The Grothendieck-Teichmüller Theory of Dessins d'Enfants

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The Grothendieck-Teichmüller Theory of Dessins d'Enfants

 08 - 12 Sep 2008
 ICMS, 14 India Street, Edinburgh

Organiser

Name
Institution
Jones, GarethUniversity of Southampton
Wolfart, JuergenGoethe-Universität, Frankfurt


Dessins d'enfants is a theory, initiated by Grothendieck in 1984, which uses maps on surfaces to create a link between on the one hand the Teichmüller theory of Riemann surfaces, and on the other hand the Galois theory of algebraic number fields and the algebraic curves defined over them. The theory thus provides surprising and powerful connections between such apparently dissimilar fields as algebraic geometry, algebraic number theory, combinatorics, Galois theory, group theory, Teichmüller theory and topology, and provides a fertile ground for collaborations between members of these different disciplines.

The main theme of the meeting will be dessins d'enfants, but this will be interpreted very broadly. The workshop will appeal to participants whose interest in the subject comes from many different areas of mathematics: in particular areas such as algebraic geometry, combinatorics, Galois theory, group theory, Riemann surfaces and Teichmüller theory, with participants who are experts (or in some cases, promising beginners) in the theoretical and practical aspects of these subjects. This workshop will involve younger mathematicians from a wide range of relevant mathematical disciplines, and from further afield, with the emphasis of the workshop being on interactive discussion and collaboration.

Numbers will be limited to a maximum of 35 and there will be a contribution to local costs (including accommodation and subsistence) from awards to ICMS. A Registration Fee of £30 will be payable by all delegates.

Participation is by invitation only: those interested in attending should contact Gareth Jones (g.a.jones@maths.soton.ac.uk).

Arrangements

Participation
Participation is by invitation only. The workshop will begin on Monday 8 September and finish on Friday 12 September 2008.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will take place at the head-quarters of ICMS, 14 India Street, Edinburgh. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The ICMS travel pages contain advice on how to travel to Edinburgh. For local information the finding ICMS page shows the location of ICMS and contains useful maps of the city centre.

The seminar room at ICMS has four whiteboards, two overhead projectors, a data projector and laptop.

Wireless access is available throughout the ICMS building. There are also three public PCs which may be used at any time for internet access and to check email.

Accommodation
ICMS will arrange single en-suite rooms in local guest houses for those who require it. Accommodation is typically about 15 to 30 minutes walk from ICMS. Participants are also free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £45.00 per night bed and breakfast. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
Buffet lunches will be provided on Monday 8 September and Friday 12 September only. For the remainder of the days, participants are free to go out for lunch and explore the many cafes, restaurants, sandwich shops and bars in the surrounding area. On arrival we will provide you with a ‘welcome’ pack which will contain information about eating places nearby.

Morning and afternoon refreshments will be provided throughout the workshop.

There will be an informal wine reception after the close of lectures on Monday 8 September.

On Tuesday 9 September there will be an informal supper at a local restaurant. The workshop dinner will take place on the evening of Thursday 11 September.

Registration
Registration will take place on Monday 8 September.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee throughout the workshop, lunch on Monday and Friday, the wine reception, the informal supper on Tuesday and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with receipts and bank details. Please note that we cannot reimburse any item without a receipt.

Under the terms of our EPSRC funding we are required to charge a 30.00 GBP registration fee to cover costs not admissible under the grant. The fee will be payable on arrival at the workshop: payment may be by cash, sterling cheque or credit/debit card. If you anticipate any difficulty covering the fee, please let us know.

 

Programme

Monday 08 September

09.00 - 10.00

Registration and coffee

10.00 - 11.00

Juergen Wolfart (Goethe-Universität, Frankfurt)
Regular Dessins, Wilson operation, and cyclotomic fields

11.15 - 11.45

Milagros Izquierdo Barrios (Linköpings Universitet)
Equisymmetric strata of the singular locus of the moduli space of Riemann surfaces of genus 4

12.00 - 12.30

David Singerman (University of Southampton)
The buckyball curve

12.30 - 14.15

Buffet lunch

14.15 - 15.15

Robert Silhol (Université Montpellier 2)
Galois actions on certain one parameter families of surfaces

15.30 - 16.00

Bernhard Koeck (University of Southampton)
About Belyi's theorem for Klein surfaces

16.00 - 16.30

Coffee/tea break

16.30 - 17.00

Grzegorz Gromadzki (University of Gdansk)
On Singerman symmetries of a class of Belyi Riemann surfaces

17.15 - 17.45

Frank Feierabend (Goethe-University)
Galois-actions on generalized Macbeath-Hurwitz curves

18.00 - 18.45

Wine Reception in the Exhibition Room at ICMS

 

Tuesday 09 September

09.30 - 10.15

Alexander Mednykh (Sobolev Institute of Mathematics)
Enumerating chiral coverings and maps

10.15 - 10.45

Coffee/tea break

10.45 - 11.30

Roman Nedela (Slovak Academy of Sciences/Matej Bel University)
Recent progress in map enunumeration

11.45 - 12.15

Martin Skoviera (Comenius University)
Regular maps with nilpotent automorphism groups

12.15 - 14.00

Lunch break

14.00 - 15.00

Jan-Cristoph Schlage-Puchta (Universität Freiburg)
Recognition of finite index subgroups in groups with few finite subgroups

15.15 - 15.45

Jozef Siran (Open University)
Regular maps of Euler characteristic -3p for prime p

15.45 - 16.15

Coffee/tea break

16.15 - 16.45

Anthony Weaver (City University of New York)
Quasiplatonic surfaces in the n-hyperelliptic locus

17.00 - 17.30

Aaron Wootton (University of Portland)
Quasiplatonic cyclic n-gonal surfaces

19.00

Informal evening meal at Athena Greek Restaurant, 89 Hanover Street

 

Wednesday 10 September

09.30 - 10.30

Gabriela Schmithuesen (Universität Karlsruhe (TH))
Origamis in Outer Space

10.30 - 11.00

Coffee/Tea

11.00 - 11.30

Frank Herrlich (Universität Karlsruhe (TH))
Relations between origamis and dessins d'enfants

11.45 - 12.15

Ernesto Girondo Sirvent (Universidad Autonoma de Madrid)
Uniform dessins in genus 2

12.15 - 14.00

Lunch break

14.00 - 15.00

Pierre Lochak (Institut de Mathématiques de Jussieu)
Grothendieck-Teichmueller theory and curve complexes

15.00

Free time

 

Thursday 11 September

09.30 - 10.30

Marston Conder (University of Auckland)
Recent progress in the study of regular maps

10.30 - 11.00

Coffee/Tea

11.00 - 11.30

Cormac Long (University of Southampton)
Hurwitz groups and their generalisations

11.45 - 12.15

Antonio Breda d'Azevedo (Universidade de Aveiro)
Bipartite-uniform hypermaps on the sphere and on the projective plane

12.15 - 14.00

Lunch break

14.00 - 15.00

Gabino Gonzalez-Diez (Universidad Autonoma de Madrid)
Quasiplatonic Riemann surfaces and Beauville complex surfaces

15.15 - 15.45

Alexandre Zvonkine (Université Bordeaux I)
On some questions related to compositions of coverings

15.45 - 16.15

Coffee/Tea

16.15 - 16.45

George Shabat (Russian State University for the Humanities)
Calculating Belyi pairs for the unicellular dessins of positive genus

17.00 - 17.30

George Shabat delivered on behalf of Vladimir Dremov
Mulase-Penkava operator and its application to Belyi pairs calculations

19.00

Workshop dinner at First Coast Restaurant, 99-101 Dalry Road, Edinburgh

 

Friday 12 September

09.00 - 09.30

Fedor Pakovich (Ben Gurion University of the Negev)
Belyi functions of Platonic solids and counterexamples to the first Ritt theorem for rational functions

09.45 - 10.15

Carlo Hamalainen (Charles University)
Latin bitrades to Belyi functions

10.15 - 10.45

Coffee/Tea

10.45 - 11.15

Cristina Sarti (Universität Frankfurt)
Difference sets, incidence graphs and dessins d'enfants

11.30 - 12.00

Daniel Pinto (Universidade de Coimbra)
Constructions for nonorientable hypermaps

12.15 - 13.15

Gareth Jones (University of Southampton)
The most symmetric dessins

13.15 - 14.15

Buffet lunch


Presentations

Breda d'Azevedo, Antonio
Bipartite-uniform hypermaps on the sphere and on the projective plane
View Abstract
A hypermap is bipartite if its set of hypervertices can be partitioned into two sets (call them bipsets) so that consecutive hypervertices (either around a hyperedge or around a hyperface) are in different bipsets. It is bipartite-uniform if all the hypervertices lying in the same bipset have the same valency, all the hyperedges have the same valency and all the hyperfaces have the same valency. We present a classification of the bipartite-uniform hypermaps on the sphere and on projective plane. They are all constructed from uniform hypermaps using two constructions. The automorphism group of a bipartite-uniform hypermap on the sphere or on projective plane acts transitively in one or two orbits of flags (that is, bipartite-uniform hypermaps on the sphere and on the projective plane are bipartite-regular). We also compute the (bipartite-) chirality group and chirality index of them, and also their closure covers and covering cores.
Conder, Marston
Recent progress in the study of regular maps
View Abstract
Following the computer-assisted determination of all regular and orientably-regular maps of characteristic -1 to -200 two years ago, a lot of new things have been discovered. We now have theorems about the genus spectra of such maps that are chiral, and such maps that have simple underlying graph. Also more is now known about regular Cayley maps (that is, orientably-regular maps whose underlying graph is a Cayley graph); for example, the curious result that a map for a cyclic group is reflexible if and only if it is anti-balanced. A detailed classification of all regular Cayley maps for cyclic groups is now in sight. I will report on many of these developments, some of which were obtained in joint work with Young Soo Kwon, Jozef Siran and Tom Tucker.
Feierabend, Frank
Galois-actions on generalized Macbeath-Hurwitz curves
View Abstract
For a prime p congruent to +-1 mod 7 there are three different torsion free normal subgroups N in the triangle group of signature (2,3,7) such that the quotient is a Hurwitz group isomorphic to PSL(2,p),and they lead to three non-isomorphic Riemann surfaces N(upper half plane) which are Galois-conjugate. We try to generalize this result for torsion free normal subgroups in (maximal) triangle groups with quotient PSL(2,q) and determine the moduli field of the related surfaces. Moreover we see that the Galois-action on these curves is compatible with a Galois-action on prime ideals in the field of definition of the curves.
Girondo Sirvent, Ernesto
Uniform dessins in genus 2
View Abstract
A basic but in general difficult problem is the following:
For two given dessins d'enfants, when do they induce the same complex
structure? We will describe in this talk how hyperbolic geometry might be useful to answer the question for the case of uniform dessins in genus 2. The talk is based in some work actually in progress by the author in collaboration with David Torres (Madrid).
Gonzalez-Diez, Gabino
Quasiplatonic Riemann surfaces and Beauville complex surfaces
View Abstract
A Riemann surface is called quasiplatonic if it admits a Belyi function given as quotient map of the action of a finite group on it. If an abstract group G acts as a quasiplatonic group simultaneously on two Riemann surfaces C and T and, moreover, the induced action on the product CxT is free then the corresponding quotient complex surface (CxT)/G is called a Beauville surface.
Beauville surfaces were introduced by Catanese in 2000 and then studied by himself jointly with Bauer and Grunewald. I shall attempt to report on the significance of these objects and on some results obtained by these authors as well as on more recent work carried out by Y. Fuertes, A. Jaikin and myself.
Gromadzki, Grzegorz
On Singerman symmetries of a class of Belyi Riemann surfaces
View Abstract
In virtue of the Belyi Theorem an algebraic curve can be defined
over the algebraic numbers if and only if the corresponding
Riemann surface can be uniformized by a subgroup of a Fuchsian
triangle group. Such surfaces are known as Belyi surfaces and an
important class of them consists of Riemann surfaces having
so-called large group of automorphisms. Necessary and sufficient
algebraic conditions for these surfaces to be symmetric
were found by Singerman in the middle of seventies and by a recent result of K"ock and Singerman, these numbers can be chosen to be
real if and only if the respective surface is symmetric. The aim
of this talk is to give, in similar like Singerman's terms, formulas for the
topological type of the corresponding symmetries, to which we refer
as to the title symmetries and which correspond the topological type
to the real forms of initial curve.
Hamalainen, Carlo
Latin bitrades to Belyi functions.
View Abstract
A latin bitrade describes the difference between two latin squares. In
this talk I will introduce latin bitrades, their representation as
hypermaps, and pose some open problems relating to Belyi functions.
Herrlich, Frank
Relations between origamis and dessins d'enfants
View Abstract
An origami is a translation surface, where the translation structure comes from a tiling by squares. The linear parts of the affine diffeomorphisms form the Veech group of the origami. It is a finite index subgroup of SL(2,Z). The quotient of the upper half plane by the Veech group is an affine algebraic curve, called the origami curve of the given origami. It sits (up to a birational map) inside a moduli space M_g, and it comes along with a Belyi map, i.e. a dessin d'enfants.
In the talk I shall also focus on a second relation between origamis and dessins d'enfants, namely a construction which associates a dessin to every cusp of an origami curve.
Izquierdo Barrios, Milagros
Equisymmetric strata of the singular locus of the moduli space of Riemann surfaces of genus 4
View Abstract
The moduli space Mg is the space of analytic equivalence classes of Riemann surfaces of a fixed genus g. The space Mg has a natural equisymmetric stratification, each stratum consists of the Riemann surfaces with an automorphisms group acting in a given topological way. Using Fuchsian groups we describe the equisimmetrical stratification for the moduli space of surfaces of genus 4. This is a joint work with Antonio F. Costa
Jones, Gareth
The most symmetric dessins
View Abstract
A dessin can be regarded as an embedding of a bipartite graph in a compact orientable surface. In many areas of mathematics it is of interest to study the most symmetric objects, so what are the most symmetric dessins? The most symmetric bipartite graphs are the complete bipartite graphs K_{n,n}, with the wreath product of the symmetric groups S_n and S_2 as automorphism group. The most symmetric orientable embeddings of any graph are its orientably regular embeddings, those for which the orientation-preserving automorphism group acts transitively on directed edges. I shall discuss joint work with Shao-Fei Du, Jin Ho Kwak, Roman Nedela and Martin Skoviera, using properties of p-groups and of finite solvable groups to classify the orientably regular embeddings of complete bipartite graphs. I shall also discuss further joint work with Manfred Streit and Juergen Wolfart, which in some of the simpler cases determines the moduli fields, the Galois conjugacy and the defining equations of these dessins.
Koeck, Bernhard
About Belyi's theorem for Klein surfaces
View Abstract
In this talk I report on a new approach to Belyi's theorem for Klein surfaces, recently developed in joint work with Eike Lau. This new approach also settles a question left open in the paper 'Real Belyi Theory' by David Singerman and myself. As time permits, the natural framework of the new main argument for varieties of arbitrary dimension and over arbitrary fields will be explained.
Lochak, Pierre
Grothendieck-Teichmueller theory and curve complexes
View Abstract
I will explain how a topologico-combinatorial structure embodied by curve complexes (and their profinite completions) ties up with automorphism groups of Teichmueller groups and ultimately with the Grothendieck-Teichmueller group.
Long, Cormac
Hurwitz groups and their generalisations
View Abstract
Since they were first discovered over a century ago, Hurwitz groups and, more generally, triangle groups, have repeatedly motivated questions about Riemann surfaces, as well as providing many interesting and beautiful examples.
Motivated by the results obtained for surfaces, it is reasonable to ask whether analogous theorems can be obtained in higher dimensions. In this talk, a brief survey of current work in this area will be presented, together with a look at some specific results in the case of hyperbolic 3-manifolds. Particular attention will be given to the Lanner groups, where a classification of all PSL(2,q) groups arising as quotients of these groups will be given.
Mednykh, Alexander
Enumerating chiral coverings and maps
View Abstract
The following problem was stated by V. A. Liskovets during the conference at Dresden University in 1996:

"Find the number of non-equivalent n-fold orientable coverings of a given non-orientable manifold with a finitely generated fundamental group."

The main purpose of this report is to give a solution of the Liskovets problem. As an
application, we enumerate reflexible coverings and chiral pairs of coverings of a non-orientable manifold with a finitely generated fundamental group. These results form a background for counting chiral pairs of maps and hypermaps on closed orientable surface. It will be done in a forthcoming paper by A. Breda, R. Nedela and A. Mednykh. The obtained formula for the number of reflexible coverings allows also to count self-dual and Petri-dual maps and other combinatorial objects.
Nedela, Roman
Recent progress in map enunumeration
View Abstract
By a map we mean a 2-cell decomposition of a (compact) surface. In our talk we give a survey on recent progress in map enumeration. Relationship to subgroup enumeration and to enumeration of coverings of surfaces will be explained.
New results are based on systematic use of maps on orbifolds. Both orientable and non-orientable case will be considered.
Pakovich, Fedor
Belyi functions of Platonic solids and counterexamples to the first Ritt theorem for rational functions
View Abstract
In 1922 J. Ritt constructed the theory of functional decompositions of polynomials. In particular, he proved (the first Ritt theorem) that any decomposition of a polynomial into a composition of indecomposable polynomials (such decompositions are called maximal) may be obtained from any other similar decomposition by a chain of some elementary transformations. In the talk, using Belyi functions of Platonic solids (for the first time calculated by F. Klein), we construct explicit counterexamples to a possible generalization of the first Ritt theorem to rational functions. In particular, we construct examples of rational functions having maximal decompositions of different length.
Pinto, Daniel
Constructions for nonorientable hypermaps
View Abstract
We will present some constructions to build an infinite number of
nonorientable hypermaps of a given genus g and type (l,m,n), when two
of the parameters are equal. The methods used in those constructions are
adapted from the ones previously developed for orientable hypermaps.
Sarti, Cristina
Difference sets, incidence graphs and dessins d'enfants
View Abstract
In this talk we will speak about uniform dessins d'enfants associated to finite projective spaces P^m(F_n) In particular, we will discuss the case of the projective space P^4(F_2). We will see that applying a special combinatorial operation (Wilson operation) to the edges of the dessin D_1 associated to P^4(F_2) we obtain a new dessin D_3 which is not only uniform but even regular. As a further nice property we obtain a covering of the surface determined by D_3 through the surface determined by D_1. We will discuss some possible generalizations of these results to other finite
projective spaces P^m(F_n). In this context Galois automorphisms of finite fields play a crucial role
Scheichl, Robert
Galois actions on certain one parameter families of surfaces
View Abstract
The orbit under the action of SL2(R) of a translation surface obtained from squares defines an algebraic curve in moduli space.
Given such an orbit the family it defines can also be described from a hyperbolic point of view (in terms of Fuchsian groups or Fenchel-Nielsen coordinates) and in some cases in terms of equations.
Using these descriptions the natural action of SL2(Z) on such families can be interpreted in terms of the hyperbolic structure and be described as an action on the equations of Galois groups of coverings.
Schlage-Puchta, Jan-Cristoph
Recognition of finite index subgroups in groups with few finite subgroups
View Abstract
Let G be a group, U a finite index subgroup. Then G acts on the cosets of U, and this action is one of the most convenient ways to do computations with subgroups. Here we study the splitting of the G-conjugacy class of a finite subgroup into U-conjugacy classes and show that the U-conjugacy classes which are G-conjugate to a given group are easily computable. As application we obtain simple and unifying proofs of results of Millington, Singerman, and Kulkarni, and generalize them to a certain class of virtually free groups.
Schmithuesen, Gabriela
Origamis in Outer Space
View Abstract
Origamis are given by purely combinatorial data and define geometric objects in Teichmüller space, in moduli space and in the mapping class group. There are a lot of similarities and relations to the theory of dessins d'enfants. One of them is that origamis can be viewed as unramified finite coverings in this case of the once-punctured torus. This covering situation does not as in the case of a dessin define a point in moduli space, but a whole algebraic curve.

We generalize this covering setting and put it in the context of Culler-Vogtmann's Outer Space, the classifying space of marked metric finite graphs with a fixed fundamental group. The study of origamis in Outer Space enables us to obtain new insights on the classical objects in Teichmüller space.
Shabat, George
Calculating Belyi pairs for the unicellular dessins of positive genus
View Abstract
Some techniques of the calculation of the clean Belyi pairs for the unicellular dessins of positive genus will be discussed. The presumably new ingredients are the special curves in Hurwitz spaces and points of second order on the jacobians. The "easy" cases will be distinguished -- those with central symmetry and bicolored structure on the graphs. The complete results for toric 4-edged dessins will be presented.
Singerman, David
The buckyball curve.
View Abstract
The most famous Riemann surface is Klein's of genus 3 which contains 7 embedded Fano planes which are imbedded as dessins. Following the famous result of Galois mentioned in his final letter we wish to pass from the prime 7 to the prime 11. We then get a curve of genus 70 that contains 11 imbedded buckyballs. This has been written about in Lieven le Bruyn's nice mathematical blog at www.neverendingbooks.org if you want to do some homework!
Siran, Jozef
Regular maps of Euler characteristic -3p for prime p
View Abstract
We will present a classification of regular maps on surfaces of Euler characteristic -3p for all primes p.
Skoviera, Martin
Regular maps with nilpotent automorphism groups
View Abstract
We study regular maps on orientable surfaces with nilpotent automorphism groups and classify those where the nilpotency class is at most two. This is a joint work with A. Malnic and R. Nedela.
Weaver, Anthony
Quasiplatonic surfaces in the n-hyperelliptic locus
View Abstract
For surfaces of genus g>4n+1, n>0, the n-hyperelliptic locus Mgn is a subvariety of the moduli space Mg consisting of surface classes admitting a unique conformal involution with quotient a surface of genus n. (When n=0, these are the classical hyperelliptic surfaces.) I will discuss conditions which gaurantee the existence or non-existence of smooth quasiplatonic surfaces in Mgn. For example, I will show that there are infinite sequences of genera in which Mg1 (the “elliptic-hyperelliptic” locus) contains no quasiplatonic surfaces, and also infinite sequences of genera in which the number of quasiplatonic surfaces is greater than any pre-assigned postive integer.
Wolfart, Juergen
Regular Dessins, Wilson operation, and cyclotomic fields
View Abstract
Report on recent joint work with Gareth Jones and Manfred Streit. A fundamental problem in the theory of Belyi surfaces is to understand how the absolute Galois group acts on dessins d'enfants. For a sufficiently rich class of regular dessins - including e.g. regular embeddings of complete graphs - this action turns out to be a combinatorial procedure on graph embeddings described many years ago by Steve Wilson. In these cases, the underlying quasiplatonic Belyi surfaces are defined over cyclotomic fields.
Wootton, Aaron
Quasiplatonic cyclic n-gonal surfaces
View Abstract
For an integer ngeq 2, a cyclic n-gonal surface is a compact Riemann surface S which admits a cyclic group of automorphisms C of order n, such that the quotient space S/C has genus 0. Such surfaces admit a plane model of the form y^n=p(x) and the first step in finding such a model is determining the full automorphism group of S. For n a prime, such results are known, but for n not prime, the problem becomes much more difficult without further assumptions. Our goal is to determine the full automorphism groups and equation for each S under the assumption that S is quasiplatonic (meaning if G is the full automorphism group of S, then S/G has genus 0 and the quotient map Srightarow S/G is branched over exactly three points). During this talk, we shall provide a brief history of these surfaces and the known results, with particular emphasis on the problems which arise for general n. We shall also discuss some of the approaches we may take to solving this problem. (Joint work with S A Broughton)
Zvonkine, Alexandre
On some questions related to compositions of coverings
View Abstract
We will discuss Ritt's theorem (a covering is decomposable if its monodromy group is imprimitive) and some algorithmic and enumerative problems related to it.

Participants

Name
Institution
Antonio, Breda d'AzevedoUniversidade de Aveiro
Marston, ConderUniversity of Auckland
Frank, FeierabendGoethe-University
Herbert, GanglUniversity of Durham
Ernesto, Girondo SirventUniversidad Autonoma de Madrid
Gabino, Gonzalez-DiezUniversidad Autonoma de Madrid
Grzegorz, GromadzkiUniversity of Gdansk
Carlo, HamalainenCharles University
Frank, HerrlichUniversität Karlsruhe (TH)
Milagros, Izquierdo BarriosLinköpings Universitet
Gareth, JonesUniversity of Southampton
Bernhard, KoeckUniversity of Southampton
Zoe, LaingUniversity oF Southampton
Pierre, LochakUniversité Paris Pierre et Marie Curie
Cormac, LongUniversity of Southampton
Alexander, MednykhSobolev Institute of Mathematics
Benjamin, MuelhbauerGoethe-University Frankfurt
Roman, NedelaSlovak Academy of Sciences/Matej Bel University
Fedor, PakovichBen Gurion University of the Negev
Daniel, PintoUniversidade de Coimbra
Cristina, SartiUniversität Frankfurt
Robert, ScheichlUniversity of Bath
Jan-Cristoph, Schlage-PuchtaUniversität Freiburg
Gabriela, SchmithuesenUniversität Karlsruhe (TH)
George, ShabatRussian State University for the Humanities
David, SingermanUniversity of Southampton
Jozef, SiranOpen University
Martin, SkovieraComenius University
Manfred, StreitDB Systel
Anthony, WeaverCity University of New York
Juergen, WolfartGoethe-Universität, Frankfurt
Aaron, WoottonUniversity of Portland
Alexandre, ZvonkineUniversité Bordeaux I