bluuebunny/crossref_metadata_2025 huggingface dataset:
Mini-Workshop: Analytical and Numerical Methods in Image and Surface Processing
Authors: Dziuk Gerhard , Reif Ulrich , Rumpf Martin , Schröder Peter
Published: December 2009

In different areas of mathematics and computer science significant progress has been achieved with respect to geometric modeling, the processing of detailed surface models and geometric methods in morphological image processing. In contrast to classical tools in approximation theory such as spline curves and surfaces these new approaches rely on methods from the calculus of variations and geometric evolution problems. As examples we would like to mention here:

Recently it has become clear that even though the applications areas differ significantly the methodological overlap is enormous. E.g. currently Willmore flow is one of the hot topics in geometric analysis and at the same time it turned out to be an adequate approach for the restoration of destroyed surfaces. Simultaneously the Willmore energy is a fundamental building block in variational methods of morphological image processing. Indeed as for surfaces it can be used for restoration and the disocclusion of images. Furthermore, it enters important fairness criteria in geometry processing and the generation of

C^1

smooth surface models.

The progress in this new, active and already very successful research field has its roots in originally completely different disciplines of mathematics and computer science. Hence, an exchange of different methodologies and a joining of activities is very promising. This was the intention of the mini workshop. To achieve this goal researchers from completely different areas of mathematics and mathematically oriented computer science were invited to share their point of view and their recent achievements with others. The intention of this mini-workshop was to initiate stronger exchange, to encourage the exploration of synergies and the building of bridges between the different disciplines and the different types of approaches for closely related research fields. In particular we intended to bring together people from

Let us unroll one of these synergy fields:

The organizers contributed introductory talks into the different fields to stimulate discussion and interaction.

In parallel to this mini workshop another mini workshop on fluid interfaces took place. Together with the organizers of the latter workshop we arranged for several joined talks on the common interest field of geometric evolution problems by participants of both workshops. This interplay turned out to be very fruitful and inspiring.

One Thursday evening we had a longer evening talk on a new approach for the discretization of fluid motion given by Peter Schröder.
Enveloping Algebras and Geometric Representation Theory
Authors: Kumar Shrawan , Littelmann Peter , Soergel Wolfgang
Published: December 2009

Since its inception in the early seventies, the study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras.

As indicated above, one of the main goals behind this meeting was to bring together a group of participants with a wide range of interests in and around the geometric and the combinatorial side of the representation theory of Lie groups and algebras. We strongly believe that such an approach to representation theory, in particular interaction between geometry and representation theory, will open up new avenues of thought and lead to progress in a number of areas.

This diversity was well reflected in the expertize represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under the following three headings:

(1) The study of the structure and representation theory through

a) Slodowy slices and the Joseph ideal,
b) Equivalence of categories of representations and certain geometric counterparts,
c) Hall algebras, Quotient schemes and canonical bases, d) Richardson elements and birationality questions,
e) free Lie algebras and current algebras,
f) Dirac cohomology,
h) Gelfand-Zeitlin theory

(2) Combinatorial Aspects of Lie Systems particularly through

a) Saturation problems and Buildings,
b) Affine and double Affine Hecke algebras, Cherednik algebras,
c) Solvable lattice models

(3) Geometric Structures including

a) Representation theoretic methods in enumerative geometry,
b) Multiplicity free symplectic reduction,
c) Monodromy actions of braid groups,
d) Associated varieties for Lie super algebras,
e) affine Grassmannians,
f) Schubert varieties and Demazure modules

The single most important development reported in this conference seems to us to be the generalization of the categorical Satake isomorphism to quantum groups at roots of unity, which allows to bring in a whole lot of new geometry into representation theory. It seems likely that it will lead to the proofs of some open conjectures of Lusztig. We were particularly happy to have quite a few very strong younger participants and some highly promising postdocs. We tried to accomodate young postdocs, as much as feasible, to present their work by devoting one afternoon to their (shorter) talks.
Free Probability Theory
Authors: Biane Philippe , Speicher Roland , Voiculescu Dan-Virgil
Published: December 2009

The workshop
Free Probability Theory
, organised by Philippe Biane (Paris), Roland Speicher (Kingston), and Dan Voiculescu (Berkeley) was held March 27th–April 2nd, 2005. This meeting was well attended with over 50 participants with broad geographic representation from Austria, Canada, Denmark, France, Germany, Hungary, Japan, Netherlands, Poland, USA.

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 who were interested in learning about free probability.

Free probability theory is a line of research which parallels aspects of classical probability, in a non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. It grew out from attempts to solve some longstanding problems about von Neumann algebras of free groups. In the almost twenty years since its creation, free probability has become a subject in its own right, with connections to several other parts of mathematics: operator algebras, the theory of random matrices, classical probability and the theory of large deviations, algebraic combinatorics, topology. Free probability also has connections with applied mathematics (wireless communication) and some mathematical models in theoretical physics.

The Oberwolfach workshop on free probability brought together a very strong group of mathematicians representing the current directions of development in the area. The diversity of the particants and the ample free time left in the programme stimulated a lot of fruitful discussions, laying the seed for many new collaborations.

The programme consisted of 13 lectures of 50 minutes, supplemented by 13 shorter contributions of 30 minutes. Because of the various backgrounds of the participants much emphasis was put on making the lectures accessible to a broad audience; most of them provided a survey on the background as well as highlighting some recent developments in connection with free probability. Instead of trying to summarize all these developments we will let the following abstracts speak for themselves.
Kommutative Algebra
Authors: Bruns Winfried , Flenner Hubert , Huneke Craig
Published: December 2009

The workshop on Commutative Algebra was very well attended by the important senior researchers in the field and many promising young mathematicians. A major subgroup of the participants was formed by researchers in affine geometry, a neighboring field that has strong interactions with commutative algebra.

The NSF Oberwolfach program made it possible to increase the number of young participants considerably, and the organizers are very grateful to the NSF for its support.

The conference took place in a very lively atmosphere, made possible by the excellent facilities of the institute. There were 53 participants and 19 talks with a considerable number of lectures given by young researchers. The program left plenty of time for cooperation and discussion among the participants. We highlight the areas in which new results were presented by the lecturers:

(a)

Characteristic

p

-methods

Starting with the landmark results of Peskine and Szpiro, characteristic

p

methods have had a extremely strong influence on the development of commutative algebra. They were crystalized by Hochster and Huneke in the notion of tight closure, and have led to remarkable results in ideal and module theory.

(b)
Combinatorial commutative algebra and effective methods
The spectacular applications of commutative algebra to enumerative combinatorics two decades ago have developed into a subfield of commutative algebra that is very active now. The main objects are algebraic structures defined by monomials, in particular face rings of simplicial complexes and affine monoid algebras. Another driving force of this development are the powerful effective methods based on Gröbner bases since they ultimately rely on monomial computations.

(c)
Homological methods and invariants
This area started from the fundamental theorems of Auslander, Buchsbaum and Serre on regular rings. Homological properties are used in the major classification of commutative rings and their modules. In the last decade considerable progress has been made through the use of differential graded algebras, derived categories and the duality between the polynomial ring and the exterior algebra.

(d)
Multiplier ideals
The multiplier ideals of an ideal

I\subset \mathbb C[X_1,\dots,X_n]

constitute subtle invariants of the singularity defined by

I

. Originally defined in analytic terms, they have now been applied to algebraic problems and have led to surprising results in ideal theory, especially on uniform bounds for the asymptotic behavior of various kinds of powers of an ideal.

(e)
Affine geometry
Affine geometry deals with algebro-geometric questions of affine varieties. The talks concerned actions of the additive and multiplicative groups

{\mathbf G_a}

and

{\mathbf G_m}

on affine varieties and, in particular, the affine space. Moreover there were lectures on the famous Jacobian problem and possible generalizations to other varieties. In many of the contributions the close ties to commutative algebra became apparent and led to fruitful discussions.
Mini-Workshop: Aspects of Ricci-Flow
Authors: Ecker Klaus , Huisken Gerhard , Ilmanen Thomas
Published: December 2009

The motion of a Riemannian metric along its Ricci curvature,

\tfrac{d}{dt}g_{ij} = - 2R_{ij} (g(t))

was proposed in 1982 by Richard S. Hamilton as a geometric version of the heat equation suitable for uniformizing and smoothing the geometry of a given initial Riemannian manifold

(M^3, g_0)

. Hamilton's work has opened up the whole area of geometric evolution equations, leading to the discovery of new phenomena in these equations and to topological applications such as the classification of 3-manifolds of positive Ricci-curvature and certain 4-manifolds. Recent work of G. Perelman has indicated how to approach a proof of the Poincaé-conjecture and the Thurston Geometrization conjecture for 3-manifolds using Hamilton–Ricciflow.

The mini workshop has concentrated on a thorough technical investigation of that part of the work of Perelman that is related to Hamilton–Ricciflow with surgeries on a finite time interval. Together with further work of Perelman and also Colding–Minicozzi this part of Perelman's work implies a proof of the Poincaré conjecture when confirmed. The efforts of the workshop were greatly helped by previous work of other mathematicians on Perelman's work., e.g. the notes of B. Kleiner and J. Lott.

The workshop was able to confirm major sections in the two papers of G. Perelman including the entropy- and reduced volume estimates, the compactness properties of ancient solutions to the flow and the surgery construction. It was also able to reinterpret several arguments involving Alexandrov spaces from the viewpoint of smooth Differential Geometry.

When the workshop had to come to an end, its participants agreed that it would be very desirable to establish self-contained expositions of the following points:

(1) The boundedness of the curvature

\sup_{B_\rho(x)} R \leq c(\rho)

in the proof of the approximation theorem I.12.1 of Perelman's first paper.

(2) The approximation of mini-max surfaces in the varifold distance for immersed surfaces in the paper of Colding–Minicozzi.

(3) The survival of the reduced volume estimate past surgeries.

(4) The uniform control of a fixed scale

\rho > 0

past all surgeries on a finite time interval, below which the approximation theorem applies.

It seems that detailed self-contained expositions of (3) and in particular (4) require more effort than (1) and (2).
Nonlinear Evolution Problems
Authors: Ecker Klaus , Shatah Jalal , Struwe Michael
Published: December 2009

The workshop ‘Nonlinear Evolution Problems’ focussed on three types of nonlinear evolution equations:

(1) Geometric evolution equations (essentially of parabolic type)

(2) Nonlinear hyperbolic equations

(3) Dispersive equations

The programme consisted of 22 talks presented by international specialists from Australia, France, Germany, Italy, Sweden, Switzerland and the United States.Generally, three lectures were delivered in the morning sessions and two in the late afternoon which left ample time for individual discussions.

In the first group of equations, in particular the Ricci flow and conformal flows such as the Yamabe flow and the Q-curvature flow were considered. A further focus was on curvature flows for hypersurfaces such as the Gauss and harmonic mean curvature flow and on geometric flows of higher order. Julie Clutterbuck presented a direct approach to certain fundamental estimates for a general class of parabolic equations. Mete Soner pointed out important relations between fully nonlinear parabolic equations and backward stochastic differential equations.

The class of hyperbolic equations was represented by wave maps, Einstein's equations of gravitation and nonlinear wave equations in waveguides and cones. Among the dispersive equations the nonlinear Schroedinger equation and the KdV equation were considered.

The individual discussions covered connections between approaches to the different equations in each group but also on common techniques used across the three classes of equations. Among such techniques are for instance the careful examination of the algebraic and geometric structure of the nonlinear terms. Another common theme appears to be the phenomenon of blow-up (singularity formation) for solutions and methods to describe the solution near these. Selfsimilar solutions and blow-up rates play an important role here.

The organizers decided to particularly encourage the young researchers among theparticipants to present their work, including a PhD student and several recent post-doctoral fellows.
Real Analysis, Harmonic Analysis and Applications to PDE
Authors: Müller Detlef , Stein Elias M.
Published: December 2009

Major areas and results represented at the workshop are:

I. Methods in harmonic analysis

(a) Multilinear analysis: this is an outgrowth of the method of “tile decomposition” which has been so successful in solving the problems of the bilinear Hilbert transform. Recent progress involves the control of maximal trilinear operators and an extension of the Carleson–Hunt theorem.

(b) Geometry of sets in

\R^d

: This includes recent progress on the interaction of Fourier analysis and geometric combinatorics related to the Falconer distance problem.

(c) Singular integrals: A break trough has been obtained on singular integrals on solvable Iwasawa

AN

- groups, which require a new type of Calderón–Zygmund decomposition, since the underlying spaces have exponential volume growth. Further significant progress involves the theory of operator-valued Calderón–Zygmund operators and their connection to maximal regularity of evolution equations.

(d) Oscillatory integrals, Fourierintegral operators and Maximal operators: This includes estimates of maximal operators related to polynomial polyhedra and their relations with higher dimensional complex analysis, sharp estimates for maximal operators associated to hypersurfaces in

\R^3,

estimates for degenerate Radon transforms and linear and bilinear estimates for oscillatory integral operators, as well as optimal Sobolev regularity for Fourier integral operators.

II. Applications to P.D.E.

(a) Dispersive linear and non-linear equations: Far reaching new approaches to dispersive estimates for Schrödinger equations via coherent state decompositions and a related new phase space transform adapted to the wave operator were introduced. Further significant progress includes

L^p

- estimates respectively blow up rates for eigenfunctions and quasimodes of elliptic operators on compact manifolds with and without boundary, and related problems for globally elliptic pseudodifferential operators, and well-posedness of the periodic KP-I equations. All these results are based in part on important ideas in harmonic analysis (such as I(d) above).

(b) Schrödinger operators with rough potentials: This includes quantitative unique continuation theorems and their relations with spectral properties and the study of embedded eigenvalues of Schrödinger operators.

The meeting was attended by 50 participants. The official program consisted of 23 lectures, and left sufficiant room for further activities, such as self-organised sessions and discussions among groups of participants. The organisers made an effort to include young mathematicians, and greatly appreciate the new joint program of the Oberwolfach Institute and the american NSF, which allowed to invite several outstanding young scientists from the United States.
Explicit Methods in Number Theory
Authors: Cohen Henri , Lenstra, Jr. Hendrik W. , Zagier Don B.
Published: December 2009

The workshop
Explicit Methods in Number Theory
was organised by Henri Cohen (Talence), Hendrik W. Lenstra (Leiden), and Don B. Zagier (Bonn) and was held July 17–23, 2005. Three previous workshops on the topic had been held in 1999, 2001, and 2003. The goal of this meeting was to present new methods and results on concrete aspects of number theory. In many cases, this included computational and experimental work, but the primary emphasis was placed on the implications for number theory rather than on the computational methods employed.

There was a ‘mini-series’ of five

1

-hour morning talks given by Bas Edixhoven, Johan Bosman, Robin de Jong, and Jean-Marc Couveignes on the topic of computing the coefficients of modular forms. Let

\Delta=q \prod_{n \geq 1} (1-q^n)^{24}=\sum_{n \geq 1} \tau(n) q^n

be Ramanujan's tau function, a newform of weight

12

for

SL_2(\mathbb Z)

. The speakers exhibited a method to compute

\tau(p)

for

p

prime in time polynomial in

\log p

.

Some of the other main themes included:

As always in Oberwolfach, the atmosphere was lively and active, providing an ideal environment for the exchange of ideas and productive discussions. This meeting was well-attended – with over 50 participants from a variety of backgrounds and with broad geographic representation from all continents, including a number of younger researchers. There were 30 talks of various lengths, and ample time was allotted for informal collaboration.
Differentialgeometrie im Grossen
Authors: Leeb Bernhard , Seidel Paul , Tian Gang
Published: December 2009

The meeting continued the biannual conference series
Differentialgeometrie im Großen
at the MFO which was established in the 60's by Klingenberg and Chern. While global Riemannian geometry with its connections to geometric analysis and topology remained an important focus of the conference, this time special emphasis was given to complex and symplectic geometry.

There were 50 participants from 8 countries, more specifically, 22 participants from Germany, 9 from other European countries, 1 from China, 19 from North-America. 10% of the participants were women. More than half of the participants (about 29) were young researchers (less than 10 years after diploma or B.A.), both on doctoral and postdoctoral level.

The official scientific program consisted of 22 one hour talks leaving plenty of time for informal discussions. The organizers would like to thank the institute staff for their great hospitality and support before and during the conference.
Mathematical Population Genetics
Authors: Baake Ellen , Ewens Warren J. , Wakolbinger Anton
Published: December 2009

One of the aims of population genetics is to explain how the genetic variation within and between biological populations is generated and maintained. Population genetics theory describes the change in the genetic composition of populations under the influence of various evolutionary processes such as genetic drift, mutation, selection, recombination, and migration. Elements of randomness turn out to be essential in the modelling of these processes. Random genetic drift, for example, is a consequence of the fact that, even without fitness differences, some individuals may, by chance, have more offspring than others, so that the offspring of one genotype may displace another.

The basic processes of evolution are known in principle, along with fundamental equations which describe the effects of interactions between genes. Indeed, of the biological sciences, genetics – and population genetics in particular – is the one with the most clearly defined mathematical models, with a strong emphasis on probability. This fact has increasingly attracted probabilists who have given the theory a new impetus through powerful modern methods. Thus there is now an increasingly large community of biologists and mathematicians speaking a common language, learning from each other, and identifying problems that should be attacked. This spirit was manifest in our meeting, with its small group of senior mathematicians and biologists, and its larger fraction of young researchers.

The topics presented during the workshop crystallized into the following thematic groups:

Genetic hitchhiking and selective sweeps (Stephan, Schweinsberg, Etheridge);
Mutation-selection systems (Evans, Stannat);
Speciation and adaptation (Bürger, Champagnat);
Spatially structured systems (Birkner, Swart, Hutzenthaler);
Stochastic processes associated with the coalescent (Krone, Pfaffelhuber);
Poisson–Dirichlet distributions (Spano, Feng);
Sampling formulae and coalescent-based inference (Ewens, Griffiths, Wakeley, Möhle, Hudson);
Recombination, gene-rearrangement and sequence alignment (M. Baake, Beresticky, Metzler)

Stimulated by these lectures, there were many discussions and a rich scientific exchange during the workshop, between generations as well as between disciplines. Last not least, the meeting also profited greatly from the friendly and serene Oberwolfach atmosphere.
Low-Dimensional Manifolds
Authors: Boileau Michel , Johannson Klaus , Scott Peter
Published: December 2009

The workshop on Low-Dimensional was organized by M. Boileau (Toulouse), K. Johannson (Frankfurt) and P. Scott (Ann Arbor). The aim was to bring together an international audience in order to expose some of the recent results in Low-Dimensional Topology and related areas. There were altogether 22 talks from a broad range of topics including Heegaard surfaces, the curve complex, hyperbolic manifolds, orbifolds and geometric group theory. There was ample opportunity for ample interaction. The organizers would like to thank the Institute for providing a pleasant and stimulating atmosphere.
Set Theory
Authors: Friedman Sy-David , Magidor Menachem , Woodin W. Hugh
Published: December 2009

This meeting was organised by Sy-David Friedman (University of Vienna), Menachem Magidor (Hebrew University, Jerusalem) and Hugh Woodin (University of California, Berkeley). Largely due to the generous EU support for the meeting, we were able to invite an unusually large number of young researchers (at most 10 years after the beginning of doctoral studies). We consequently chose an unusual format for the meeting, in which young people were given priority in the scheduling of lectures. This turned out to be a good choice, as the quality of these lectures was extremely high, giving us an excellent opportunity to learn about the new talent that has recently come into our field.

45-minute lectures were delivered by Krueger (Vienna), Dobrinen (Vienna), Camerlo (Torino), Zoble (Toronto), Geschke (Berlin), König (Paris), Viale (Paris), Asperó (Barcelona), Sharon (Irvine) and Lopez-Abad (Paris). These young people presented striking new results concerning combinatorial set theory, descriptive set theory, set-theoretic analysis and forcing axioms. One of the most impressive of these new results was that of Viale, who established a connection between two important set-theoretic principles, by showing that the singular cardinal hypothesis follows from the proper forcing axiom.

Shorter lectures (20 to 30 minutes) were given by the more senior researchers. In total, lectures were held by 29 of the 59 participants. By restricting the number and length of these lectures, we were able to leave substantial time for private discussion. The 15-minute breaks between lectures were especially appreciated, giving the meeting a relaxed feeling.

The traditional Wednesday afternoon walk was a success. The Schwarzwaldkirschtorte in Wolfach was superb, and despite the rain and getting (slightly) lost in the dark, everyone made it safely back to the institute in time for dinner.
Mathematics in the Physical Sciences, 1650-2000
Authors: Guicciardini Niccolo , Hoff Kjeldsen Tinne , Rowe David E.
Published: December 2009

The workshop was organised by Niccolò Guicciardini (Siena), Tinne Hoff Kjeldsen (Roskilde), and David Rowe (Mainz). During the five days of the conference 25 talks were given and one special evening lecture was organised.

The organizers developed the idea for this meeting in consultation with several other colleagues who attended the conference on early modern mathematics held in Oberwolfach January 5–11, 2003. That meeting brought together historians with considerable expertise on developments outside pure mathematics. Afterwards there was a general consensus among the participants that this format had produced fruitful interactions and some promising new perspectives. The idea behind the present workshop called for a similar, open-ended framework, but covering a broader expanse of time reaching far into the twentieth century. By focusing on the interplay between mathematics and the physical sciences the aim was to gain an insight into developments that had a crucial impact on modern mathematics.

This was achieved by inviting experts on the role of mathematics in the physical sciences who were able to approach this subject from a variety of different perspectives. The speakers addressed major developments relating to the overall theme of the conference which focused on thematic issues structured around three time periods: 1650-1800, 1800-1920, and 1920 up to recent times. Three particular topics emerged as central themes of interest:

The workshop brought together the core community of historians of mathematics, many of whom have attended past meetings in Oberwolfach, along with a number of historians and philosophers of science with strong interests in mathematical issues. The meeting was characterized by open discussions which, together with the talks, shed more light on the interplay between mathematics and the physical sciences and gave new insights into developments that had a crucial impact on the development of modern mathematics.

The organizers and participants thank the “Mathematisches Forschungsinstitut Oberwolfach” for making the workshop possible in the usual comfortable and inspiring setting.
Mechanics of Materials
Authors: Kienzler Reinhold , McDowell David L. , Werner Ewald A.
Published: December 2009

The workshop
Mechanics of Materials
, organised by Reinhold Kienzler (Bremen), David L. McDowell (Atlanta) and Ewald A. Werner (München) was held January 22nd–January 28th, 2006. The workshop attracted some 40 participants with a wide geographic spread. Special attention was devoted to increasing the participation of younger members of the related research community.

Mechanics of Materials is a broad, interdisciplinary subject that focusses on the intersection of Applied Mathematics, Continuum Mechanics, Material Physics and applications. To address important, emerging topics related to these interdisciplinary areas, several themes were pursued in distinct sessions, each with keynote addresses and extended discussions. The following main topics were treated:

Although these fields appear to be quite unconnected, certain physical properties and numerous mathematical approaches were identified as common structures. This includes basic balance and conservation laws as well as variational principles for establishing and solving the evolving partial differential equations. Bridging scales from electrons to macroscopic structures by various consistent methods we were able to arrive at a more complete picture of modelling material behavior and the associated mathematical challenges.

The unique atmosphere at the Institute offered an extraordinary opportunity for intense, amiable exchange of currently emerging, detailed and conceptual ideas. The significant amount of time devoted to fruitful discussion is certainly an element that made this meeting in Oberwolfach distinct from other outstanding technical venues. Many new collaborative relationships were initiated.

The following abstracts very well summarize both the keynote lectures and the additional contributions to the discussion.

It was our great pleasure to celebrate during an informal gathering the 50th anniversary of Horst Lippmann as participant, organizer and long-term intellectual contributor to many Oberwolfach workshops.
Miniworkshop: L2-Spectral Invariants and the Integrated Density of States
Authors: Dodziuk Jozef , Lenz H. Daniel , Schick Thomas , Veselić Ivan
Published: December 2009

Both the study of

L^2

-spectral invariants in geometry and the investigation of the integrated density of states in mathematical physics have attracted much attention in recent years. While the two topics are strongly related, the corresponding communities are rather unaware of each others work and methods. The main aim of this mini-workshop was to bring together people from both fields and provide a basis for interaction.

Accordingly, the first two days of the conference were spent with survey talks solicited by the organizers to highlight concepts and methods. There were 9 such talks with durations between 60 and 90 minutes. The second half of the conference was devoted to more detailed investigations. Most participants used the opportunity to present their current research in the area of the meeting. There were 13 such talks.

The results presented in those talks contained significant contributions e.g. to the Atiyah conjecture about integrality of

L^2

-Betti numbers for a completely new class of groups by Peter Linnell, a mathematically rigorous derivation using von Neumann traces of the asymptotics of the specific heat near absolute zero by Mikhael Shubin, and approximation results for the integrated density of states in various new contexts.

Altogether the conference was attended by 17 participants.
Mini-Workshop: On Isaac Newton's Work
Authors: Grattan-Guinness Ivor , Pulte Helmut
Published: December 2009

“Helpful” is not quite the word that comes to mind when considering Isaac Newton’. (Alan Shapiro)

A widely known legend is encapsulated by Alexander Pope's famous two-liner: ‘Nature, and Nature's laws lay hid in night. God said, let Newton be! and all was Light.’ The built in claim is somewhat oversimplified even for Britain; and it certainly does not capture the complex web of enthusiasm, acceptance, doubt, emulation, objection and opposition that characterises the Continental reactions. The need to consider the actual states of affairs within Newtonianism in general is especially significant, since from the 1740s onwards Britain became breathtakingly mediocre in mathematical research, so that almost everything that mattered was done in Europe.

The purpose of the workshop was to outline and study the details of these complexities. The summaries below show that we divided up the post-Newtonian corpus largely by subject matter and topic. Two presentations dealt with important contemporaries; Leibniz, and the Bernoulli family with Hermann: the book will contain more chapters of this kind. In addition, only one chapter tackled a community as such, namely the French mathematicians from 1780 to 1830: several chapters of the book will be of this type.

While we tried to take Newton's own contributions to be more or less known and clear, Newton's own presentations of his theories, often cryptic in the extreme, forced us back to his texts on many occasions. An important hierarchy of interpretation was exposed: what we today think that Newton was (not) saying, and what we today think that Newton's successors thought that he was (not) saying.

The mini-workshop brought together 15 participants from seven countries, including the two editors of the book. Funds made available by the US Junior Oberwolfach Fellows Funds allowed one of our younger members to take part. Most sessions lasted 75 minutes, including up to 30 minutes of discussion. The measure of further consultation in the evenings and on afternoon walks was considerable. Some subsets of us were already quite well known to each other, but all of us made new contacts and friendships, which will help signally in the preparation of the book over this and the next year.
Representations of Finite Groups
Authors: Kleshchev Alexander S. , Linckelmann Markus , Malle Gunter , Rickard Jeremy
Published: December 2009

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.
Interactions between Algebraic Geometry and Noncommutative Algebra
Authors: Happel Dieter , Small Lance W. , Stafford J. Toby , Van den Bergh Michel
Published: December 2009

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

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.
Classical Algebraic Geometry
Authors: Eisenbud David , Harris Joseph , Schreyer Frank-Olaf
Published: December 2009

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:
Algebraic K-Theory
Authors: Grayson Daniel R. , Huber-Klawitter Annette , Jannsen Uwe , Levine Marc
Published: December 2009

Vaguely speaking, K-theory is a way of examining features of systems of polynomial equations by considering the possible ways to associate vector spaces to each solution. While the objects of study come from algebraic geometry or number theory the methods come from algebraic topology. In the last decade motivic cohomology has matured to provide a useful tool for computing algebraic

K

-theory. The conference covered a wide range of interconnected topics from classical Milnor

K

-theory, cyclic homology, algebraic

K

-theory (in the narrow sense), Chow groups, regulators, homotopy theory of schemes with a certain stress in motivic cohomology and triangulated motives. Algebraic

K

-theory is the theme holding these questions together. This was reflected both in the selection of the participants as well as in the choice of talks. The 48 participants came mostly from Europe and North America, but also from Japan and China. There was a good mix of participants, from leading experts of the field to younger researchers and even a couple of graduate students. There were 17 mostly 1-hour talks. Priority was given to young researchers, to allow them to present their results. The schedule left enough time for interesting discussions between participants.
Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry
Authors: Knieper Gerhard , Polterovich Leonid , Potyagailo Leonid
Published: December 2009

The workshop
Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry
, organized by Gerhard Knieper, Leonid Polterovich and Leonid Potyagailo brought together leading mathematicians working in these rapidly developing fields. The scientific goal of the workshop was twofold: first, to present some recent breakthroughs, and, second, to discuss and digest a surprisingly fruitful interaction of these seemingly remote mathematical disciplines. The topics of talks included rigidity of group actions (A. Katok, D. Witte Morris, M. Kanai), ergodic properties of group actions ( U. Hamenstädt, V. Kaimanovich), dynamics on homogeneous spaces (S. Mozes), hyperbolic groups and hyperbolic manifolds (G. Courtois, F. Dahmani, M. Kapovich, I. Mineyev, N. Peyerimhof), hyperbolic dynamics ( D. Kotschick, G. Paternain, M. Pollicott), geodesic flows (H. Koehler, E. Leschinsky ), asymptotic geometry of finitely generated groups and Lie groups (E. Breuillard, G. Margulis).

Interestingly enough, quasi-morphisms – a group theoretical notion which was originated in the framework of bounded cohomology theory, arose in the lectures of several researchers in quite a different context. Quasi-morphisms were discussed by M. Burger and A. Iozzi from the viewpoint of representation theory and by D. Calegari and K. Fujiwara who studied commutator length on hyperbolic groups and mapping class groups. At the same time M. Entov, G. Ben Simon and P. Py discussed their appearance and applications in the framework of symplectic topology.

The workshop witnessed another interesting trend. It seems that geometric group theory is trying to conquer a new area: bi-invariant geometries on groups of diffeomorphisms were discussed by D. Burago and D. McDuff. Some talks reflected a fruitful application of a powerful tool of modern symplectic topology, Floer homology, to “non-traditional” (from the symplectic viewpoint) areas such as the study of topological entropy of Reeb flows (F. Schlenk) and theory of the mapping class groups (A. Fel'shtyn). Some very recent developments in the foundation of Floer theory were presented by O. Cornea.

An important feature of the workshop was the presence of many excellent students from several countries who had an indispensable opportunity to profit from the intense interaction with the established mathematicians. In fact the students contributed a lot to the success of the workshop by active participation in the discussions. Some of the students gave high quality talks on their results during informal “late night” sessions.

One of the hightlights of the meeting was the open problem session on Wednesday night, skillfully moderated by Anatole Katok, where many participants formulated questions and research programs which, as we believe, will develop our fields.
Affine Algebraic Geometry
Authors: Flenner Hubert , Russell K. Peter , Zaidenberg Mikhail
Published: December 2009

Affine geometry deals with algebro-geometric questions of affine varieties. In the last decades this area has developed into a systematic discipline with a sizeable international group of researchers, and with methods coming from commutative and non-commutative algebra, algebraic, complex analytic and differential geometry, singularity theory and topology. The meeting was attended by 48 participants, among them the most important senior researchers in this field and many promising young mathematicians.

Especially helpful were the programs for young researchers: the NSF Oberwolfach program, the EU grant and JAMS grant. They allowed to increase the number of young participants considerably.

The conference took place in a very lively atmosphere, made possible by the excellent facilities of the institute. There were 24 talks with a considerable number of lectures given by young researchers at the beginning of their careers. Moreover there were 4 invited lectures that gave an overview over some of the most vivid subfields of Affine Geometry. The program left plenty of time for cooperation and discussion among the participants. We highlight the areas in which new results were presented by the lecturers:

In the first 4 areas there were overview talks given by D. Wright, M. Miyanishi, D. Daigle and Sh. Kaliman. Finally, in a problem session there were presented a number of open questions and problems, which are listed in at the end of this report.
Computational Electromagnetism and Acoustics
Authors: Hiptmair Ralf , Hoppe Ronald H.W. , Joly Patrick , Langer Ulrich
Published: December 2009

The Oberwolfach Conference on Computational Electromagnetism and Acoustics, held Feb 5–9, 2007, at the Mathematisches Forschungsinstitut Oberwolfach, was already the second in series, following the 2004 Oberwolfach Conference on Computational Electromagnetism. This time the scope was slightly broader, because there has been substantial progress in numerical techniques for high-frequency wave propagation. In this regime, the mathematical and numerical challenges in acoustics are very similar to those arising in the simulation of electromagnetic fields. Usually new ideas and methods are first explored for acoustics and then adapted to electromagnetics. Moreover, many researchers are active in both fields. So it is natural to cover them in the context of a single event.

Like the previous meeting, the 2007 conference brought together more than fifty researchers active in the field, hailing from more than 10 countries. Among them were a large number of leading experts, but also quite a few students and postdoctoral fellows. At this point the organizers acknowledge the generous support of the European Union and NSF that covered expenses for young European and overseas participants. The majority of the participants were applied mathematicians, but a sizable number of people with a background in engineering also attended, as appropriate for a field with close ties to engineering and the sciences.

A total of 28 presentations was given at the conference, eight of which were meant to give a survey of particular numerical techniques. They covered the following topics:

As is typical of an event that is focused on an area of applications rather than a specific mathematical technique, talks varied a lot in style and scope, ranging from discussions of specific theoretical issues to general discussion of a computational method. Nevertheless, all of them managed to convey exciting new insights and developments. Thus the conference made it possible to catch a glimpse of the future of computational electromagnetism and acoustics. The following extended abstracts will also enable the reader to keep abreast of many crucial trends in the field.