bluuebunny/crossref_metadata_2025 huggingface dataset:
Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n)
Authors: Conti Sergio , Dolzmann Georg , Kirchheim Bernd , Müller Stefan
Published: December 2009

The Cauchy–Born rule provides a crucial link between continuum theories of elasticity and the atomistic nature of matter. In its strongest form it says that application of affine displacement boundary conditions to a monatomic crystal will lead to an affine deformation of the whole crystal lattice. We give a general condition in arbitrary dimensions which ensures the validity of the Cauchy–Born rule for boundary deformations which are close to rigid motions. This generalizes results of Friesecke and Theil [J. Nonlin. Sci.
12
(2002), 445–478] for a two-dimensional model. As in their work the key idea is to use a discrete version of polyconvexity (ordinary convexity of the elastic energy as a function of the atomic positions is ruled out by frame-indifference). The main point is the construction of a suitable discrete null Lagrangian which allows one to separate rigid motions. To do so we observe a simple identity for the determinant function on

\operatorname{SO}(n)

and use interpolation to convert ordinary null Lagrangians into discrete ones.
Existence of pulsating waves in a model of flames in sprays
Authors: Constantin Peter , Domelevo Komla , Roquejoffre Jean-Michel , Ryzhik Lenya
Published: December 2009

A one-dimensional system describing the propagation of low Mach number flames in sprays is studied. We show that pulsating waves may exist when the droplet distribution in the unburnt region is spatially periodic. The range of possible propagation speeds may be either bounded or unbounded, depending on the threshold temperatures of the burning and vaporization rates.
Characteristic points, rectifiability and perimeter measure on stratified groups
Authors: Magnani Valentino
Published: December 2009

We establish an explicit formula between the perimeter measure of an open set

E

with

C^1

boundary and the spherical Hausdorff measure

\mathcal S^{Q-1}

restricted to

\mathrm{der} E

, when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and

Q

denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of

E

is less than or equal to

\mathcal S^{Q-1}(\mathrm{der} E)

up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli, Garofalo and Nhieu. The crucial ingredient of this result is the negligibility of “characteristic points” of the boundary. We introduce the notion of “horizontal point”, which extends the notion of characteristic point to arbitrary submanifolds and we prove that the set of horizontal points of a

k

-codimensional submanifold is

\mathcal S^{Q-k}

-negligible. We propose an intrinsic notion of rectifiability for subsets of higher codimension, namely,

(\mathbb G,\mathbb R^k)

-rectifiability and we prove that Euclidean

k

-codimensional rectifiable sets are

(\mathbb G,\mathbb R^k)

-rectifiable.
Optimal mass transportation and Mather theory
Authors: Bernard Patrick , Buffoni Boris
Published: December 2009

We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather's minimal measures.
Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation
Authors: Serfaty Sylvia
Published: December 2009

We study vortices for solutions of the perturbed Ginzburg-Landau equations

\Delta u+ \frac{1}{\varepsilon^2} u(1-|u|^2)=f_\varepsilon

where

f_\varepsilon

is estimated in

L^2

. We prove upper bounds for the Ginzburg–Landau energy in terms of

\|f_\varepsilon\|_{L^2}

, and obtain lower bounds for

\|f_\varepsilon\|_{L^2}

in term of the vortices when these form “unbalanced clusters” where

\sum_i d_i^2\neq (\sum_i d_i)^2

.

These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat-flow, which allow to study various phenomena occurring in this flow, among which vortex-collisions; allowing in particular to extend the dynamical law of vortices passed collisions.
Sum-product theorems and incidence geometry
Authors: Chang Mei-Chu , Solymosi Jozsef
Published: December 2009

In this paper we prove the following theorems in incidence geometry.

The main ingredients used are the subspace theorem, Balog–Szemerédi–Gowers Theorem, and Szemerédi–Trotter Theorem. We also generalize the theorems to high dimensions, extend Theorem 1 to

\Bbb F_p^2

, and give the version of Theorem 2 over

\Bbb Q

.
On the zero-temperature or vanishing viscosity limit for certain Markov processes arising from Lagrangian dynamics
Authors: Anantharaman Nalini
Published: December 2009

We study the zero-temperature limit for Gibbs measures associated to Frenkel-Kontorova models on (R d) Z/Z d . We prove that equilibrium states concentrate on configurations of minimal energy, and, in addition, must satisfy a variational principle involving metric entropy and Lyapunov exponents, a bit like in the Ruelle-Pesin inequality. Then we transpose the result to certain continuous-time stationary stochastic processes associated to the viscous Hamilton-Jacobi equation. As the viscosity vanishes, the invariant measure of the process concentrates on the so-called Mather set of classical mechanics, and must, in addition, minimize the gap in the Ruelle-Pesin inequality.
Poisson geometry and deformation quantization near a strictly pseudoconvex boundary
Authors: Leichtnam Eric , Tang Xiang , Weinstein Alan
Published: December 2009

Let

X

be a complex manifold with strongly pseudoconvex boundary

M

. If

\psi

is a defining function for

M

, then

-\log\psi

is plurisubharmonic on a neighborhood of

M

in

X

, and the (real) 2-form

\sigma = i \partial \bar \partial (-\log \psi)

is a symplectic structure on the complement of

M

in a neighborhood in

X

of

M

; it blows up along

M

.

The Poisson structure obtained by inverting

\sigma

extends smoothly across

M

and determines a contact structure on

M

which is the same as the one induced by the complex structure. When

M

is compact, the Poisson structure near

M

is completely determined up to isomorphism by the contact structure on

M

. In addition, when

-\log\psi

is plurisubharmonic throughout

X

, and

X

is compact, bidifferential operators constructed by Engliš for the Berezin–Toeplitz deformation quantization of

X

are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on

M

, along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.
The homotopy groups of the automorphism group of Kirchberg algebras
Authors: Dadarlat Marius
Published: December 2009

We compute the homotopy groups of the automorphism group of Kirchberg algebras. More generally, we calculate the homotopy classes [X,Aut(A)] for a Kirchberg algebra A and a path connected metrizable compact space X satisfying a natural continuity property.
Partial translation algebras for trees
Authors: Brodzki Jacek , Niblo Graham A. , Wright Nick
Published: December 2009

In [1] we introduced the notion of a partial translation C*-algebra for a discrete metric space. Here we demonstrate that several important classical C*-algebras and extensions arise naturally by considering partial translation algebras associated with subspaces of trees.
The geometry of determinant line bundles in noncommutative geometry
Authors: Chakraborty Partha Sarathi , Mathai Varghese
Published: December 2009

This article is concerned with the study of the geometry of determinant line bundles associated to families of spectral triples parametrized by the moduli space of gauge equivalence classes of Hermitian connections on a Hermitian finite projective module. We illustrate our results with some examples that arise in noncommutative geometry.
Combinatorics
Authors: Lovasz Laszlo , Prömel Hans Jürgen
Published: December 2009

The conference was organized by László Lovász (Redmond) and Hans
Jürgen Prömel (Berlin). The programme consisted of 15 lectures, supplemented
by 21 shorter contributions, and covered many areas in Combinatorics such as partition
theory, discrete geometry, homomorphisms and lattices, extremal combinatorics, graph
theory, random structures, and additive number theory. The aim of the workshop was to
emphasize the underlying methods that are common to many of these combinatorial branches
and that act as both driving forces and organizing principles of the field. The diversity
of the topics and par-ticipants stimulated a lot of fruitful discussion between the
different branches and gave rise to new collaborations, in particular for the younger
generation of researchers.
Mini-Workshop: Multiscale Modeling in Epitaxial Growth
Authors: Voigt Axel
Published: December 2009

Thin film epitaxy is a modern technology of growing single crystals that inherit atomic structures from substrates. Various mathematical models and numerical algorithms are proposed to be used for describing epitaxial growth processes. Due to the underlying multiscale phenomena, which range from the interaction of single atoms at steps up to an engineering scale, on which the transport of material to the surface in a MBE (molecular beam epitaxy) furnace needs to be described, the models can be distinguished by the relevant length scales they are living on

(a) discrete atomic models: Individual atoms are the basic degrees of freedom and single hoppings to neighbouring lattice sites are simulated by kinetic Monte Carlo methods. A n example are the so-called Solid-on-Solid models.

(b) discrete-continuous models: The atomic distance in the growth direction is discrete, but the atomic distance in the lateral direction is coarse grained. The steps are assumed to be smooth curves and serve as free boundaries for an adatom diffusion equation on terraces. These models are known as Burton-Cabrera-Frank models.

(c) continuous models: The atomic processes at steps are neglected , the overall surface is assumed to be smooth and phenomenological equations describe directly the height of the growing film. An example is the Villain equation.

The main goal in modelling epitaxial growth is to bridge the gap between these different models and to describe growth process on a continuous scale by incorporating atomic effects. The focus of this workshop was to bring together materials scientists, theoretical physicists and applied mathematicians to exchange ideas on the three different regimes (a),(b) and (c). The mini-workshop consisted of three introducing lectures, one for each approach and several lectures which focus on connections both in an analytical and numerical fashion. The contributions ranged from quantum-chemistry, molecular dynamics and kinetic Monte Carlo to step flow and continuum models. Several multiscale approaches have been considered to combine at least two of these models.

Besides the mathematical aspects of modelling epitaxial growth also the connection to experimental results was dealt with in order to drive the recent theoretical developments into a direction which is relevant for a large variety of industrial applications. The mini-workshop was also used to give young researchers the opportunity to be introduced into such an actual interdisciplinary field.
Finite and Infinite Dimensional Complex Geometry and Representation Theory
Authors: Huckleberry Alan T. , Neeb Karl-Hermann , Wolf Joseph Albert
Published: December 2009

As the theme of the conference indicates, one of the organizers' main goals was to put together a group of participants with a wide range of interests in and around the complex geometric side of the representation theory of Lie groups and algebras. It is their belief that a hybrid approach to representation theory, in particular interaction between complex geometers and harmonic analysts standing on a strong foundation of finite- and infinite-dimensional Lie theory, will open up new avenues of thought and lead to progress in a number of areas. Since the previous Oberwolfach conference (Represention Theory and Complex Analysis, April 2000), there has been quite a positive development toward these goals. A number of breakthroughs were achieved, and of course these were reported at this year's conference. More than half of the 47 participants (from 15 countries) are now working in some middle ground between complex geometry and representation theory. Furthermore, it was clear from the discussions both after the talks and in the evenings that most participants now understand each other's language well enough to discuss high level research projects. A basic new component, infinite-dimensional complex geometry and related representation theory, was added this year. This quickly developing subject is already attracting wide attention. A goal for the future is to better integrate this with the more classical finite-dimensional theory. One consequence of the broad range of backgrounds of the participants is that, without prodding by the organizers, virtually all speakers gave quite comprehensive introductions to their subjects before going into their most recent results. This was greatly appreciated by all! Instead of attempting to summarize these talks we will let the following abstracts speak for themselves.
Funktionentheorie
Authors: Bergweiler Walter , Ruscheweyh Stephan , Saff Edward B.
Published: December 2009

The present conference was organized by Walter Bergweiler (Kiel), Stephan Ruscheweyh (Würzburg) and Ed Saff (Vanderbilt). The 24 talks gave an overview of recent results and current trends in function theory.
Algebraische Gruppen
Authors: Brion Michel , Jantzen Jens Carsten
Published: December 2009

The workshop was organized by Michel Brion (Grenoble) and Jens Carsten Jantzen (Aarhus). The schedule comprised twenty talks from a broad range of areas connected to algebraic groups, including but not limited to: arithmetic invariants of algebraic groups, compact Lie groups, embeddings of homogeneous spaces, Hilbert schemes, invariant theory, moduli spaces, nilpotent orbits, quantum groups, quotient singularities, rationality questions, reductivity and reducibility of algebraic groups, Schubert varieties, and symmetric varieties.
Discrepancy Theory and Its Applications
Authors: Chazelle Bernard , Chen William W.L. , Srivastav Anand
Published: December 2009

The meeting was organized by Bernard Chazelle (Princeton), William Chen (Sydney) and Anand Srivastav (Kiel), and was attended by some twenty participants from over ten countries and three continents.

The purpose of the meeting was to encourage and enhance dialogue and collaboration between the theoretical and practical aspects of discrepancy theory. The topics covered included:

Nineteen talks were presented, including a few of a survey nature as well as others that concentrated on specific recent results. These talks demonstrated the diversity on all four areas and their inter-relationships, as well as the vitality of these areas of research.

The organizers and participants would like to take this opportunity to thank again the “Mathematisches Forschungsinstitut Oberwolfach” for having provided a comfortable and inspiring environment for the meeting and the scientific work. The pleasant atmosphere and superb facilities contributed to the overall success of the meeting.

We include the abstracts of all the talks in alphabetical order of the speakers.
Mini-Workshop: Amalgams for Graphs and Geometries
Authors: Ivanov Alexander A. , Shpectorov Sergey V.
Published: December 2009

This meeting was well attended with 17 participants with broad geographic representation from 3 continents. There were 16 talks during the workshop including an invited talk by Anda Degeratu a participant in the competing String Theory workshop. The method of group amalgams is a highly effective way of classifying mathematical objects possessing high degrees of symmetry. The idea of the method is separation of the study of the local structure of the acting group from the question of it's global isomorphism type. The method of group amalgams has been successfully applied to problems in graph theory and diagram geometry. It also featured prominently in group theory. For example, the fact that the Monster sporadic simple group is the universal completion of the amalgam associated with the tilde geometry formed the foundation of the solution of the famous

Y

-group conjecture that the

Y_{555}

presentation defines the Bimonster (the direct product of two copies of the Monster sporadic simple group extended by a group of order 2). J.H. Conway coined for this theorem the name 'NICE' where \textbf{N} is for \textbf{N}orton, \textbf{I} for \textbf{I}vanov, \textbf{C} for \textbf{C}onway and \textbf{E} for anyone \textbf{E}lse. The proof of the NICE theorem based on the method of group amalgams is presented in the two volume monograph of the organisers published by Cambridge University Press. \par Recently a dramatic progress was made within the study of flag-transitive diagram geometries. The importance of the notion of \textit{constrained} completions of amalgams was realised. Within this framework many geometries of sporadic groups were characterised as the constrained completions of suitable amalgams. This approach also gives a general criterion about possible shapes of diagrams of flag-transitive geometries. This enables the area of diagram geometries to leave it's "botanical" stage of example collection and enter the stage of theory building. \par Among other applications we would like to mention recent applications of the amalgam method to the cohomologies of finite groups. These ideas were described in the notes of M. Aschbacher on calculation of the Schur multiples of some finite simple groups. \par During the workshop we had discussed in detail the proofs of a number of results obtained along the lines of the amalgam method, as well as of directions of future research. We believe that the abstract of the talks given at the workshop facilitate for the younger mathematicians access to these extremely important, yet very technically complex tools of mathematical research.
Mini-Workshop: Local Time-Space Calculus with Applications
Authors: Eisenbaum Nathalie , Kyprianou Andreas E. , Matache Ana-Maria , Peskir Goran
Published: December 2009

Itô's formula has been celebrated as a fundamental extension of the classical Leibnitz-Newton formula and forms a cornerstone to what is known today as stochastic calculus. In recent years, there have been further extensions of this formula which essentially weaken further the smoothness of the function to which the formula is applied at the cost of an additional term involving integration over a two dimensional field which has been labeled local time-space. As of yet the full potential of local time-space calculus and its applications has neither been realized nor properly understood. The current proposal aimed at bringing together researchers who in some way have touched on this currently unchartered territory in order to examine its importance as both a pure and applied field of study as well as to establish the state of the art in the current literature and identify new road map of research.
Approximation Algorithms for NP-Hard Problems
Authors: Kannan Ravindran , Karpinski Marek
Published: December 2009

It is an interesting artifact that most computational tasks today that arise in realistic scenarios are intractable, at least if one insists on delivering exact solutions with certainty within a strict deadline. An important mean for surmounting this intractability barrier is that of
approximate computation
, where the answer is guaranteed to be within some small fraction of optimality. One of the great recent successes in that area has been the discovery of a new paradigm connecting probabilistic proof verification theory to the theory of approximate computation as well as some new probabilistic combinatorial and algebraic paradigms in designing efficient approximation algorithms.

The workshop was concerned with the most important recent developments in the area of
efficient approximation algorithms
for NP-hard optimization problems as well as with new techniques for proving
intrinsic lower bounds
for efficient approximation.

In addition to 25 lectures delivered at general sessions, there were several additional lectures given at the special sessions and the evening problem session. The Program of the meeting and Abstracts of all talks are listed in the subsequent sections of this report. The special sessions were on the following topics:

The meeting was held in very informal and stimulating atmosphere. The
ad hoc
organized special sessions were extraordinarily interesting and intensive venues for communicating most recent results. Thanks to everybody who contributed to the success of this meeting and made it such an enjoyable event!

The organizers and participants thank the
Mathematisches Forschungsinstitut Oberwolfach
for its help in organizing this conference.

Special thanks go to Mathias Hauptmann for his help in editing this proceedings. We are also indebted to Christiane Andrade, Cornelia Kaufmann, Martin Löhnertz, Ignatios Souvatzis and Claus Viehmann for their continuous support in organizing this conference.
Classical Algebraic Geometry
Authors: Eisenbud David , Harris Joseph , Schreyer Frank-Olaf
Published: December 2009

The worshop on Classical Algebraic Geometry, organized by David Eisenbud (Berkeley), Joe Harris (Harvard) and Frank-Olaf Schreyer (Saarbrücken) was well attended by important senior researchers and many gifted young mathematicians.

Classical Algebraic Geometry is characterized by having very basic and concrete problems. However development of the abstract, sophisticated tools of algebraic geometry has often lead to remarkable progress on these problems. It was our intention to make a conference emphasizing progress on the classical problems, and featuring the new tools and their applications. We emphasize a few results:

Geometry of the Moduli Spaces of Curves. Based partly on motivations from mathematical physics, the enumerative geometry of the moduli spaces of curves has been an extremely active area for some time. Several of the major conjectures in the area have seen major progress, and may be close to resolution. The extension of Brill–Noether theory for line bundles to the case of bundles of higher rank is a long-standing problem (Mukai's conjecture), and this plays now a novel role in the search for interesting divisors on the moduli space in the work of Gavril Farkas. In the talk of Brendon Hassett we learned how other birational models of the Moduli space can be interpreted in terms of curves with nodes, cusps and tacnodes.

Derived Categories. Originally introduced in the late fifties as a tool for generalizing duality theory, derived categories have arisen in a several striking new contexts, including the characterization of birational transformations in the minimal model program, the analysis of the Bernstein–Gel'fand–Gel'fand and Fourier–Mukai transforms. One of the speakers of the conference (Mihnea Popa) suggested an amazing series of parallels between Castelnuovo theory and the Schottky problem, based on these last two contexts.

Enumerative Geometry. The high-tech tools of modern geometry have led to a better understanding of degenerations, which has in turn led to the solution of many classical enumerative problems. We heard talks on a solution via specialization of the long-standing problem of intersection theory on flag manifolds, and on the maximal degneration of complex structures through amoebas and tropical geometry.

We were particularly pleased by the number of extremely strong young (i.e. untenured) participants, among them for example Gavril Farkas and Mihnea Popa.
Combinatorial Commutative Algebra
Authors: Peeva Irena , Welker Volkmar
Published: December 2009

There has been very fruitful interaction between the fields of Combinatorics and Commutative Algebra since the 70's [1, 2, 3, 4]. Notably, there has been a surge in interest and research during the last 8 years: a great variety of new ideas and techniques were introduced, and substantial progress was made. Projects in this direction have been undertaken by both established mathematicians and graduate students or postodcs. Currently, the main centers for such research are Germany, Italy, Japan, and USA. The toolset from Commutative Algebra that helps to solve combinatorial problems ranges from Hilbert-series to local cohomology. On the other hand, Combinatorics enriches Commutative Algebra by supplying questions, methods and results that ask for a more general setting, a setting which in many cases has a ring theoretic framework.

The Oberwolfach workshop on “Combinatorial Commutative Algebra” was organized as an attempt to gather researchers from Combinatorics and Commutative Algebra in order to announce the latest developments, spread new problems, and spark further interaction. For that purpose only very few and only longer talks were scheduled, and they all reported on exciting recent developments. The talks covered a wide spectrum of topics ranging over f-vector theory, algebraic shifting, simplicial complexes, polytopes, Gröbner basis, free resolution, powers of ideals, Hilbert–Kunz functions, and related questions in Classical Algebraic Geometry.

The talks were confined to the morning session and the afternoons were kept free for research. Existing teams continued their collaboration and new teams were formed; conjectures announced during a lecture before lunch did not exist anymore at dinner time. Mathematics was on the move.

Gil Kalai envisaged in his talk yet another round of progress through the interaction of Commutative Algebra and Combinatorics. We hope that this conference has made a contribution for this vision to become true. The success of a conference is determined by many factors, one of them is the atmosphere at the conference location. The Oberwolfach staff created the perfect atmosphere and we are very grateful for their hospitality.
Arithmetic Algebraic Geometry
Authors: Faltings Gerd , Harder Günter , Katz Nicholas M.
Published: December 2009

At this workshop various aspects of Arithmetic Algebraic Geometry have been discussed. It was organized by G. Faltings, G. Harder (Max-Planck-Institute for Mathematics Bonn) and N. Katz (Princeton university)

The main goal of this field is, to obtain information on the solution of diophantine problems by applying the tools provided by algebraic geometry.

The workshop was attended by 42 participants and we had the total number of 18 talks.

A very interesting diophantine problem, which has a very geometric flavour, is the investigation of the structure of the Brauer group of a scheme or more in geometric terms of a complex algebraic variety. This aspect has been discussed in the talks by Gabber, who reported on some recent progress in direction of the purity conjecture. It has also been discussed in the talks of Lieblich and de Jong, in which some more geometric questions have been discussed.

Another tool to obtain information on diophantine problems is provided by

p

-adic methods. Here certain analogies between classical analytic theory over

\mathbf{C}

and

p

-adic analytic theory have to be developed. We have to understand the meaning of local systems. This topic was discussed in the talks of Deninger and Ramero. Minhyong Kim outlined a program how to used a

p

-adic unipotent Albanese map to prove finiteness in diophantine geometry, for instance the classical theorem of Siegel.

Shimura varieties are certainly interesting objects in Arithmetic Algebraic Geometry, they provide interesting examples of algebraic varieties. Ben Moonen's talk was on the borderline between

p

-adic methods and Shimura varieties. Laumon reported on the fundamental Lemma for

U(n)

. Various talks discussed the Galois representations attached to automorphic forms and abelan varieties (Bültel, Edixhoven, Harris, Diamond). In the talks of Pink and Ullmo Wildeshaus, Rapoport, M. Kings and Ullmo some other aspects of this field were discussed.

Goncharov discussed some interesting aspects of higher Teichmüller theory.
Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry
Authors: De Loera Jesús , Haase Christian
Published: December 2009

The mini-workshop Ehrhart Quasipolynomials: Algebra, Combinatorics, and Geometry, organised by Jesús De Loera (Davis) and Christian Haase (Durham), was held August 15th–21st, 2004. A small group of mathematicians and computer scientists discussed recent developments and open questions about
Ehrhart quasipolynomials
. These fascinating functions are defined in terms of the lattice points inside convex polyhedra. More precisely, given a rational convex polytope

P

for each positive integer

n

, the Ehrhart quasipolynomials are defined as

i_P ( n ) = \# \left( n P \cap {\mathbb Z}^{ d } \right)

. This equals the number of integer points inside the dilated polytope

n P = \{ nx : x \in P \}

. The functions

i_P(n)

appear in a natural way in many areas of mathematics. The participants represented a broad range of topics where Ehrhart quasipolynomials are useful; e.g. combinatorics, representation theory, algebraic geometry, and software design, to name some of the areas represented.

Each working day had at least two different themes, for example the first day of presentations included talks on how lattice point counting is relevant in compiler optimization and software engineering as well as talks about tensor product multiplicities in representation theory of complex semisimple Lie Algebras. Some special activities included in the miniworkshop were (1) a problem session, a demonstration of the software packages for counting lattice points (by P. Clauss), (by J. De Loera et al.), and (by S. Verdoolaege), (2) a guest speaker from one of the research in pairs groups (by R. Vershynin),and (3) a nice expository event where each of the three mini-workshops sharing the Oberwolfach facilities had a chance to introduce the hot questions being pursued to the others. The atmosphere was always very pleasant and people worked very actively. For instance, two of the talks reported on new theorems obtained during the miniworkshop. The organizers and participants sincerely thank MFO for providing a wonderful working environment, perhaps unique around the world. We also thank Günter M. Ziegler for his support and encouragement. In what follows we present the abstracts of talks following the order in which talks were presented.
Large Scale Stochastic Dynamics
Authors: Landim Claudio , Olla Stefano , Spohn Herbert
Published: December 2009

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

We had 49 participants from 11 countries, mostly probabilists, but also experts from partial differential equations and statistical physics. They all enjoyed tremendously the unique and stimulating atmosphere at the Mathematische Forschungsinstitut Oberwolfach and hope to return some day.
Nichtkommutative Geometrie
Authors: Connes Alain , Cuntz Joachim , Rieffel Marc A.
Published: December 2009

Noncommutative geometry applies ideas from geometry to mathematical structures determined by noncommuting variables. Within mathematics, it is a highly interdisciplinary subject drawing ideas and methods from many areas of mathematics and physics. Natural questions involving noncommuting variables arise in abundance in many parts of mathematics and quantum mathematical physics. On the basis of ideas and methods from algebraic topology and Riemannian geometry, as well as from the theory of operator algebras and from homological algebra, an extensive machinery has been developed which permits the formulation and investigation of the geometric properties of noncommutative structures. Areas of intense research in recent years are related to topics such as index theory, K-theory, cyclic homology, quantum groups and Hopf algebras, the Novikov- and Baum-Connes conjectures as well as to the study of specific questions in other fields such as number theory, modular forms, topological dynamical systems, renormalization theory, theoretical high-energy physics and string theory. The meeting was attended by 45 participants, including a fair number of young mathematicians but also many of the leading experts in the field. The exchange of ideas was very lively and quite a few significant new results were presented in the talks, which covered many of the aspects of noncommutatve geometry mentioned above.
Polylogarithms
Authors: Bloch Spencer , Kings Guido , Wildeshaus Jörg
Published: December 2009

The

k

-th polylogarithm function is defined on

|z|<1

by

Li_k(z) = \sum_{n\geq 1} \frac{z^n}{n^k}.

In the past 25 years or so, polylogarithms have appeared in many different areas of Mathematics. The following list is taken, for the most part, from [Oe]: volumes of polytopes in spherical and hyperbolic geometry, volumes of hyperbolic manifolds of dimension 3, combinatorial description of characteristic classes, special values of zeta functions, geometry of configurations of points in

\mathbb P^1

, cohomology of

GL_n(\mathbb C)

, calculation of Green's functions associated to perturbation expansions in quantum field theory, Chen iterated integrals, regulators in algebraic

K

-theory, differential equations with nilpotent monodromy, Hilbert's problem on cutting and pasting, nilpotent completion of

\pi_1(\mathbb P^1-\{0,1,\infty\})

, Bethe's Ansatz in thermodynamics, and combinatorial problems in quantum field theory.

Of course, these problems are not all unrelated. One common thread is that values of polylogarithms appear naturally as periods of certain mixed Hodge structures associated to mixed Tate motives over cyclotomic fields. How these periods are related to special values of

L

-functions is a part of the Beilinson conjectures, which were discussed in a previous Arbeitsgemeinschaft. Since that time, the general picture has clarified. A number of talks are devoted to aspects of this more general philosophy (talk 2-5, 9, 10 13-17). The

p

-adic aspects of the theory have been studied and will be explained in the eleventh talk. In addition, a vast generalization, multiple polylogarithms of the form

Li_{s_1,\dotsc,s_k} := \sum_{n_1>\ldots > n_k \geq 1} \frac{z_1^{n_1}\cdots z_k^{n_k}}{n_1^{s_1}\cdots n_k^{s_k}},

have come to play a role as presented in the fourth and twelfth talk.

The connection between volumes of hyperbolic manifolds and polylogarithms is described in the sixth talk. The seventh talk introduces higher torsion and explains why polylogarithms occur in that setting.

Polylogarithms play a role in physics. The eighth talk explains how zeta values, polylogarithms, and multiple polylogarithms appear in calculations of perturbative expansions in quantum field theory.

The final talks 13-17 return to the relationship with special values of

L

-functions. The thirteenth and fourteenth talk are devoted to the Zagier conjectures for number fields. The fifteenth talk concerns the elliptic polylogarithm sheaves and Zagier's conjecture for elliptic motives. Finally, talk 16 and 17 describe how polylogarithms relate to Euler systems and to the Bloch-Kato conjectures on special values of

L

-functions.

The
Arbeitsgemeinschaft
was organized by Spencer Bloch (University of Chicago), Guido Kings (Universität Regensburg) and Jörg Wildeshaus (Université Paris 13). It was held October 3rd – October 9th, 2004 with 46 participants.
Nonlinear Waves and Dispersive Equations
Authors: Kenig Carlos E. , Koch Herbert , Tataru Daniel
Published: December 2009

Nonlinear dispersive equations are models for nonlinear waves in a wide range of situations. Mathematically they display an interplay between linear dispersion and nonlinear focusing or defocusing effects. They are linked to diverse areas of mathematics and physics, ranging from nonlinear optics over oscillatory integrals and integrable systems to algebraic geometry. The conference did focus on the analytic (PDE) aspects with a view towards applications.

Major results and areas are:

Motivated by recent developments there has been a series of lectures by T. Kappeler and P. Topalov on their application of the inverse scattering transform to rough initial data for the Korteweg-de-Vries equation.

This meeting was attended by 45 participants. The organizers made an effort to include young mathematicians and to give them the opportunity of a shorter talk.
Entanglement and Decoherence: Mathematics and Physics of Quantum Information and Computation
Authors: Albeverio Sergio , Dell'Antonio Gianfausto , De Martini Francesco
Published: December 2009

The meeting was attended by over 50 participants from more than 10 countries and three continents. It brought together mathematicians, physicists and computer scientists working on quantum information and computation. Experts were present as well as young participants, both at predoctoral level as well as postdoctoral level. The focus was on the problems of entanglement and decoherence, aiming at the formulation and discussion of precise concepts, developments of models and their interrelations, and the discussion of experiments in relation with the theory. Special attention was given to recent developments, and to furthering interaction and co-operations between the different groups. The scientific programm focused mainly on the following subjects: 1) Quantum Entanglement and Nonlocality: including Bell-type inequalities (theoretical and experimental studies), equivalence of quantum states under local unitary transformations. 2) Quantum Separability: separability criteria for multiple quantum mixed states in arbitrary dimension. 3) Decoherence: study of models; quantum error corrections, fault tolerant computation, multiparticle problems. 4) Sources of Quantum Entanglement. 5) Quantum Information: quantum cloning, teleportation, key distribution, algorithms. 6) Quantum Measurement and Quantum Optics; Holonomic quantum gates; Quantum semigroups; Continuum Observation. 7) Mathematical Quantum and non commutative structures in connection with quantum information theory. 20 lectures were presented, including 15 surveys and 5 concentrating on specific recent results. There were 10 informal evening lectures on 3 topics: entanglement, control and decoherence. Besides 7 informal talks were presented in 2 round table discussions. All presentations and discussions amply demonstrated the vitality and actuality of this area of research and the fascinating interrelations between the different specialities it involves. The organizers and participants would like to take this opportunity to thank the Mathematisches Forschungsinstitut Oberwolfach for having provided a comfortable and inspiring environment for the meeting and the scientific work.
Komplexe Algebraische Geometrie
Authors: Catanese Fabrizio , Kawamata Yujiro , Tian Gang , Viehweg Eckart
Published: December 2009

The Workshop “Komplexe algebraische Geometrie”, organized by Fabrizio Catanese (Bayreuth), Yujiro Kawamata (Tokyo), Gang Tian (Princeton), and Eckart Viehweg (Essen), drew together 49 participants. In an age range between the early 20's and the late 50's, there were young PhD students, together with established leaders of the fields which are related to the not too specialized thematic title of the workshop. There were 23 talks, each one lasting 50 minutes but followed by a lively 10-15 minutes discussion, which as usual continued outside the lecture room and throughout the day and the night. The Conference fully fulfilled its purported aim, of setting in contact mathematicians with different specializations and non uniform background, of presenting new fashionable topics alongside with new insights on long standing classical open problems, and also cross-fertilizations with other research topics as arithmetic and physics (for the last, cf. the survey of Jun Li on Gromow Witten invariants). The more detailed contents of the several expositions can be suitably grouped under the following general headings :

The style of the talks varied considerably, ranging from general overviews like the one by Corti, to “experimental” conjectures, like the ones given by Schreyer, but the variety of striking results and very interesting and challenging proposals is so great that certainly each organizer and each participant had its favourite ones. We believe however that the quality of the expositions in the abstracts collected here is such that we do not deem it necessary to further procrastinate the anticipated pleasure of the reader.
Regelungstheorie
Authors: Allgöwer Frank , Helmke Uwe , Kwakernaak Huibert
Published: December 2009

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering. It therefore covers a wide variety of topics, ranging from fundamental mathematical aspects to real world engineering applications of industrial relevance. In particular, it has deep connections to different branches of pure and applied mathematics, including e.g. operator theory, real and complex analysis, probability theory, commutative algebra, as well as algebraic and differential geometry.

The Oberwolfach workshop “Regelungstheorie” has the traditional goal of bringing active researchers with both a mathematical and an engineering background together in order to stimulate a fruitful interaction between these communities. This diversity of our field was once again nicely reflected in the expertise of the 42 participants of the 2005 workshop from all over the world. A particular effort has been devoted to inviting newcomers from the somewhat younger generation which had the refreshing side-effect of an increase in the number of female participants.

The particular mission of the workshop was as well reflected in the various themes that have been addressed in the 29 stimulating presentations. The program has been clustered into rather coherent sessions of three lectures each revolving around

In addition to regular session talks of thirty minutes, each day was initialized with a somewhat longer presentation that was followed by a five-minute discussant statement. This novel feature provided an extra stimulus for lively general discussions during the subsequent coffee breaks and beyond.

As a particular highlight, Vincent Blondel and Jan Willems organized and chaired a session on open problems in systems and control on Tuesday evening. Eight contributors had the opportunity to present concrete research questions in seven minutes, followed by a brief two minutes discussion with the audience. Six of the corresponding abstracts related to the open problems have been collected at the end of this report and should, in view of the raised challenges, stimulate lots of new research initiatives.

In addition to the excellent scientific program most participants could as well enjoy the traditional Wednesday afternoon walk to St. Roman, with beautiful weather and in a wonderful winter landscape. The day was concluded by an impressive musical evening with excerpts from Schubert's “Winterreise” presented by P. Hippe (bariton) and P. Lohmann (piano).
Groups and Geometries
Authors: Liebeck Martin W. , Mühlherr Bernhard , Stroth Gernot
Published: December 2009

The workshop \textit{Groups and Geometries} was one of a series of Oberwolfach workshops which takes place every 3 years. It focused on finite simple groups, Lie-type groups and their interactions with geometry. There were 47 participants and 25 talks. The themes of the latter centered around the classification of the finite simple groups and its applications, simple algebraic groups and group-theoretic applications of Moufang buildings. There were several 1-hour talks describing recent major results concerning these topics. For instance, D. Segal gave a talk about a proof of a long standing conjecture on finitely generated pro-finite groups which uses the classification. The topic of B. Martin's was the answer to a question of Serre about the notion of complete reducibility for algebraic groups. R. Weiss determined the Whitehead group for certain algebraic groups of type

E_6

and

E_7

as an application of his new approach to exceptional Moufang quadrangles. There had been also important contributions by young participants. For instance, by D. Bundy on the

C(G,T)

-theorem, by P.-E. Caprace on the solution of the isomorphism problem for Kac-Moody groups, by T. De Medts on rank-1 groups and by R. Gramlich on local recognition. The conference showed that the theory of simple groups and their geometries is a very active area and that there is a lot of interaction with an increasing number of other areas. There is in particular growing impact of algebraic groups on the theory of finite groups. One of the aims of the conference was, to bring together people from these different areas, so that they can speak with each other and possibly work together on common problems. This interaction was indeed very lively and thus the meeting stands in the tradition of very successful meetings on \textit{Groups and Geometries} at Oberwolfach.
Discrete Geometry
Authors: Henk Martin , Matoušek Jiří , Welzl Emo
Published: December 2009

The
Discrete Geometry
workshop was attended by 53 participants from a wide range of geographic regions, many of them young researchers (some supported by a grant from the European Union). The morning sessions consisted of survey talks providing an overview of recent developments in Discrete Geometry:

There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants.