bluuebunny/crossref_metadata_2025 huggingface dataset:
Geometric cycles, index theory and twisted K-homology
Authors: Wang Bai-Ling
Published: December 2009

We study twisted

\mathrm{Spin}^c

-manifolds over a paracompact Hausdorff space

X

with a twisting

α : X → K(ℤ, 3)

. We introduce the topological index and the analytical index on the bordism group of

α

-twisted

\mathrm{Spin}^c

-manifolds over

(X, α)

, taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this article is to establish the equality between the topological index and the analytical index for closed smooth manifolds. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of

(X, α)

analogous to Baum–Douglas's geometric cycles. As an application of our twisted index theorem, we discuss the twisted longitudinal index theorem for a foliated manifold

(X, F)

with a twisting

α : X → K(ℤ, 3)

, which generalizes the Connes–Skandalis index theorem for foliations and the Atiyah–Singer families index theorem to twisted cases.
Explicit HRS-tilting
Authors: Noohi Behrang
Published: December 2009

For an abelian category A equipped with a torsion pair, we give an explicit description for the tilted abelian category B introduced in [HaReSm], and also for the categories Ch(B) and \mathcal{D}(B). We also describe the DG structure on Ch(B). As a consequence, we find new proofs of certain results of [ibid.]. The main ingredient is the category of
decorated
complexes.
Fibrations with noncommutative fibers
Authors: Echterhoff Siegfried , Nest Ryszard , Oyono-Oyono Hervé
Published: December 2009

We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C*-algebra bundles. We then derive an analogue of the Leray–Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature.
Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets
Authors: Pearson John , Bellissard Jean
Published: December 2009

An analogue of the Riemannian Geometry for an ultrametric Cantor set

(C,d)

is described using the tools of Noncommutative Geometry. Associated with

(C,d)

is a weighted rooted tree, its Michon tree [29]. This tree allows to define a family of spectral triples

(\mathcal{C}_{\rm Lip}(C),\mathcal{H},D)

using the

ℓ^2

-space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here

\mathcal{C}_{\rm Lip}(C)

denotes the space of Lipschitz continuous functions on

(C,d)

. The family of spectral triples is indexed by the space of
choice functions
, which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the family of these spectral triples allows to recover the metric on

C

. The corresponding

ζ

-function is shown to have abscissa of convergence,

s_0

, equal to the
upper box dimension
of

(C,d)

. Taking the residue at this singularity leads to the definition of a canonical probability measure on

C

, which in certain cases coincides with the Hausdorff measure at dimension

s_0

. This measure in turn induces a measure on the space of choices. Given a choice, the commutator of

D

with a Lipschitz continuous function can be interpreted as a directional derivative. By integrating over all choices, this leads to the definition of an analogue of the Laplace–Beltrami operator. This operator has compact resolvent and generates a Markov semigroup which plays the role of a Brownian motion on

C

. This construction is applied to the simplest case, the triadic Cantor set, where: (i) the spectrum and the eigenfunctions of the Laplace–Beltrami operator are computed, (ii) the Weyl asymptotic formula is shown to hold with the dimension

s_0

, (iii) the corresponding Markov process is shown to have an anomalous diffusion with

\mathbb{E}(d(X_t,X_{t + δt})^2) ≃ δt \ln(1/δt)

as

δt ↓ 0

.
Periodic cyclic homology of reductive $p$-adic groups
Authors: Solleveld Maarten
Published: December 2009

Let

G

be a reductive

p

-adic group,

\mathcal{H}(G)

its Hecke algebra and

\mathcal{S}(G)

its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This provides an alternative proof of the Baum–Connes conjecture for

G

, modulo torsion.

As preparation for our main theorem we prove two results that have independent interest. Firstly, a general comparison theorem for the periodic cyclic homology of finite type algebras and certain Fréchet completions thereof. Secondly, a refined form of the Langlands classification for

G

, which clarifies the relation between the smooth spectrum and the tempered spectrum.
The first Hochschild cohomology group of quantum matrices and the quantum special linear group
Authors: Launois Stéphane , Lenagan Thomas H.
Published: December 2009

We calculate the first Hochschild cohomology group of quantum matrices, the quantum general linear group and the quantum special linear group in the generic case when the deformation parameter is not a root of unity. As a corollary, we obtain information about twisted Hochschild homology of these algebras.
The Goodwillie Calculus of Functors
Authors: Goodwillie Thomas , McCarthy Randy
Published: December 2009

The purpose of this meeting was to introduce the participants to calculus of functors, a theory aimed at “approximating” functors in algebra and topology. The focus of the talks was on two related branches of the theory, homotopy and manifold calculus. T

he organizers, Thomas Goodwillie and Randy McCarthy, scheduled 16 talks. The first was given by Goodwillie who introduced the main ideas and outlined the plan for the rest of the meeting. In talks 2–5, speakers explained the most important terminology, techniques, and results used in both versions of the theory. In talks 6–8, participants learned about manifold calculus and how it applies to spaces of knots (talk 8), while talks 9–15 dealt with homotopy calculus. In the last four of those lectures, some applications, elaborations on results established in previous talks, and different versions of certain ideas and proofs encountered so far were given. Goodwillie explored some connections between the two versions of calculus of functors (as they apply to the embedding and identity functors) and gave the concluding remarks in the last talk.

It should be noted that a third version of the theory, orthogonal calculus, was not discussed due to time constraints, but Goodwillie explained some of its most salient features in an extra evening session. Other evening events took place as well, such as Rainer Vogt's elaboration of an aspect of his talk and informal research reports given by some participants.
Multiplier Ideal Sheaves in Algebraic and Complex Geometry
Authors: Kohn Joseph J. , Schumacher Georg , Siu Yum-Tong
Published: December 2009

The conference was organized by Joseph Kohn (Princeton), Georg Schumacher (Marburg), and Yum-Tong Siu (Harvard), and was attended by 44 participants. Its aim was to put together a group from both complex analysis and algebraic geometry, reflecting recent developments, where the title of the conference stands for phenomena and methods, closely related to both of these areas. The original approach involving the theory of partial differential equations and subelliptic estimates was addressed in several contributions, including estimates for the

\overline\partial

-Neumann problem, subelliptic PDE's and sub-Riemannian Geometry, and subelliptic estimates from an algebraic-geometric point of view. Main areas were also applications to the abundance conjecture, pseudoeffective bundles, and the use of the twisted Nakano identity to investigate the cohomology of multiplier ideals. Critical points of sections of holomorphic vector bundles were discussed from a probabilistic viewpoint. Concerning the hyperbolicity of complex manifolds, for entire analytic curves the multiplicity of the associated current with respect to subsets was studied, and holomorphic curves in semi-abelian varieties. Another main topic was recent results on invariants arising from multiplier ideal sheaves, and critical exponents of analytic functions related to jumping coefficients, applications to local analytic geometry, and also results concerning the Fujita conjecture. Finally recent progress on transcendental Morse inequalities was presented.
Geometrie
Authors: Bangert Victor , Burago Yury Dmitri , Pinkall Ulrich
Published: December 2009

The workshop was organized by V. Bangert (Freiburg), Yu. D. Burago (St. Petersburg) and U. Pinkall (Berlin). Out of the 47 participants 22 came from Germany, 8 from the United States, 7 from Switzerland, 4 from Russia, 4 from England and 2 from France.

The official program consisted of 20 lectures and included four lectures by B. Kleiner (Ann Arbor) on “Perelman's work on Ricci flow'”. The program covered a wide range of new developments in geometry. To name some of them, we mention the topics “Metric space geometry in the style of Alexandrov/Gromov”, “Finsler geometry”, “Constant mean curvature surfaces in Thurston geometries”.
New Inference Concepts for Analysing Complex Data
Authors: Fan Jianqing , Müller Klaus-Robert , Spokoiny Vladimir
Published: December 2009

The workshop
New Inference Concepts for Analysing Complex Data
, organised by Jianqing Fan (Chapel Hill ), Klaus-Robert Müller (Berlin) and Vladimir Spokoiny (Berlin) was held November 14st–November 20th, 2004. This meeting was well attended with about 45 participants with broad geographic representation from all continents. This workshop was a nice blend of researchers with various backgrounds from the areas of statistics and machine learning.

The main purpose of this workshop was to assemble international leaders from statistics and machine learning in the Institute: to identify important research problems, to cross-fertilize between the disciplines, and to ultimately start coordinated research efforts towards better solutions. The program included more than 25 talks organized in sections on various topics:
Support Vector Machines
(B. Schölkopf, A.J. Smola, G.Wahba),
mathematical finance
(H. Dette, W. Härdle, S.X. Chen),
dimension reduction
(Y. Xia, A. Dalalyan, M. Kawanabe, E. Mammen, H. H. Zhang),
non-parametric smoothing
(D. Belomestny, J. Polzehl),
Boosting
(P. Bühlmann, Y. Ritov),
genomics
(A. Nobel, Hepping Zhang),
classification
(S. van de Geer, G. Blanchard),
statistical inverse problem
(M. Reiss, A. Goldenshluger),
Functional regression
(H.-G. Müller, T.Cai),
Clustering
(J. M. Buhmann, A. Nobel) among others. P. Bühlmann gave an overview on statistical methods of
boosting
while D. Yekutieli made an extended presentation on
False Discovery Rate
.

There was one organized discussion on
Smoothing methods in classification
. The workshop gave an excellent opportunity for exchanging the opinions and expertise as well as for discussing various topics in different areas of modern mathematical statistics and machine learning theory. The discussion revealed a lot of common ideas and principles in these two fields but also differences in the methodology and approaches. An exchange of ideas can clearly contribute to the both fields. Already during workshop some new projects were originated that involve both statisticians and people from machine learning society.

The workshop was attended by a number of young statisticians and gave an excellent opportunity for training: both by attending the high level presentation, by presenting their own results and by participating at the numerous informal discussions.
Optimization and Applications
Authors: Jarre Florian , Lemaréchal Claude , Zowe Jochem
Published: December 2009

This workshop was attended by 47 participants with broad geographic representation from all continents. It was enlightened by this cheerful atmosphere special to Oberwolfach, and which the participants contributed to by their mutual friendship and esteem.

The theme was optimization problems, their theory and resolution, and above all applications of them. Some emphasis was put on the still recent subject of optimization over the cone of positive semidefinite matrices (SDP). In the following 20 extended abstracts, one finds

We thank Gail Pieper (Argonne National Laboratory) for her careful proofreading of every abstract; her editorial work substantially improved the presentation of this material.
Mini-Workshop: Interface Problems in Computational Fluid Dynamics
Authors: Bänsch Eberhard , Tobiska Lutz , Walkington Noel J.
Published: December 2009

This workshop was concurrent with the workshop on Analytical and Numerical Methods in Image and Surface Processing (see: Report No. 10/2005), and since the participants interests overlapped several combined talks were organized. In particular, we thank P. Schroder, U. Reif, G. Dziuk, C. Elliot, and F. Cirak, for their contributions.

Currently there are three major techniques for tracking or characterizing interfaces present in a CFD computation: (i) The level set method, (ii) Phase field approximations, and (iii) Explicit parameterization using Lagrangian variables. Each of these techniques, and various hybrids, were presented by the participants to solve a variety of challenging CFD problems involving interfaces. Several trends emerged during the workshop.
Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics
Authors: Buss Samuel R. , Moschovakis Yiannis N. , Schwichtenberg Helmut
Published: December 2009

The workshop
Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics
, was held March 20th–March 26th, 2005 and had several aims.

To promote
interaction between traditional proof theory and a more structural mathematical proof theory. It is hoped to encourage the application-oriented to consider their tools more abstractedly and those with foundational leaning to focus on possible applications. Questions of feasibility should play an essential role here.

To further develop
constructive mathematics. For instance, there has been recent progress in designing some central notions for a constructive treatment of algebraic topology (like that of a scheme, in Peter Schuster's Habilitationsschrift 2003). An essential tool is the so-called formal or point-free topology, developed by Sambin, Coquand and others. Type theory offers some unifying concepts for a useful dicussion of the notions involved.

To explore
the relevance of classical mathematics to algorithms. Recent work of Kohlenbach, Lombardi, Roy and others showed, in very different ways, that mathematical proofs that use a priory highly non computational concepts, such as Zorn's lemma or compactness principles, may contain implicitely very interesting computational information. For instance, recent work of Kohlenbach – using a modification of Gödel's Dialectica interpretation – could extract not only algorithmic information but also new theorems, surprising to the expert (here in the field of metric fixed point theory).

To understand
in depth mathematical concepts in connection with algorithms and proofs, and also to further develop the notion of a certificate, aimed at unifying attempts to connect proof systems and computer algebra systems.

To develop
connections between proof theory and computational complexity. Specifically to understand the connections between the complexity of formal proofs, computational complexity and descriptive complexity.

The variety of these aims is well reflected by the many talks given. As can be seen from their abstracts, presented in chronological order, they cover a broad range of topics – without losing their common theme, that is, mathematical logic and the formal reasoning about proofs and computations.

In order to provide the participants with an overview of some of the recent developments in some of the covered topics two invited lecture series were given. Both highlighted applications of proof theory to other areas of mathematics. Ulrich Kohlenbach presented proof mining as an area of applications of proof theory to analysis; Thierry Coquand's talk on infinite objects in constructive mathematics showed applications of proof theory to algebra.
Optimal Control of Coupled Systems of PDE
Authors: Kunisch Karl , Leugering Günter , Sprekels Jürgen , Tröltzsch Fredi
Published: December 2009

Current research in the control of partial differential equations is driven by a multitude of applications in engineering and science that are modelled by coupled systems of nonlinear differential equations. Associated optimal control problems need efficient numerical methods to deal with the resulting very large problems. There is a fast development of numerical methods and the associated analysis must keep track to justify them and to prepare the basis for further research. It has been the main intention of this Conference to tighten the links between applications, numerics and analysis with some emphasis on the analytic aspects. The meeting was attended by about 50 participants from Europe and the US. \vspace{0.25cm} The scientific program consisted of 30 talks that covered various topics such as controllability, feedback control, optimality conditions, analysis and control of Navier-Stokes equations, model reduction of large systems, optimal shape design, and applications in crystal growth, chemical reactions and aviation. It showed that Optimal Control of Partial Differential Equations is a very lively and active mathematical field. Well known experts with long standing experience, Postdocs and PhD students contributed to the program. In particular, 4 PhD students from US took part who received full support from the NSF. This diversity of topics and mix of participants stimulated an extensive and fruitful discussion and initiated new collaborations, in particular of younger researchers.
Algebraische Zahlentheorie
Authors: Deninger Christopher , Schneider Peter , Scholl Anthony J.
Published: December 2009

The meeting was organized by Christopher Deninger (Münster), Peter Schneider (Münster) and Anthony Scholl (Cambridge). The subject of the conference was Algebraic Number Theory and Arithmetic Algebraic Geometry. It brought together internationally leading experts in these areas as well as a considerable number of young researchers. Correspondingly the talks were a mixture of lectures on recent important progress by experts and of reports about outstanding dissertations. This created a very stimulating atmosphere for discussions and exchange of ideas.
Arakelov Geometry
Authors: Bost Jean-Benoît , Künnemann Klaus , Roessler Damian
Published: December 2009

Arakelov geometry studies the geometry and arithmetic of schemes of finite type over Spec

{\bf Z}

, i.e. systems of polynomial equations with integer coefficients. It combines methods from algebraic geometry, number theory, and hermitian differential geometry.

The workshop was organized by Jean-Benoît Bost (Orsay), Klaus Künnemann (Regensburg) and Damian Roessler (Paris). It brought together internationally leading experts in the area as well as a considerable number of young researchers. The talks covered various aspects of Arakelov geometry from analytic torsion over adelic and non-archimedean analytic spaces to modular forms and diophantine geometry.

A non-mathematical complement was a piano recital by Harry Tamvakis on Thursday night featuring Bach, Beethoven and Chopin.
Analysis and Quantum Theory
Authors: Bach Volker , Dereziński Jan , Solovej Jan Philip
Published: December 2009

The MFO-Workshop 0538
Analysis and Quantum Theory
, organized by V. Bach (U Mainz), J. Dereziński (U Warsaw), and J. P. Solovej (U Copenhagen), was held from September 18 through September 24, 2005. The workshop's general theme was mathematical physics, it especially focussed on

Besides these lectures of more tutorial and overview character, several seminars on current research results were delivered; see the abstracts below. The purpose of this workshops was twofold.

Besides the lectures present during daytime, an evening discussion on open problems and future developments took place on September 26. This discussion was chaired by J. Fröhlich who emphazised the importance that mathematical physics addresses
physically relevant
problems. This criterion in competition with the criterion of
feasability of its solution
should be the guiding principles of our choice of research topics. There was a general agreement of the audience to this statement.

It is a pleasure to thank the director G.-M. Greuel, the administrative staff, and the kitchen staff of MFO, as well as, the state of Baden-Würtemberg (the main source of funding of MFO), for their work which made this workshop possible.
Deformations and Contractions in Mathematics and Physics
Authors: Fialowski Alice , Novikov Sergey , Schlichenmaier Martin , de Montigny Marc
Published: December 2009

Deformations of mathematical structures are not only important in most parts of mathematics but also to a large extend in physics. Contractions are in some respect dual to deformations. The aim of the proposed workshop was to bring together world experts in these complementary topics of deformations and contractions of various algebraic structures. Deformations and contractions have been investigated by researchers who had different approaches and goals. Tools such as cohomology, gradings, etc. which are utilized in the study of one concept, are likely to be useful for the other concept as well. At this meeting there were mathematicians, mathematical physicists and physicists as well. The organizers hope that the meeting was of benefit to all groups.

Because various fields in mathematics and physics exist in which deformations are used, it was necessary to focus the topic of the workshop. The meeting mainly considered deformations of algebras (in particular, of Lie algebras), groups, and related algebraic structures, the corresponding contractions, and their applications to problems in physics. Nevertheless, other fields with strong relations to the central topic were present too. One such field, discussed in detail at the workshop, with tight interaction was deformation quantization. But also other topics like quantum groups, deformation of Hopf algebras, q-deformed physics, fuzzy spaces, quantum systems as deformations of classical systems, etc showed up.

As the workshop had an interdisciplinary character it was considered to be useful to start with some introductory talks on

with the aim to introduce the necessary concepts which were not always well known to all the different communities present. For more details on the concepts, see the corresponding extended abstracts in this Oberwolfach report.

The following is a (non-exhaustive) list of topics which were discussed at the workshop.

The workshop was attended by 44 participants from all over the world. The official program consisted of 24 lectures. Two evening sessions of informal presentations were organised. Beside the official program, there was ample time for the participants for further activities, such as self-organised sessions and discussion groups.
Algorithmic Graph Theory
Authors: Czumaj Artur , Jansen Klaus , Meyer auf der Heide Friedhelm , Schiermeyer Ingo
Published: December 2009

The workshop
Algorithmic Graph Theory
, organized by Artur Czumaj (New Jersey), Friedhelm Meyer auf der Heide (Paderborn), Klaus Jansen (Kiel) and Ingo Schiermeyer (Freiberg) was held February 12th–February 18th, 2006. This meeting was attended by 46 participants from a wide range of geographic regions, many of them young researchers. In the morning sessions survey talks providing an overview of recent developments in Algorithmic Graph Theory were presented.

Geometry and Graphs

Graph Algorithms

Game Theory and Graphs

Graph Structures

There were 10 shorter talks and two special sessions on “Graph Algorithms” (organized by Christian Sohler) and “Graph Colourin” (organized by Ingo Schiermeyer). The contributions showed progress in the field provided in recent years. Furthermore, several open problems and conjectures were presented, some of them still far from being resolved. Beyond the program there was plenty of time for the participants to use the stimulating atmosphere of the Oberwolfach Institute for fruitful discussions.
Discrete Differential Geometry
Authors: Bobenko Alexander I. , Kenyon Richard , Sullivan John M. , Ziegler Günter M.
Published: December 2009

The workshop
Discrete Differential Geometry
, organized by Alexander I. Bobenko (Berlin), Richard W. Kenyon (Vancouver), John M. Sullivan (Berlin) and Günter M. Ziegler (Berlin), was held March 5th to March 11th, 2006. The meeting was very well attended, with almost 50 participants, from as far away as Australia and China.

Discrete differential geometry is a new and active mathematical terrain where differential geometry (providing the classical theory for smooth manifolds) and discrete geometry (concerned with polytopes, simplicial complexes, etc.) meet and interact. Problems of discrete differential geometry also naturally appear in (and are relevant for) other areas of mathematics. Moreover, the process of discretizing notions, problems and methods from the smooth theory often brings out new connections and interrelations between different areas.

The workshop at Oberwolfach brought together researchers with a wide variety of backgrounds, including of course discrete geometry and differential geometry, but also integrable systems, combinatorics, mathematical physics and geometry processing. The exchange of ideas among different subfields helped to build new bridges between these mathematical communities.

Discrete differential geometry can be said to have arisen from the observation that when a notion from smooth geometry (such as the notion of a minimal surface) is discretized “properly”, the discrete objects are not merely approximations of the smooth ones, but have special properties of their own which make them form in some sense a coherent entity by themselves. The discrete theory would seem to be the more fundamental one: The smooth theory can always be recovered as a limit, while there seems to be no natural way to predict from the smooth theory which discretizations will have the nicest properties.

One case where these ideas seem particularly well-developed is for geometries described by integrable systems. The notion of a discrete integrable system as given by consistency on a cubic lattice has already shed new light on classical, smooth integrable systems.

Another theme which arose repeatedly during the workshop was that of circle patterns and sphere packings. These can be used to discretize conformal maps, isothermic surfaces, and elastic bending energy.

Since a computer works with discrete representations of data, it is no surprise that many of the applications of discrete differential geometry are found within computer science, particularly in the areas of computational geometry, graphics and geometry processing. The workshop brought theoreticians together with people interested in these and other applications.
Topological and Geometric Methods in Group Theory
Authors: Bridson Martin R. , Kropholler Peter H. , Vogtmann Karen
Published: December 2009

The focus of this meeting was the use of topological and geometric methods to study infinite discrete groups. These methods are increasingly being supplemented by powerful new techniques from analysis. Key topics included group actions on CAT(0) and tree-like spaces, filling invariants, cohomology, K-theory and

\ell^2

-cohomology, amenability and Kazhdan's property (T), and deformation spaces, curve complexes and Teichmüller space. More specific information is contained in the abstracts which follow in this volume.

The meeting was organized around a series of 23 lectures each of 50 minutes' duration representing the major recent advances in the area. The first day's lectures were selected from the abstracts and talk proposals which had been submitted in advance of the meeting, and the remainder were decided on Monday and Tuesday morning; in this way were able to take advantage of a full range of abstracts and also to incorporate last-minute information about exciting developments in the field. We had interesting proposals from virtually every participant but lecture slots for fewer than half to speak. We posted all abstracts on the wall of the lecture building and drew attention to them by running a poster event on Tuesday evening, at which every participant took the opportunity to introduce their research and poster. This worked very effectively and was valuable especially to younger people or those visiting the Forschungsinstitut for the first time. It also led to small groups getting together for unofficial lecture sessions in the evening and afternoon breaks on subsequent days.

There were 51 participants from a wide range of countries, including Germany, France, the United States, the United Kingdom, Greece, Russia, Poland, Switzerland and Australia. We are grateful to the European Union, which provided funds to support 6 advanced graduate students, a number of recent Ph. Ds and a few senior researchers.

We feel that the meeting was exciting and highly successful. The quality of the lectures was outstanding, and outside of lectures there was a constant buzz of intense mathematical conversations. One indication of the high degree of current activity and interest in the subject is the fact that four of the participants (including two of the organizers) have been invited to speak at the upcoming International Congress of Mathematicians in Madrid.
Mini-Workshop: The Hauptvermutung for High-Dimensional Manifolds
Authors: Pedersen Erik Kjaer , Ranicki Andrew
Published: December 2009

The Mini-Workshop
The Hauptvermutung for High-Dimensional Manifolds
, organised by Erik Pedersen (Binghamton) and Andrew Ranicki (Edinburgh) was held August 13th–18th, 2006. The meeting was attended by 17 participants, ranging from graduate students to seasoned veterans.

The manifold Hauptvermutung is the conjecture that topological manifolds have a unique combinatorial structure. This conjecture was disproved in 1969 by Kirby and Siebenmann, who used a mixture of geometric and algebraic methods to classify the combinatorial structures on manifolds of dimension

>4

. However, there is some dissatisfaction in the community with the state of the literature on this topic. This has been voiced most forcefully by Novikov, who has written “In particular, the final Kirby–Siebenmann classification of topological multidimensional manifolds therefore is not proved yet in the literature.” ()

At this conference we discussed a number of questions concerning the Hauptvermutung and the structure theory of high-dimensional topological manifolds. These are our conclusions:

We found nothing fundamentally wrong with the original work of Kirby and Siebenmann [4], which is solidly grounded in the literature. Their determination of

\textit{TOP}/\textit{PL}

depends on Kirby's paper on the Annulus Conjecture and his ‘torus trick’. It was noted that Kirby's paper is based on the well-documented work on

\textit{PL}

classification of homotopy tori (Hsiang and Shaneson, Wall) and Sullivan's identification of the

\textit{PL}

normal invariants with

[-,G/\textit{PL}]

, but does not depend on any other work of Sullivan, documented or undocumented. This classification can be reduced to the Farrell Fibering Theorem [1], the calculation of

\pi_i(G/\textit{PL})

(Kervaire and Milnor [3]), and Wall's non-simply connected surgery theory [7].

There are modern proofs determining the homotopy type of

\textit{TOP}/\textit{PL}

using either the bounded surgery of Ferry and Pedersen [2] or a modification of the definition of the structure set.

Sullivan's determination of the homotopy type of

G/\textit{PL}

, which is well documented (for instance, in Madsen and Milgram [5]) is used to determine the homotopy type of

G/\textit{TOP}

and is fundamental to understanding the classification of general topological manifolds.

The 4-fold periodicity of the topological surgery sequence established by Siebenmann [4, p.283] contains a minor error having to do with base points. This is an easily corrected error, and the 4-fold periodicity is true whenever the manifold has a boundary.

The equivalence of the algebraic and topological surgery exact sequence as established by Ranicki [6] was confirmed. Sullivan's characteristic variety theorem, however it is understood, is not essential for the Kirby–Siebenmann triangulation of manifolds.

The following papers have been commissioned:
Singularities
Authors: Steenbrink Joseph H.M. , van Straten Duco , Vassiliev Victor A.
Published: December 2009

The Oberwolfach workshop
Singularity Theory
was held in the week September 10th–September 15th, 2006. It was organised by J. Steenbrink (Nijmegen), D. van Straten (Mainz) and V. Vassiliev (Moskow) and continued the sequence of workshops
Singularitäten
that take place regularly at Oberwolfach.

The morning sessions consisted of three talks, which were complemented with two talks in the afternoon. So by taking the traditional Oberwolfach hike on wednesday into account, a total of

23

full hour talks was presented during the workshop.

A broad variety of topics was covered by the speakers. Normal surface singularities continue to pose some big questions which were adressed in the talks of A. Nemethi and A. Melle. The theory of motivic integration has brought the arc structure of singularities into the focus of current interest. A. Reguera and P. Popescu-Pampu reported on recent progress related to the original Nash-problem. F. Loeser reported on the first steps in non-archimedian local analytic geometry, A. Parushinski talked on conormal geometry. There has been an increased activity around global aspects of singularity theory, characteristic classes, Thom-polynomials and their applications. These topics were adressed by M. Kazarian, M. Weiss J. Schürmann and V. Sedykh. Applications of ideas from singularity theory to the geometry of the fundamental group were presented in the talks by A. Dimca, V. Kulikov and A. Libgober. S. Shadrin presented a conjectural method to produce tautological relations in the cohomology of the moduli space

\mathcal{M}_{g,n}

and O. Tommasi reported on the rational cohomology of

\mathcal{M}_{4}

. A. Gorinov talked about bounds for the size of the automorphism group of complete intersections that arise from the computation of the cohomology of the discriminant-complement. The recent interest in the Hodge/Twistor structures associated to isolated hypersurface germs and tame polynomials on affine varieties, reported on by C. Sevenheck and C. Hertling, stems partly from the interpretation of quantum cohomology via mirror symmetry in terms of so-called Landau–Ginsburg models. D. Mond talked about his idea to relate these to the theory of functions on a free divisor. J. Christophersen showed us a Calabi–Yau space appearing in the deformation theory of a degenerate abelian surface. A. Gabrielov talked about the maximally inflected rational curves and its relation to the Bethe Ansatz. A. Takahasi presented his results for exceptional collections in the category maximal Cohen–Macaulay modules on hypersurface singularities and its mirror interpretation in the case of the exceptional singularities.

An informal concert by J. Steenbrink on piano und M. Weiss on flute formed the musical coda of the workshop.

The organisers would like to use this occasion to thank the Oberwolfach staff for their efficient handling of boundary conditions, which created the unique Oberwolfach athmosphere that helped to make the workshop a success.
Geometrie
Authors: Bangert Victor , Burago Yury Dmitri , Pinkall Ulrich
Published: December 2009

The workshop was organized by V. Bangert (Freiburg), Yu. D. Burago (St. Petersburg) and U. Pinkall (Berlin). Out of the 47 participants 22 came from Germany, 8 from the United States, 7 from Switzerland, 4 from Russia, 4 from England and 2 from France.

The official program consisted of 21 lectures and therefore left plenty of space for fruitful informal collaboration. As a tradition in this meeting there always is a series of talks that constitute a small course on a chosen topic of current interest. This time this course consisted of three lectures by V. Schroeder (Zürich) and S. Buyalo (Sankt-Petersburg) on “Asymptotic geometry of Gromov hyperbolic spaces”. In addition there were several informal talks organized by the participants, among them an evening devoted to topics related to visualization.

The program covered a wide range of new developments in geometry. These came from four major topics:

The wide range of topics provided a particularly pleasing environment for the young participants (among them six Phd-students).
Dynamische Systeme
Authors: Eliasson Håkan , Hofer Helmut W. , Yoccoz Jean-Christophe
Published: December 2009

This workshop, organised by Hakan Eliasson (Paris), Helmut Hofer (New York), and Jean-Christophe Yoccoz (Paris) continued the biannual series at Oberwolfach on Dynamical Systems that started as the “Moser & Zehnder meeting” in 1981. The workshop was attended by more than 50 participants from 12 countries. The main themes of the workshop were the new results and developments in the area of classical dynamical systems, in particular in celestial mechanics and Hamiltonian systems. The workshop covers a large area of dynamical systems and the following samples give an idea about the scope. The topic of Arnold Diffusion was treated in great detail by talks of M. Levi and J. Mather. In the classical field of celestial mechanics new insight has been gained about two interesting families of relative periodic solutions of the spatial n-body problem (the

P_{12}

-family and the hip-hop-family) as they share the property of being global continuations of Lyapunov families which bifurcate from a relative equilibrium solution in the direction orthogonal to the plane of motion (A. Chenciner). K. Kuperberg reported on the construction of flows on three-manifolds where every nonconstant trajectory is wild in a sense related to the Artin–Fox example of an exotic arc in Euclidean three-space. Other results were concerned with one of the main problems in Hamiltonian dynamic, namely the stability of motions in nearly-integrable systems (L. Niedermann). Y. Pesin outlined the construction of hyperbolic volume-preserving flows on manifolds of dimension at least three and V. Ginzburg described the recent developments concerning the Conley Conjecture for periodic points of Hamiltonian symplectic maps. John Franks described his recent results about group actions on surfaces. In addition several talks covered new Floer-theoretic methods in the study of Hamiltonian systems and it will be interesting to see in the future how these symplectic methods can be merged with the more classical dynamical systems methods.
Mini-Workshop: Projective Normality of Smooth Toric Varieties
Authors: Haase Christian , Hibi Takayuki , Maclagan Diane
Published: December 2009

The mini-workshop on
Projective normality of smooth toric varieties
, organized by Christian Haase (Berlin), Takayuki Hibi (Osaka), and Diane Maclagan (New Brunswick), was held August 12th–18th, 2007. A small group of researchers with backgrounds in combinatorics, commutative algebra, and algebraic geometry worked on the conjecture that embeddings of smooth toric varieties are projectively normal. This very basic question appears in different guises in algebraic geometry, commutative algebra, and integer programming, but specific cases also arise in additive number theory, representation theory, and statistics. See the summary by Diane Maclagan for three versions of the same question.

There were a limited number of contributed talks in the mornings, setting the theme for the afternoon working groups. Monday morning began with Diane Maclagan describing the problem, and Winfried Bruns surveying the known results in the polyhedral formulation. This was followed on Tuesday morning by Benjamin J. Howard and Hidefumi Ohsugi on special cases of normality, and an
introductory talk by Milena Hering on the geometric vanishing theorem approach to the problem. On Wednesday morning Hal Schenck described a commutative algebra approach developed on site together with Greg Smith, while Sam Payne explained the Frobenius splitting approach. The commutative algebra approach, with optimization notes, continued in the talk of Ngô Viêt Trung on Thursday morning. Najmuddin Fakhruddin also explained his proof of the extended two-dimensional case on Thursday morning. Finally, on Friday we heard from Christian Haase and Andreas Paffenholz on some techniques for showing normality in special cases, and Francisco Santos on lattice Delaunay simplices which are potential starting points in search for a counterexample.

In the afternoons we split into working groups which then reported on their findings before dinner. These discussions continued through breaks, and in gaps between talks. The atmosphere of the group was very energetic, and we hope that the momentum generated during the meeting will continue with some of the ideas developed being pursued by the participants.
As a direct outcome of the workshop, we would like to mention

The organizers and participants sincerely thank the institute for providing excellent working conditions and the unique Oberwolfach spirit. We are also grateful for funding from the NSF grant supporting young US-based participants, which allowed an extra participant to attend.

In what follows we present, in addition to summaries of the talks, brief accounts on the outcome of brainstorming sessions and working groups.
Analysis and Geometric Singularities
Authors: Brüning Jochen , Mazzeo Rafe , Piazza Paolo
Published: December 2009

The conference on “Analysis and Geometric Singularities” took place from August 19 to August 25, 2007. The meeting was a popular one, with all spots taken, and its atmosphere was lively and full of activity. Participants ranged from many experts and old hands in the field to many young postdocs and graduate students, and the interaction between researchers at all levels was clearly productive for everyone. The talks were notably very well attended throughout the week, and many were accompanied by lively discussions.

Each day was structured thematically: every day except Wednesday began with a longer (70 minute) survey talk on some general theme in the subject; the other talks each day were shorter (55 minutes) and focused on that theme. Wednesday morning, however, was devoted to five half-hour lectures by some of the younger participants. The first day's theme was spectral geometry; the lecture by Iosif Polterovich surveyed many recent results connecting spectral asymptotics and dynamics; later talks that day were given by Mueller, Kordyukov, Paycha and Grieser. Tuesday's survey talk by Mihalis Dafermos on the mathematics of black holes was followed by the lectures of Rodnianski, Vasy, Baer and Perry; the first two of these especially focusing on mathematical relativity. Thursday's survey by Jared Wunsch on propagation of singularities for the Schroedinger equation was followed by talks by Lesch, Ammann, Moscovici and Dai. Friday's theme was index theory, with a survey by Richard Melrose and other lectures by Carron, Albin, Bunke and Richardson. The young researchers who spoke on Wednesday were Azzali, Degeratu, Mazzieri, J. Mueller and Rochon.

The scientific level of the talks was uniformly high, as will be seen from the abstracts which follow; many interesting results were announced, and the survey talks helped provide a focus which in turn facilitated communication between researchers in the various different fields represented. Overall, the conference gave clear evidence that the field of analysis on singular spaces is very active, and indeed is extending in many new directions. The many strong young researchers here indicate that the field has a bright future!
Noncommutative Geometry
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. This includes K-theory, cyclic homology and the theory of spectral triples. Areas of intense research in recent years are related to topics such as index theory, 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. Many results elucidate important properties of fascinating specific classes of examples that arise in many applications.

The talks covered substantial new results and insights in several of the different areas in Noncommutative Geometry. The workshop was attended by 53 participants including 6 young researchers supported by the European Union.
Material Theories
Authors: DeSimone Antonio , Luckhaus Stephan , Truskinovsky Lev
Published: December 2009

Physical Sciences and, in particular, Mechanics, have always played an important role as a source of inspiration for Mathematics. They have suggested fundamental problems, and stimulated the development of new techniques and tools. Now it is a particularly exciting time for the interaction of Mathematics and Mechanics: experimental developments in Mechanics and Biophysics at nano scales are posing challenging questions and giving new opportunities to Mathematics.

This became very clear at this meeting. A broad spectrum of problems from such diverse fields of applications as biology and biophysics, mechanics of nanostructures, and crystal plasticity was presented.

From theoretical physics, recent concepts and paradigms such as self-organized criticality, and non equilibrium statistical mechanics as applied to living systems were discussed.

Stochastic processes, calculus of variations, and pde's, up to the theory of Boltzmann equation are the mathematical foundations of the recent results presented at the workshop.

A new topic of attention with respect to the glorious tradition of this conference series was the attempt to establish the foundations of engineering theories at the mesoscopic scales from ab-initio calculations (quantum mechanics).

An important aspect of the meeting was also the contribution from experimentalists, from wetting and contact angle hysteresis phenomena, to intermittency in plasticity, to electronic transport in semiconductors.

The success of the workshop was mostly due to the strong interaction among participants, bridging different fields of expertise, particularly in the discussion on classical fields such as dislocation mechanics, plasticity, kinetic theory and even buckling phenomena in structural mechanics. We have witnessed that several participants were coming out of the meeting with new ideas and new research projects to pursue. Many expressed their appreciation for the open-minded spirit with which problems were presented, and for the constructive spirit of the discussions. The atmosphere at the Institute certainly contributed in a decisive way to the success of the Workshop.