Survey Talk Research Articles

Algorithmic Graph Theory

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.

The Arithmetic of Fields

The fifth conference with the title The Arithmetic of Fields , organized by Wulf-Dieter Geyer (Erlangen), Moshe Jarden (Tel Aviv), and Florian Pop (Philadelphia), was held February 5–11th, 2006. In contrast to the fourth conference held in February 3–9th, 2002, this conference was a “full” one, namely as many participants were invited as the Institute could host. Due to support from the European Union, more young people were invited in the last few weeks prior to the conference, so that the total number of participants reached 54. The participants came from 13 countries: Germany (20), USA (10), Israel (7), France (7), Denmark (2), Austria (1), Brazil (1), Canada (1), Hungary (1), Japan (1), Romania (1), Russia (1), and South Africa (1). Among the participants there were 9 graduate students and 8 young researchers. Six women attended the conference. The organisers asked four people before the conference to give surveys of one hour on recent progress made by other colleagues in Field Arithmetic. Tamás Szamuely (Budapest) reported on the solution by János Kollár of a Problem due to Ax from 1968: Every PAC field of characteristic 0 is C_1 . The case where the characteristic is positive remains open. Alexandra Shlapentokh (Greenville) described the progress Bjorn Poonen made on Hilbert's Tenth Problem: There exists a recursive set T_1 of prime numbers of natural density 0 and a set T_2 of prime numbers of natural density 1 such that T_1\subseteq T_2 and for each set S with T_1\subseteq S\subseteq T_2 Hilbert's Tenth Problem for O_{\mathbb Q,S} has a negative solution. Here O_{\mathbb Q,S} is the ring of all rational numbers whose denominators are divided only by primes in S . Whether Hilbert's Tenth Problem for \mathbb Q has a negative solution is still open. Alexander Prestel (Konstanz) presented a theorem of Jochen Koenigsmann: If a p -Sylow extension P of a field K is Henselian and P is neither separably closed nor real closed, then K itself is Henselian. Pierre Débes (Lille) surveyed Fried's problem on Modular Towers. He mentioned that the main conjecture is close to completion in the case of 4 branch points (Bayley–Fried). He also reported on a result of Anna Cadoret: The dihedral group has a regular realization over \mathbb Q_{\rm ab} with only inertia groups of order 2 . In addition to these survey talks seventeen participants were invited to report on their own achievements in 45 minutes talks. Altogether, the talks presented the impressive progress made in Field Arithmetic in recent years. The reader may find here extended abstracts of all talks. We hope they will be to the benefit of all of the participants as well as the fans of Field Arithmetic. The organisers: Wulf-Dieter Geyer, Moshe Jarden, Florian Pop

Miniworkshop: L2-Spectral Invariants and the Integrated Density of States

Both the study of L^2 -spectral invariants in geometry and the investigation of the integrated density of states in mathematical physics have attracted much attention in recent years. While the two topics are strongly related, the corresponding communities are rather unaware of each others work and methods. The main aim of this mini-workshop was to bring together people from both fields and provide a basis for interaction. Accordingly, the first two days of the conference were spent with survey talks solicited by the organizers to highlight concepts and methods. There were 9 such talks with durations between 60 and 90 minutes. The second half of the conference was devoted to more detailed investigations. Most participants used the opportunity to present their current research in the area of the meeting. There were 13 such talks. The results presented in those talks contained significant contributions e.g. to the Atiyah conjecture about integrality of L^2 -Betti numbers for a completely new class of groups by Peter Linnell, a mathematically rigorous derivation using von Neumann traces of the asymptotics of the specific heat near absolute zero by Mikhael Shubin, and approximation results for the integrated density of states in various new contexts. Altogether the conference was attended by 17 participants.

Analysis and Geometric Singularities

This workshop was a successful and happy meeting of 23 researchers working on various aspects of partial differential equations on singular spaces and noncompact manifolds. This field encompasses such diverse topics as L^2 cohomology, harmonic analysis and spectral analysis on locally symmetric spaces, spectral geometry, and constructions of metrics with special geometry. All of these topics, and several others, were covered extensively at this meeting. One noteworthy feature was a series of four expository survey talks by leading experts. These served to stimulate extensive discussion which carried over into all the other sessions. Lizhen Ji presented a view of the extensive compactification theory for locally symmetric spaces; Werner Müller gave a detailed survey of harmonic analysis on locally symmetric spaces, leading up to a presentation of his important new work in this area; Thomas Schick discussed the topological and C^* algebraic techniques now used extensively in index theory; finally, Michael Singer explained some of the analytic aspects of the burgeoning field of extremal Kähler metrics. The rest of the talks complemented these surveys. For example, Leslie Saper presented an overview of his recent and ongoing work on the L^2 cohomology of locally symmetric spaces and the algebraic machinery of {\mathcal L} -modules he has developed for this goal. Gilles Carron talked about the L^2 cohomology of QALE spaces and Eugenie Hunsicker's talk explored further relationships between L^2 and intersection cohomology theory on manifolds with fibred boundaries, while Ulrich Bunke discussed his recent work on twisted K-theory. There were several talks on more traditional problems in spectral geometry, including one by Iosif Polterovich on new methods to obtain lower bounds for the spectral function of the Laplacian, one on estimates for the first positive eigenvalue of the Dirac operator by Bernd Ammann, Daniel Grieser's talk on the ‘max flow min cut’ method adapted from graph theory applied to the estimation of the Cheeger constant, and Sergiu Moroianu's talk on the disappearance of the essential spectrum for certain magnetic Laplacians. Talks focusing on metrics with special geometry included the one by Robin Graham on the Dirichlet-to-Neumann operator for Poincare–Einstein metrics, Hartmut Weiss' results on deformations of three-dimensional constant curvature cone metrics, and Frank Pacard's presentation of his construction, with Arezzo, of new constant scalar curvature Kähler metrics. Finally, Nader Yeganefar discussed his work on topological restrictions associated with quadratic curvature decay. One of the things that stood out about this meeting was the spirited participation by a number of young researchers, many of whom are clearly poised to take their places amongst the next generation of leaders in this field. The quality of the talks was uniformly high, and their clarity certainly furthered the main goal, which was to further stimulate the discussion between the various groups of researchers at this meeting. Several major results were announced and discussed here. These illustrate that this field is now reaching a certain maturity. A number of these problems have been outstanding for many years, but new ideas have finally allowed for more serious attacks on problems of a truly global nature, and on problems involving iterated singularities. In any case, the meeting certainly demonstrated the current vitality of this field.

Discrete Geometry

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.

Gitter und Anwendungen

Respecting the different backgrounds of the participants, most of the talks were aimed to a broad audience, among those nine survey talks invited by the organizers. This concept was very successful and opened many discussions and interdisciplinary interactions. It was also very fruitful for the many young participants. Also most of the non speakers took the opportunity to present their recent work via posters. The theory of lattices has many applications and interactions with various other mathematical and technical disciplines such as information technology, topology, algebraic geometry, representation theory, combinatorics, number theory and modular forms to name only the most prominent ones. One of the most classical problems is the construction of dense lattice sphere packings. The densest lattice sphere packings are only known in dimensions up to 8 and, due to a recent work by H. Cohn and A. Kumar, also in dimension 24. A. Kumar gave a very illuminating talk on the strategy of their proof of this great result which uses certain linear programming techniques based on the fact that the minimal vectors of the Leech lattice form a very good design. Design techniques have also been used by B. Venkov to define the notion of strongly perfect lattices, which realize certain local maxima of the density function and which are now classified up to dimension 12. The theory of modular forms is one well established tool in the investigation of lattices as shown in the nice introduction by Krieg. Bannai applies this theory to bound the strength of designs provided by layers of unimodular lattices. On the other hand, lattices provide one important tool for the construction of modular forms. The last talk by Böcherer showed that in many situations all modular forms are linear combinations of theta-series. Of increasing interest but computationally very difficult is the investigation of thin lattice sphere coverings. Vallentin and Schürmann developed new algorithms to find local optimal coverings and discovered lattice coverings better than the ones previously known. Nguyen presented new reduction algorithms for lattices in small dimensions motivated by practical applications to cryptosystems. For certain applications in information technology not only the density of the lattice but also other properties that minimize the fading error play a role. It turns out that lattices constructed from algebraic number fields yield good codes for Rayleigh fading channels. These ideal lattices are also used to investigate the ring of integers in algebraic number fields as shown in the talk by Schoof and also in the applications of ideal lattices introduced by Bayer-Fluckiger. Strongly related to this is the application to Arakelov theory as illustrated in the talks by Bost and Künnemann. Also Voronoi's classical theory of perfect forms (see Martinet's talk for an introduction) has new fruitful applications in number theory more precisely in the calculation of the homology of GL_n({\bf Z}) as presented by Elbaz-Vincent. Last but not least one should mention the talk by J.-P. Serre on BL-bases and unitary groups in characteristic 2.

Typical case complexity of Satisfiability Algorithms and the threshold phenomenon

This is a written record of a survey talk on the topic of typical case complexity of Satisfiability algorithms which was presented at the LICS workshop on this subject in June, 2003, Ottawa, Canada.

Typical case complexity of Satisfiability Algorithms and the threshold phenomenon

This is a written record of a survey talk on the topic of typical case complexity of Satisfiability algorithms which was presented at the LICS workshop on this subject in June, 2003, Ottawa, Canada.

Mathematics and Physics of Disordered Systems

It was the aim of this workshop to bring together researchers from different fields, working on various aspects of the theory of disordered systems in theoretical and mathematical physics. On the one hand, this means triggering an interaction between researchers working on these phenomena using methods from theoretical physics and those working with mathematical tools on the subject, as well as scientists mainly interested in the mathematical methods themselves. On the other hand, the conference also tried to bring together researchers working on related but well distinguished physical phenomena like Anderson localization in random systems, the theory of aperiodic order, the random matrix theory its applications (e.g., in quantum chaos). To do justice to this mix of people with different background, about half the talks were survey talks (with an extended time for the speaker), giving an introduction to the corresponding topic in general as well as to recent results. We found it important to give enough time for physicists to explain their ideas in this way. About a similar number of mathematical and physical review talks were given. It was a nice experience that there was always a very lively and intense discussion after the talks and in many cases even during the talks. Many of these discussions were continued outside the lecture hall in the free time in the afternoons and the evenings. In this way, many interactions started during the week, in particular between people who would otherwise not have met. An important topic in the meeting was the theory of “quasiperiodic” order in various forms. Consequently, a survey talk on quasicrystals was scheduled. It triggered interesting discussions with various other disciplines. One highlight was the demonstration that and how Delone sets and Schödinger operators defined on them form a natural bridge between the world of perfect (periodic) order and that of stochastic phenomena. Though this is still in its infancy, it became clear that a high potential for unified approaches is still to be unraveled. One of the main topics of the conference was the theory of Anderson localization for random Schrödinger operators from different points of view. There were reviews by theoretical physicists on the subject explaining basic ideas about Anderson localization, its applications to the quantum Hall effect and the supersymmetric approach to this field. There was also special emphasis on the phenomenon of “weak localization” both from physicists and mathematicians. Another interesting topic was the explanation of the Aizenman–Molchanov method to prove Anderson localization in the continuum. This extension of the method is rather new and is considered to be very important for future developments in the area. In the field of Anderson localization/delocalization, there were also presentations of new developments. There was also a survey talk on threshold phenomena for the random Landau Hamiltonian which is not only an interesting result by itself, but may also lead to progress toward a proof for the existence of delocalized states. A number of talks were devoted to the theory of quantum Hall conductance. Both the physical theory and recent mathematical progress were explained. Presumably this topic encouraged the highest amount of discussion among the various disciplines. The phenomena are not yet understood from a physical point of view, even the model itself is under discussion. The theory of random matrices is a discipline that evolved rather independently of the theory of random Schrödinger operators although there are obvious intersections between these fields. It was therefore an important task of the conference to bring together people from these fields. As a result of this attempt there are joint research papers in preparation triggered by the meeting, e.g., on Hamiltonians on random graphs which can be interpreted both as examples of random matrices and as discrete Schrödinger operators. The use of poster presentations and the official allocation of one evening session to this proved successful in that it sparked lots of discussions that went on during the meeting. It became evident that some results can actually profit from such a presentation. In view of the fact that time for talks is limited, it might be a reasonable alternative, long used by other sciences, and some more wall space might help. The unique atmosphere of the institute did its magic once again, and the combination with the excellent library makes Oberwolfach still one of the best possible places for meetings such as this one, also in view of the recent competition from places like BIRS in Canada.

On Matrices with Perron–Frobenius Properties and Some Negative Entries

We consider those n-by-n matrices with a strictly dominant positive eigenvalue of multiplicity 1 and associated positive left and right eigenvectors. Such matrices may have negative entries and generalize the primitive matrices in important ways. Several ways of constructing such matrices, including a very geometric one, are discussed. This paper grew out of a recent survey talk about nonnegative matrices by the first author and a joint paper, with others, by the second author about the symmetric case [Tarazaga et al. (2001) Linear Algebra Appl. 328: 57].

Noncommutative geometry and fundamental interactions: the first ten years

AbstractThis is the full text of a survey talk for nonspecialists, delivered at the 66th Annual Meeting of the German Physical Society in Leipzig, March 2002. We have not taken pains to suppress the colloquial style. References are given only insofar as they help to underline the points made; this is not a full‐blooded survey. The connection between noncommutative field theory and string theory is mentioned, but deemphasized. Contributions to noncommutative geometry made in Germany are emphasized.

Noncommutative geometry and fundamental interactions: the first ten years

This is the full text of a survey talk for nonspecialists, delivered at the 66th Annual Meeting of the German Physical Society in Leipzig, March 2002. We have not taken pains to suppress the colloquial style. References are given only insofar as they help to underline the points made; this is not a full-blooded survey. The connection between noncommutative field theory and string theory is mentioned, but deemphasized. Contributions to noncommutative geometry made in Germany are emphasized.

Some polynomially solvable subcases of the detailed routing problem in VLSI design

There are plenty of NP-complete problems in very large scale integrated design, like channel routing or switchbox routing in the two-layer Manhattan model (2Mm, for short). However, there are quite a few polynomially solvable problems as well. Some of them (like the single row routing in 2Mm) are “classical” results; in a past survey [36] we presented some more recent ones, including: 1. a linear time channel routing algorithm in the unconstrained two-layer model; 2. a linear time switchbox routing algorithm in the unconstrained multilayer model; and 3. a linear time solution of the so called gamma routing problem in 2Mm. (This latter means that all the terminals to be interconnected are situated at two adjacent sides of a rectangular routing area, thus forming a Γ shape. Just like channel routing, it is a special case of switchbox routing, and contains single row routing as a special case.) In the present survey talk we also mention some results from the last three years, including 4. some negative results (NP-completeness) in the multilayer Manhattan model and a channel routing algorithm if the number of layers is even; 5. an interesting relation between channel routing and multiprocessor scheduling; and 6. some improvements of 1–3 above. We present the (positive and negative) results in a systematic way, taking into account two hierarchies, namely that of geometry (what to route) and technology (how to route) at the same time.

Science and Engineering in Knowledge Representation and Reasoning

As a field, knowledge representation has often been accused of being off in a theoretical no-man's land, removed from, and largely unrelated to, the central issues in AI. This article argues that recent trends in KR instead demonstrate the benefits of the interplay between science and engineering, a lesson from which all AI could benefit. This article grew out of a survey talk on the Third International Conference on Knowledge Representation and Reasoning (KR-92) (Nebel, Rich, and Swartout 1992) that I presented at the Thirteenth International Joint Conference on Artificial Intelligence (IJCAI-93).

The semigroup of continuous selfmaps ofIhas infinitely many ideals

LetIdenote the interval [0,1] in its usual topology and letS=S(I)be the semigroup of continuous mappings ofIinto itself with function composition as the semigroup operation. In a survey talk given at Oberwolfach in 1989 (see [5]), K. D. Magill Jr. pointed out that some elementary algebraic properties ofSare still unknown. We shall answer one of the questions he raised (which appears as Problem 4.6 in [5] and which he had asked earlier, as long ago as 1975 [3]) by showing thatShas infinitely many distinct two-sided ideals. In fact, we shall produce an infinite descending sequence of distinct ideals. As Magill points out, this also solves his Problem 4.5:Shas infinitely many distinct congruences. We believe thatSmust have c distinct ideals, but we have been unable to prove this.

United Kingdom: Coalite health survey talks

United Kingdom: Coalite health survey talks

Neutral Injectors (IAEA Workshop, Culham, 1978)

The first IAEA-sponsored Workshop on Neutral Injectors was held at Culham Laboratory from 26th to 30th June 1978. Approximately 60 people attended representing essentially all the major laboratories with extensive programmes in the field. The subject area was limited to neutral-beam sources based on positive ions, specifically for injection into fusion systems, although we did have one survey talk by Dr. Sluyters on negative-ion systems.