environment to the participants.
Mathematics education
General
1285
1318
10.4171/OWR/2006/22
http://www.ems-ph.org/doi/10.4171/OWR/2006/22
Interactions between Algebraic Geometry and Noncommutative Algebra
Dieter
Happel
Technische Universität Chemnitz, CHEMNITZ, GERMANY
Lance
Small
University of California, San Diego, LA JOLLA, UNITED STATES
J. Toby
Stafford
The University of Manchester, MANCHESTER, UNITED KINGDOM
Michel
Van den Bergh
Hasselt University, HASSELT, BELGIUM
This meeting had over 45 participants from 11 countries (Australia, Belgium, Canada, France, Germany, Italy, Israel, Norway, Russia, UK and the US) and 26 lectures were presented during the five day period. The sponsorship of the European Union allowed the organizers to invite and secure the participation of a number of young investigators. Some of these young mathematicians presented thirty minute lectures. As always, there was a substantial amount of mathematical activity outside the formal lecture sessions. This meeting explored the applications of ideas and techniques from algebraic geometry to noncommutative algebra . Several lecturers presented open problems to stimulate the interest of researchers in other areas. Areas covered include \begin{itemize} \item{} noncommutative projective algebraic geometry, \item{}Hopf algebras, \item{}combinatorial ring theory, \item{} symplectic reflection algebras, \item{} representation theory of quivers and preprojective algebras \item{} homological techniques and derived categories \end{itemize} The sweep of the meeting can be seen from de Jong's contribution that uses contemporary algebraic geometry to prove a theorem in the classical theory of finite dimensional division algebras to the works of Keller-Reiten and Ingalls on cluster algebras. Additionally, de Jong notes a result obtained during the workshop with van den Bergh. Looking to the future, Goodearl and Zelmanov propose a number challenging problems. Zelmanov discusses both an interesting Lie algebra example and a possible connection to an old problem of Kurosh. The previous paragraph represents just a sampling of the scope and variety of the mathematics at the meeting. The abstracts following will give the whole story.
Algebraic geometry
General
1319
1384
10.4171/OWR/2006/23
http://www.ems-ph.org/doi/10.4171/OWR/2006/23
Mathematical Biology
Emmanuele
DiBenedetto
Vanderbilt University, NASHVILLE, UNITED STATES
Benoît
Perthame
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Angela
Stevens
Universität Heidelberg, HEIDELBERG, GERMANY
This meeting on Mathematical Biology tied in with the long tradition of these workshops in Oberwolfach and at the same time aimed to account for the fast growing synergy between biology and mathematics of the recent years. The use of new instrumentation and visualization methods at the molecular scale in biological and medical experiments allows for measurements which have not been possible a few years ago. Major questions for theoreticians and experimentalists are how to tackle this vast complexity of biological information and data, and, more important, if underlying principles can be found. Finding these would enable the field to become more predictive. Here is exactly where mathematical modeling, analysis, and simulation can contribute. On the other hand, mathematical biologists and mathematicians are now providing first new models to explain the measurements, and these models are ready for mathematical analysis. The synergy between mathematics and physics, chemistry, engineering, and material sciences, has already proven to greatly advance the respective sciences and mathematics itself. To further deepen the connections between mathematics and biology, a group of experimental biologists, mathematical biologists, and mathematicians - especially many young scientists - met, joining the lively talks and discussions in this workshop. The meeting intentionally focussed on some specific biological topics this time. Among those were cell movement, where results on the dynamics of the cellular cytoskeleton were presented, as well as on chemotaxis and cell adhesion. Questions of pattern and structure formation in cell systems were discussed for self-organizing microorganisms and cancer invasion. The analysis of structured population models in this context is new, but has a long tradition in ecology and epidemiology. Further topics of interest with clear mathematical challenges were transport and molecular motors, the organization of cell membranes, and the process of photo-transduction Wherever possible, the experimentalists talks were placed in tandem with related presentations on mathematical modeling. Mathematical topics were: reaction-diffusion equations, parabolic and hyperbolic chemotaxis equations, fluid dynamics, variational principles and methods based on the Wasserstein distance, homogenization, singular perturbations, bifurcation analysis, and numerical simulations. Besides the lectures, two discussion groups were organized, one on mathematical results for chemotaxis equations and one on cell motility. A round table discussion on `mathematical modeling in biology, aims and scopes' rounded off the meeting, not to forget the nice concert, organized by some of the participants. We would like to express our sincere thanks to the very dynamic and kind support of the Oberwolfach team before and during the workshop.
Biology and other natural sciences
General
1385
1462
10.4171/OWR/2006/24
http://www.ems-ph.org/doi/10.4171/OWR/2006/24
Pro-p Extensions of Global Fields and pro-p Groups
Nigel
Boston
University of Wisconsin, MADISON, UNITED STATES
John
Coates
University of Cambridge, CAMBRIDGE, UNITED KINGDOM
Christian
Liedtke
Universität Bonn, BONN, GERMANY
The meeting \emph{Pro-$p$ Extensions of Global Fields and pro-$p$ Groups} was organised by Nigel Boston (Madison), John Coates (Cambridge) and Fritz Grunewald (D\"{u}sseldorf). As the name of the meeting conveys, a primary aim was to bring together group theorists working in the field of pro-$p$ groups and number theorists interested in pro-$p$ extensions of global fields. The workshop consisted of over $25$ talks, supplemented by informal discussions. Topics included: Galois groups of extensions with restricted ramification, self-similar and automata groups, non-commutative Iwasawa theory, groups acting on rooted trees. This meeting was well attended with over $50$ participants; more than $30$ of these came from countries other than Germany. The range of topics and the diverse backgrounds of the participants led to a stimulating exchange of recent results, challenging problems and general ideas. The organisers and participants thank the \emph{Mathematisches Forschungsinstitut Oberwolfach} for providing the setting for this successful workshop. The following extended abstracts appear in the chronological order of the talks; they were collected and edited by Benjamin Klopsch (D\"{u}sseldorf).
Group theory and generalizations
General
1463
1536
10.4171/OWR/2006/25
http://www.ems-ph.org/doi/10.4171/OWR/2006/25
Teichmüller Space (Classical and Quantum)
Shigeyuki
Morita
University of Tokyo, TOKYO, JAPAN
Athanase
Papadopoulos
Université de Strasbourg et CNRS, STRASBOURG CEDEX, FRANCE
Robert
Penner
Aarhus University, AARHUS C, DENMARK
In a broad sense, the subject of Teichm\"uller theory is the study of moduli for geometric structures on surfaces. The progenitor of the subject is usually considered to be G. F. B. Riemann, who in a famous paper on Abelian functions, studied the moduli space of algebraic curves and stated that the space of deformations of equivalence classes of conformal structures on a closed orientable surface of genus $\mathfrak{g} \geq 2$ is of complex dimension $3\mathfrak{g}-3$. This was explicated by O. Teichm\"uller who laid the foundations of the theory in a series of famous papers (during a remarkably brief period). Many prominent mathematicians including L. Ahlfors and L. Bers continued developing the theory over several decades. In the 1970s, W. Thurston introduced techniques of hyperbolic geometry in the study of Teichm\"uller space and its asymptotic geometry. In the 1980s, new combinatorial treatments of Teichm\"uller and moduli spaces evolved with a concurrent interplay of ideas from string theory in high-energy physics. Teichm\"uller theory is one of those precious subjects in mathematics which have the advantage of bringing together, at an equally important level, fundamental ideas coming from different fields. Among the fields associated to Teichm\"uller theory, one can surely mention complex analysis, hyperbolic geometry, discrete group theory, algebraic geometry, low-dimensional topology, Lie groups, symplectic geometry, dynamical systems, topological quantum field theory, string theory, and many others. Teichm\"uller theory is growing at a fantastic rate, and the fact that it involves all these areas is probably a consequence of the fact that Teichm\"uller space itself carries a diversity of rich structures. As a matter of fact, this space can be seen from at least three points of view: as a space of equivalence classes of hyperbolic metrics, as a space of equivalence classes of conformal structures, and as a space of equivalence classes of representations of the fundamental group of a surface into a Lie group. Each of these points of view endows Teichm\"uller space with various structures, including several interesting metrics, a natural complex structure, a symplectic structure, a real analytic structure, an algebraic structure, cellular structures, various boundary structures, a natural discrete action by the mapping class group, interesting geodesic and horocyclic flows on the quotient Riemann moduli space, a quantization theory of its Poisson structure, and the list goes on and on. The quantization of Teichm\"uller space was developed in the last few years by L. Chekhov and V. Fock and independently in work by R. Kashaev. This theory produces noncommutative families of deformations of the Poisson or symplectic structure of Teichm\"uller space in the form of $*$-algebras, with an action of the mapping class group of the surface as an outer automorphism group. In particular, quantization of Teichm\"uller space leads to new invariants of hyperbolic three-manifolds. The conference brought together people in almost all of the active areas of Teichm\"uller theory. The fact that Teichm\"uller theory is a living and rich subject connecting several areas of mathematics was reflected in the richness of the talks that were presented, and in the variety of the new perspectives that were discussed at the problem session, on which we report separately. We note that many other attendees were ready to give interesting talks than time permitted. As a general rule, younger researchers were given the opportinity to present their own work. In this short report, we have divided the talks that were delivered in five groups: {\it 1) Metric theory.} U. Hamets\"adt reported on her recent work on the behaviour in moduli space of images of certain closed geodesics for the Teichm\"uller metric, namely, for every compact set $K$, one can find such images which do not intersect $K$. G. Schmith\"usen reported on Teichm\"uller disks, which are embeddings of the Poincar\'e disks which are isometric with respect to the Teichm\"uller metric. G. Th\'eret gave a talk on Thurston's asymmetric metric on Teichm\"uller space and presented results on the convergence of certain geodesics to points on Thurston's boundary. {\it 2) Mapping class groups and the associated simplicial complexes.} V. Markovich gave a review of several realization problems for the mapping class group and he reported on his result stating that for any closed surface $S$ of genus $\geq 6$, the natural projection from the space of homeomorphisms to the mapping class group has no section. This result answered a famous open problem. D. Kotschick gave a survey of his work on quasi-homomorphisms with applications to the mapping class group. J. McCarthy gave a talk in which he described the automorphism group of a recently introduced simplicial complex, the complex of domains of a surface (joint work with A. Papadopoulos). E. Irmak described recent results on superinjective simplicial maps of the curve complex, on the automorphism group of the complex of nonseparating curves, and on the Hatcher-Thurston complex of cut systems of curves (some of this work is joint with J. McCarthy and with M. Korkmaz). N. Wahl described a stability theorem for the homology of the mapping class group of non-orientable surfaces which is analogous to Harer's theorem for orientable surfaces. K. Fujiwara spoke on the geometry of the curve graph showing that the asymptotic dimension of this graph is finite, and that for surfaces of genus $\geq 2$ with one bundary component, the dimension is at least two. A description of symplectic structures on Lefschetz fibrations using algebraic properties of the mapping class group was given by M. Korkmaz. At a more algebraic level, N. Kawazumi described recent work on characterictic classes in the mapping class group, in which he constructs higher analogues of the period matrix in order to obtain ``canonical" differential forms that represent all the Morita-Mumford classes and their higher relations. R. Cohen gave a talk on joint work with I. Madsen on a generalized Mumford conjecture on the stable cohomology of the mapping class group and a general version of homology stability for that group in the setting of spaces of Riemann surfaces with appropriate boundary conditions in a simply connected target manifold. G. Mondello reported on his work relating the tautological classes to cycles of Witten and Kontsevich, which are constructed combinatorially (using fatgraphs). {\it 3) Quantum theory.} R. Kashaev described a new and elegant quantization of a homology bundle over Teichm\"uller space related to his earlier work. L. Chekhov reported (on his joint work with Penner) quantizing Thurston's projective lamination space for the once-punctured torus. V. Fock discussed an example from cluster algebras giving an explicit relationship between Teichm\"uller geometry and representation theory which is related to his recent work with A. Goncharov on higher Teichm\"uller spaces. Y. Gerber described a new construction of a collection of surface mapping classes with computable quantum invariants leading to new invariants for fibered knots. F. Bonsante gave a report on his recent work with R. Benedetti on constant curvature Lorentzian structures on manifolds that are topologically the product of a hyperbolic surface with the real line. {\it 4) Dynamics.} M. M\"oller reported on joint work with I. Bouw on billiards in relation to Veech surfaces, projective affine groups and Teichm\"uller curves in the moduli space of curves characterizing these curves by properties of the variation of Hodge structures. The talk by U. Hamenst\"adt, mentioned in {\it 1)} above, involved the Teichm\"uller geodesic flow on quotient of the space of quadratic differentials by the action of the mapping class group. M. Mirzakhani studied the ergodic properties of natural flows on moduli space in relation to the asymptotic behaviour of simple closed geodesics on hyperbolic surfaces. {\it 4) Complex geometry.} Y. Imayoshi gave a talk on joint work with T. Nogi on the complex analytic structure of moduli space together with its Deligne-Mumford compactification. Continuing ideas that originate in work of Kodaira, he described a cut-and-paste construction which produces holomorphic families of closed Riemann surfaces of genus two over a four-punctured torus, to which they associate two holomorphic sections. {\it 5) Higher Teichm\"uller theory} A. Wienhard gave a talk on her recent work with M. Burger and A. Iozzi on representations of the fundamental group of the surface into simisimple Lie groups of Hermitian type. G. McShane described geometric identities for surfaces that are related to Hitchin's a component of the representation variety of the fundamental group of a compact surface into $SL(n, R)$. D. Dumas and S. Kojima spoke on complex projective structures on surfaces, the space $\mathcal{P}(S)$ of which (equivalence calsses) can be considered as a higher-analog of Teichm\"uller space. $\mathcal{P}(S)$ is a fibre bundle over Teichm\"uller space, and like Teichm\"uller space itself, can be studied from different points of view: complex analysis (via the Schwarzian derivative) and hyperbolic geometry(via Thurston's {\it grafting} map). Some of the most interesting questions in the theory of projective structures relate the two points of view, and the talk by Dumas focused on this relation. Kojima described a geometric parametrization of the moduli space of projective surfaces by cross ratios. He develops (together with S. Mizushima and S. P. Tan) a theory of circle packings in projective geometry which can be traced back to works by Andreev and by Thurston.
Differential geometry
General
1537
1614
10.4171/OWR/2006/26
http://www.ems-ph.org/doi/10.4171/OWR/2006/26
Classical Algebraic Geometry
David
Eisenbud
Mathematical Sciences Research Institute, BERKELEY, UNITED STATES
Joseph
Harris
Harvard University, CAMBRIDGE, UNITED STATES
Frank-Olaf
Schreyer
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
The Workshop on Classical Algebraic Geometry was notable for a relaxed atmosphere (18 talks) and an abundance of young people. A wide variety of themes related to classical topics were discussed with a very modern point of view. Although it is tempting to summarize each of the talks, we limit ourselves to four highlights: \begin{itemize} \item There has been a great deal of interest in the question: are there structural characterizations of rational varieties in higher dimensions? Rationality itself is elusive: the notion of ``rationally connected variety'' (a variety where any two points can be connected by a rational curve) seems much more tractable. Brendan Hassett described work of his with Yuri Tschinkel showing that these varieties exhibit an analogue of a famous arithmetic property of quadrics: if a family of varieties has smooth rationally connected fibers, then given a collection of share at least some properties of quadrics in low dimensions. In many cases, a collection of ``local sections'' can be connected by a global section. \item A central theme of algebraic geometry is that the set of algebraic varieties of a particular kind often itself is naturallly an algebraic variety. It was classically assumed that such families would be generically nice in some sense. This has turned out not to be the case: to understand them, one must accept non-reduced components. The first examples of this phenomenon were given by David Mumford in a very famous paper. A second highlight of our conference was a re-interpretation and generalization of Mumford's example by Shigeru Mukai. \item A number of combinatorial and computational applications in algebraic geometry have recently come from what the statisticians have for a long time called the ``max-plus'' algebra---in algebraic geometry it now goes under the name ``tropical''. Sean Keel showed off a new application of these ideas, found jointly with Paul Hacking and Eugene Tevelev: the ``tropical fan'' associated with certain toric varieties provides an extremely nice and natural compactification of these varieties. Among the remarkable classical examples that Keel gave is that of the moduli space of smooth cubic surfaces. \item Among the algebraic varieties of algebraic varieties, the moduli space of curves of genus $g$ and some of its variants is by far the most important, with applications ranging from string theory in physics (Witten) to new versions of resolution of singularities (de Jong). Constructions of Severi and others from the early part of the 20th century showed that for low genus ($\leq 10$) the moduli space is rational, and Severi believed that he had proved rationality for all genera. The error in his argument was soon found, and it has been an important problem to decide which moduli spaces were actually rational. The importance of this work comes as much from the technique involved --- studying the divisor class group of the moduli space, which really means describing conditions on an algebraic curve that are locally given by just one equation --- as from the results. At our conference Farkas spoke on a very far-reaching generalization of what was known systematically using syzygies to describe conditions on curves that lead to new divisors. \end{itemize}
Algebraic geometry
General
1615
1662
10.4171/OWR/2006/27
http://www.ems-ph.org/doi/10.4171/OWR/2006/27
Applications of Asymptotic Analysis
Rupert
Klein
Freie Universität Berlin, BERLIN, GERMANY
Evariste
Sanchez-Palencia
Université Pierre et Marie Curie, PARIS, FRANCE
Jan
Sokolowski
Université Henri Poincaré, VANDOEUVRE LES NANCY, FRANCE
Barbara
Wagner
Applied Analysis and Stochastics, BERLIN, GERMANY
The workshop \emph{Applications of Asymptotic Analysis}, organised by Rupert Klein (Potsdam), Evariste Sanchez-Palencia (Paris), Jan Sokolowski (Nancy) and Barbara Wagner (Berlin) was held June 18th--June 24th, 2006. This meeting was well attended with 46 participants with a broad geographic representation. This workshop was a nice blend of young and senior researchers with various mathematical backgrounds. The objective of this workshop was to present the new developments of multiple scale asymptotics as they are developed for problems in various fields of application. It brought together experts working in different areas of asymptotic analysis and application fields and initiated exchange of new ideas and discussions of parallel developments. On the whole the atmosphere of this workshop was very cheerful and characterized by the mutual interest into each others expertise and approach. The themes of the workshop included: \begin{itemize} \item[-] Applications in shape optimization, where we discussed new tools like the internal and external topological derivatives and topological variations and their close relationship to basic research in asymptotic analysis of elliptic problems under singular perturbations of boundaries. \item[-] Also, new asymptotic problems that arise in thin shell theory were discussed. \item[-] A major field of application of asymptotic analysis in this workshop turned out to be numerical analysis. Examples within this workshop included new procedures for problems that involve multiple time and spatial scales, where adaptive tools alone are not robust. For boundary layer problems anisotropic finite element methods and the method of asymptotic decomposition of domains point to promising directions to tackle these complex problems. Furthermore, application of asymptotic analysis to finite difference schemes, where the grid spacing is the small parameter, could be shown to be very useful for studying consistency, stability and long-time behavior. \item[-] Naturally, new emerging singular perturbation methods were another focus of discussions. These included methods such as {\em Gevrey asymptotics} to treat phenomena of {\em boundary layer resonance} and {\em logarithmic switchback}. Other new developments that require {\em asymptotics beyond all orders} analysis were demonstrated for problems that exhibit the Stokes phenomenon, such as for the finger selection problem for Hele-Shaw flow with kinetic undercooling. \end{itemize} Apart from these themes, various other lectures on applications of asymptotic analysis were given, ranging from applications in biology, solid mechanics to fluid mechanics, in particular thin liquid films. Most of the lectures were given in the morning, with one hour overview lectures, followed by shorter half-hour talks. In the afternoon ample time was left for discussion sessions. Wherever possible the lectures and discussion sessions were grouped according to specific topics, which concerned the interplay of asymptotics and shape optimization (monday), numerical analysis and asymptotics (tuesday), mathematical theory and asymptotic methods (wednesday, thursday) and some new developments in homogenization theory (friday).
Numerical analysis
General
1663
1730
10.4171/OWR/2006/28
http://www.ems-ph.org/doi/10.4171/OWR/2006/28