ICMS

Lie algebras, vertex algebras and automorphic forms

Aug 31, 2009 - Sep 04, 2009

15 South College Street, Edinburgh EH8 9AA

Organisers

Name	Institution
Höhn, Gerald	Kansas State University
Nikulin, Viacheslav	University of Liverpool
Scheithauer, Nils	Technische Universität Darmstadt

Group photograph taken outside ICMS at 15 South College Street is available here.

In 1968, Kac and Moody introduced a new class of infinite dimensional Lie algebras, called Kac-Moody algebras, which generalize the finite dimensional simple Lie algebras in a natural way. These Lie algebras became very important in many areas of mathematics. In 1986, Borcherds defined vertex algebras as an axiomatic framework for conformal field theories. Certain subquotients of these algebras have a Lie algebra structure. The Lie algebras obtained in this way are very similar to Kac-Moody algebras but might have imaginary simple roots. These generalized Kac-Moody algebras describe symmetries in string theory. The most interesting generalized Kac-Moody algebras have denominator identities which are automorphic forms on orthogonal groups.

The relations between generalized Kac-Moody algebras, vertex algebras and automorphic forms have led to fascinating results e.g. Borcherds' proof of Conway and Norton's moonshine conjecture for the monster. However, many important problems are still open including:

• Generalized Moonshine
• Moonshine for other groups
• Classification of vertex algebras and generalized Kac-Moody algebras
• Construction of vertex algebras and generalized Kac-Moody algebras
• Automorphic forms related to vertex algebras and generalized Kac-Moody algebras
• Related areas as hyperbolic reflection groups and moduli spaces

In this workshop we present the recent achievements in these problems. We also want to discuss possible new approaches to them.

PDF file of full report.

Arrangements

Venue
The workshop will be held at 15 South College Street, Edinburgh. All lectures will be held in the Newhaven Lecture Theatre. To view this room and a list of the visual equipment available click here. In addition, an overhead projector will be available to use and two blackboards have recently been installed. Follow this link for a map showing the location of 15 South College Street.

Computer Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network. If you wish to use a computer while in 15 South College Street please ask a member of the ICMS staff.

Travel
Information about travel to the UK and Edinburgh is available here. You may find this map useful.
Lothian buses charge 1.20 GBP for a single journey and 3.00 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 15.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight the train at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map.

Accommodation
ICMS has arranged accommodation for participants who require it. En-suite rooms have been booked in two locations (unless otherwise requested): The Kenneth MacKenzie Suite in the city centre, and Chancellor's Court in the Pollock Halls area. A map of the location of the two accommodation venues and the location of the workshop venue can be found here.

Participants are also free to make their own arrangements. A list of Edinburgh accommodation of various sorts and prices can be found on the Accommodation Page on the ICMS web site. Section 4 is particularly relevant. September is busy in Edinburgh and early booking is recommended.

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop.

On Monday 31 August, the first day of the workshop, a light buffet lunch will be provided free of charge to participants in the in the Chapterhouse on the ground floor of 15 South College Street. 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.

Following the last talk on Tuesday 1 September, an informal wine reception, to which all all participants are invited, will take place in the Chapterhouse on the ground floor of 15 South College Street. The workshop dinner will take place at 19.30 at Blonde, 75 St. Leonards's Street, Edinburgh, EH8 9QR, on the evening of Thursday 3 September. The workshop grant will cover the cost of this catering.

Registration
Registration will take place on Monday 31 August from 9.00 to 10.00 in the Chapterhouse on the ground floor of 15 South College Street.

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 only, the wine reception, 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 and will involve payment directly into your bank account. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world, their IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts.

Unless otherwise stated in your invitation or further correspondence, there will be a registration fee of 40.00 GBP for the workshop. We ask that this is paid in advance by using this credit or debit card payment form. The form should be printed out, completed and faxed back (as email is not a secure way of sending credit card information). The fax number is on the form. If it is not possible for you pay in advance, you may print out the credit/debit card form above and bring the completed form along to Registration. We prefer not to handle cash at Registration. Please note we do not accept American Express or Discover Card. Alternatively, we can accept sterling cheques. The cheque should be made payable to Heriot-Watt University and can be handed over at Registration.

Knowledge transfer event

Date 1 September
Time 16.45 to 20.00 (See the below programme)

As part of the workshop, Victor Kac (MIT) will deliver a lecture on Tuesday 1 September from 17.30 to 18.30 in the Newhaven lecture theatre of the e-Science building, 15 South College Street, Edinburgh. This talk will be opened up to mathematicians and mathematical physicists from all Scottish Universities.

Those interested in attending only the lecture (and not the rest of the workshop) should register their interest by emailing Johanna Kytöharju at ICMS, giving the following information:

• your name and institution

A wine reception for all attendees will follow the lecture.

Please note the workshop participants have been automatically registered for this talk.

Programme

Monday 31 August

09.00 - 10.00	Registration and coffee, Ground Floor, 15 South College Street
10.00 - 10.15	Welcome
10.15 - 11.00	Viacheslav Nikulin (University of Liverpool) PDF of Presentation On the classification of arithmetic hyperbolic reflection groups
11.15 - 12.00	Rudolf Scharlau (Technische Universität Dortmund) PDF of Presentation On the explicit construction of arithmetic hyperbolic reflection groups
12.00 - 14.00	Lunch provided in the Chapterhouse, Ground Floor, 15 South College Street
14.00 - 14.45	Daniel Allcock (University of Texas at Austin) On the classification of reflective Lorentzian lattices
15.00 - 15.45	Tathagata Basak (University of Chicago) PDF of Presentation A complex hyperbolic reflection group and the bimonster
15.45 - 16.15	Tea/Coffee
16.15 - 17.00	Robert Griess (University of Michigan) Automorphism groups of VOAs

 

Tuesday 1 September

09.00 - 09.45	Gerald Höhn (Kansas State University) PDF of Presentation Vertex algebra constructions of generalized Kac-Moody algebras
09.45 - 10.15	Tea/Coffee
10.15 - 11.00	Scott Carnahan (Massachusetts Institute of Technology) PDF of Presentation Singular commutative rings in a braided category
11.15 - 12.00	Antun Milas (University at Albany, SUNY) W-algebra extensions of minimal models
12.00 - 14.00	Lunch break
14.00 - 14.45	Masahiko Miyamoto (University of Tsukuba) PDF of presentation Semirigidity and flatness of C_2-cofinite VOAs
15.00 - 15.45	Ching Hung Lam (Academia Sinica, Taipei) PDF of Presentation Virasoro frames and structure codes of the moonshine vertex operator algebra
16.00 - 16.45	John McKay (Concordia University) Absorbing sporadics
	Knowledge Transfer event
16.45 - 17.30	Tea/Coffee
17.30 - 18.30	Victor Kac (MIT) PDF of Presentation Quantization and chiralization
18.30 - 20.00	Wine Reception in the Chapterhouse, 15 South College Street, Edinburgh

 

Wednesday 2 September

09.00 - 09.45	Nils Scheithauer (Technische Universität Darmstadt) PDF of Presentation On the classification of automorphic products and generalized Kac-Moody algebras
10.00 - 10.45	Valery Gritsenko (Laboratoire Paul Painlevé) The baby functions of the Borcherds form Phi_12
10.45 - 11.15	Tea/Coffee
11.15 - 12.00	Shigeyuki Kondo (Nagoya University) PDF of Presentation The moduli spaces of Enriques, Coble surfaces and automorphic forms
12.15 - 13.00	Nils-Peter Skoruppa (Universität Siegen) Weil representations of finite quadratic modules and automorphic forms of critical weight
13.00 -	Free afternoon

 

Thursday 3 September

09.00 - 09.45	Urmie Ray (Université de Reims) On Cartan subalgebras
09.45 - 10.15	Tea/Coffee
10.15 - 11.00	Chris Cummins (Concordia University) Torsion free, genus one congruence subgroups, eta products and monstrous moonshine
11.15 - 12.00	John Duncan (Harvard University) Rademacher sums, monstrous moonshine and black holes
12.00 - 14.00	Lunch break
14.00 - 14.45	Miranda Cheng (Harvard University) PDF of Presentation Wall-crossing, black holes and generalized Kac-Moody algebras
15.00 - 15.45	Christoph Schweigert (Universität Hamburg) PDF of Presentation Mapping class group representations and conformal boundary conditions
15.45 - 16.15	Tea/Coffee
16.15 - 17.00	Shrawan Kumar (University of North Carolina) PDF of Presentation The Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras and its generalization to symmetric spaces
19.30 -	Workshop Dinner at Blonde Restaurant, 75 St Leonard's Street, Edinburgh

 

Friday 4 September

09.00 - 09.45	Peter Bantay (Eötvös Loránd University) PDF of Presentation Vector-valued modular forms
09.45 - 10.15	Tea/Coffee
10.15 - 11.00	Terry Gannon (University of Alberta) Mostly moonshine and vector-valued modular forms
11.15 - 12.00	Michael Tuite (National University of Ireland, Galway) PDF of Presentation Vertex operator superalgebras on a genus two Riemann surface
12.00 - 14.00	Lunch break
14.00 - 14.45	Hiroshi Yamauchi (Aichi University of Education) PDF of Presentation n-transposition groups and vertex operator algebras
15.00 - 15.45	Chongying Dong (University of California, Santa Cruz) PDF of Presentation A characterization of the vertex operator algebra L(1/2,0) otimes L(1/2,0)
15.45 - 16.15	Tea/Coffee
16.15 - 17.00	Geoffrey Mason (University of California, Santa Cruz) PDF of Presentation Vector-valued modular forms and logarithmic field theories

 

Presentations:

Presentation Details
Allcock, Daniel
On the classification of reflective Lorentzian lattices
View Abstract Hide Abstract
A Lorentzian lattice means an integer quadratic form of signature (n,1). These are important in hyperbolic geometry and in Kac-Moody algebras beyond the affine ones. Reflective means that the group generated by reflections has finite index in the full isometry group; another way to say this is a finite-volume Weyl chamber. There are only finitely many, as in the celebrated results of Nikulin and Esselmann. I will speak on recent progress on enumerating them.
Bantay, Peter
Vector-valued modular forms
View Abstract Hide Abstract
We present an overview, with explicit examples, of the approach to the theory of vector-valued modular forms, developed in joint work with T. Gannon.
Basak, Tathagata
A complex hyperbolic reflection group and the bimonster
View Abstract Hide Abstract
Let R be the reflection group of the complex leech lattice plus a hyperbolic cell. Let D be the incidence graph of the projective plane over the finite field with 3 elements. Let A(D) be the Artin group of D: generators of A(D) correspond to vertices of D. Two generators braid if there is an edge between them, otherwise they commute. It is surprising that both the bimonster and the reflection group R are quotients of A(D) when the generators are mapped to elements of order 2 and 3 respectively. A conjecture by Daniel Allcock seeks to explain this connection between R and the bimonster. We shall try to explain some of the evidences for the conjecture so far. We shall see that D behaves like the Coxeter-Dynkin diagram for the reflection group R. This imprecise analogy with Weyl groups actually makes our proofs work. There is a parallel story for a quaternionic hyperbolic reflection group where the analogies repeat.
Carnahan, Scott
Singular commutative rings in a braided category
View Abstract Hide Abstract
One may encounter objects, for example in orbifold CFT, that look like vertex algebras but do not quite obey locality. These objects are often treated using some notion of singular associative ring, possibly with a braiding endomorphism. We introduce a setting which in many cases allows one to promote them to commutative structures, and we describe an obstruction theory.
Cheng, Miranda C.N.
Wall-crossing, black holes and generalized Kac-Moody algebras
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I will explain how certain generalized Kac-Moody algebras play an important role in certain black hole systems in string theory, and how it is related to the wall-crossing phenomenon of certain geometric invariants of K3 surfaces. The talk will be based on arXiv.org/pdf/0806.2337, arXiv.org/pdf/0809.4258.
Cummins, Chris
Torsion free, genus one congruence subgroups, eta products and monstrous moonshine
View Abstract Hide Abstract
In 2001 Sebbar gave a classification of the torsion free, genus zero congruence subgroups. By a congruence subgroup, we mean here a discrete subgroup of PSL(2,R) which contains some principal congruence subgroup. Sebbar's proof made use of "Larcher subgroups". Using similar ideas we classify the torsion free, genus one congruence subgroups. A curious connection with eta products, monstrous moonshine groups and McKay's second E_8 correspondence is pointed out.
Dong, Chongying
A characterization of the vertex operator algebra L(1/2,0) otimes L(1/2,0)
View Abstract Hide Abstract
We will present our recent results on the characterization of the vertex operator algebra L(1/2,0) otimes L(1/2,0).
Duncan, John
Rademacher sums, monstrous moonshine and black holes
View Abstract Hide Abstract
In 1939 Rademacher derived a conditionally convergent series expression for Klein's j-invariant, and used this expression - the first Rademacher sum - to verify its modular invariance. We will explain how to associate Rademacher sums to arbitrary groups commensurable with the modular group, and we will show how these Rademacher sums can be used to reformulate the group theoretic characterization of the discrete groups of monstrous moonshine due to Conway-McKay-Sebbar. A physical interpretation of the Rademacher sums comes to light when we consider black holes in the context of three dimensional quantum gravity. We will explain how this leads to new connections between moonshine and physics, and important new directions for future research.
Gannon, Terry
Mostly moonshine and vector-valued modular forms
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In joint work with Peter Bantay, we've been developing a basic theory of vector-valued modular forms. In this talk I apply the theory to monstrous moonshine (mostly), but also sketch some of the other potential applications.
Griess, Robert
Automorphism groups of VOAs
View Abstract Hide Abstract
Lately, Ching Hung Lam and the author have classified configurations of EE_8 lattices in order to study Miyamoto involutions on lattice type VOAs and generalizations. We shall report on the current status.
Gritsenko, Valery
The baby functions of the Borcherds form Phi_12
View Abstract Hide Abstract
In this talk I present our joint result with B. Grandpierre about a new family of reflective modular forms related to the Borcherds modular form Phi_12. Many reflective modular forms were constructed by R. Borcherds in 2000. There are two other reasonable classes of reflective forms. The first class in signature (1,2) was obtained in the papers of Gritsenko-Nikulin. The second class of reflective modular forms of singular weight was described by N. Scheithauer. In our new class we do not fix dimension or weight but we fix the main source, i.e, we consider the Borcherds Phi_12 as an universal function. The new family contains many nice functions. For example its first member is the classical Igusa modular form of weight 35 with respect to the Siegel modular group of genus two. We give some applications of the new reflective modular forms to algebraic geometry and to Lorentzian Kac-Moody algebras
Höhn, Gerald
Vertex algebra constructions of generalized Kac-Moody algebras
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I will review the definitions of vertex algebras and generalized Kac-Moody algebras and explain how vertex algebras can be used to construct generalized Kac-Moody algebras using ideas from string theory. Applications to monstrous moonshine will also be discussed.
Hung Lam, Ching
Virasoro frames and structure codes of the moonshine vertex operator algebra
View Abstract Hide Abstract
In this talk, we will report the recent progress on the characterization of the Virasoro frames and structure codes of the moonshine vertex operator algebra.
Kac, Victor
Quantization and chiralization
View Abstract Hide Abstract
The four fundamental frameworks of physics theories are classical mechanics, quantum mechanics, classical field theory, and quantum field theory. The related fundamental algebraic structures are Poisson algebras, associative algebras, Poisson vertex algebras, and vertex algebras. In my talk I will explain the connections between these structures and discuss some of their applications to various fields of mathematics.
Kondo, Shigeyuki
The moduli spaces of Enriques, Coble surfaces and automorphic forms
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Several people (Allcock, Freitag, Gritsenko, Hulek, Nikulin, Sankaran, Salvati-Manni et al.) have studied the moduli of some algebraic varieties by using Borcherds theory on automorphic forms. In this talk, I shall introduce this application in case of Enriques surfaces. The moduli of marked Enriques surfaces is an arithmetic quotient of a 10-dimensional bounded symmetric domain of type IV. Let q be a 10-dimensional quadratic form over Z/2Z of even type. Then the orthogonal group O(q) naturally acts on the moduli. For each maximal totally singular subspace of q we associate an automorphic form. These automorphic forms give a projective model of the moduli.
Kumar, Shrawan
The Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras and its generalization to symmetric spaces
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Let g be a finite dimensional simple Lie algebra over the complex numbers. Consider the exterior algebra R on two copies of g. The diagonal adjoint action of g gives rise to a g-algebra structure on R compatible with the bigrading. There are three ’standard’ copies of the adjoint representation of g in the total degree 2 component R^2. Let J be the (bigraded) ideal of R generated by these three copies and define the bigraded g-algebra A := R/J. The Killing form gives rise to a g-invariant S in A^{1,1}. Motivated by supersymmetric gauge theory, Cachazo-Douglas-Seiberg-Witten made the following conjecture. Various cases of the conjecture were proved by Cachazo-Douglas-Seiberg-Witten, Witten, Etingof-Kac, Etingof. Conjecture (i) The subalgebra A^g of g-invariants in A is generated, as an algebra, by the element S. (ii) S^h = 0, where h is the dual Coxeter number of g. (iii) S^(h-1) is nonzero. The aim of this talk is to give a uniform proof of the above conjecture part (i). We will also give an analog of this result for symmetric spaces. The main ingredients in the proof are: Garland's result on the Lie algebra cohomology of u^ := g otimes tC[t]; Kostant's result on the `diagonal' cohomology of u^ and its connection with abelian ideals in a Borel subalgebra of g; and a certain deformation of the singular cohomology of the infinite Grassmannian introduced by Belkale-Kumar.
Mason, Geoffrey
Vector-valued modular forms and logarithmic field theories
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Until just a few years ago, studies of CFT were usually restricted to regular VOAs V of CFT-type, i.e. rational, C_2-cofinite theories with no negative weight spaces and nondegenerate vacuum. Two developments - the discovery of logarithmic CFTs and the chiral de Rham complex (Gorbounov-Malikov-Schechtman) - suggests a program to extend the older VOA theory to cover these more general situations. Thus we will consider C_2-cofinite VOAs with no negative weight spaces. In this context, some parts of the existing theory extend without difficulty, some extend with some effort, and other parts of it are quite false. The study of genus 1 correlation functions requires a general theory of logarithmic vector-valued modular forms, which is just getting under way. We will discuss results old and new that are relevant to this enterprise. This will lead to a modular-invariance conjecture for logarithmic CFTs.
McKay, John
Absorbing sporadics
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Are the "sporadic" finite simple groups essentially sporadic? Burnside (1911) named the five Mathieu groups such and today there are 26 of them. I sketch independent approaches with the intent of examining whether they may suggest a more encompassing construction than we have now. The 24-dim Witten spin manifold. dKP reduced. The cusps. Chern classes. Contact with physics?
Milas, Antun
W-algebra extensions of minimal models
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Virasoro minimal models have many applications in vertex algebra theory. For instance, the (3,4) minimal models are building blocks of the moonshine module. While the vertex algebra of the (p,q) minimal model is rational, surprisingly it should admit a quasi-rational extension, denoted by W_{p,q}. These vertex algebras are obtained by quantum reduction from the Felder's BRST complex and are believed to be related to certain quantum groups at root of unity via KL-correspondence. In my talk I will focus mostly on W_{p,2}-algebras and their modules. Several new results about these algebras will be presented (joint work with D. Adamovic). Relevant super extensions and modular properties of the (generalized) characters will be also discussed.
Miyamoto, Masahiko
Semirigidity and flatness of C_2-cofinite VOAs
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We introduce a concept of semirigidity and prove the flatness property of modules under semirigidity and C_2-cofiniteness and some other conditions. Then we describe a relation between semirigidity and modular invariance.
Nikulin, Viacheslav
On the classification of arithmetic hyperbolic reflection groups
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In 1980 and 1981 I showed that the numbers of reflective hyperbolic lattices and maximal hyperbolic reflection groups are finite for a fixed dimension n>9 of the hyperbolic space. In 1981, Vinberg showed that n<30. Only in 2005 similar finiteness was proved for n=2 by Long, Maclachlan and Reid, and for n=3 by Agol. In 2006, using these results and my methods of 1980, 1981, I showed finiteness in remaining dimensions 3<n<10. In 2007, I showed the effective finiteness in all dimensions. Thus, now, in principle, we can effectively classify the finite numbers of reflective hyperbolic lattices and maximal arithmetic hyperbolic reflection groups in all dimensions and over all ground fields (totally real) together. Reflective hyperbolic lattices are important for moonshine. In my lecture, I shall review these results.
Ray, Urmie
On Cartan subalgebras
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In this talk, I will try to develop a general theory of Cartan subalgebras in the context of Borcherds-Kac-Moody algebras and of locally finite Lie algebras, in particular locally reductive ones.
Scharlau, Rudolf
On the explicit construction of arithmetic hyperbolic reflection groups
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Arithmetic (in particular: discrete) groups acting on hyperbolic n-space and generated by reflections in hyperplanes can be obtained by (at least) two different approaches, well known from the work of Vinberg and Nikulin: Firstly, one can directly work with the Gram matrix of the fundamental domain, which is a Coxeter polytope. This allows to prescribe the combinatorial type of the fundamental domain. The difficult part then is to bound effectively the parameters (degree, discriminant) of the ground field. Secondly, one can start from integral quadratic forms, or lattices over totally real number fields, subject to the appropriate signature conditions. The task then is to identify the "reflective" lattices, for which the subgroup generated by all reflections stabilising the lattice has finite index in the full orthogonal group of the lattice. I shall present (partly unpublished) results and open questions for both approaches.
Scheithauer, Nils
On the classification of automorphic products and generalized Kac-Moody algebras
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The denominator identities of generalized Kac-Moody algebras are sometimes automorphic forms on orthogonal groups. Under certain conditions these Lie algebras can be classified. In this talk we give a review of these results.
Schweigert, Christoph
Mapping class group representations and conformal boundary conditions
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We show how traces of intertwiners of representations of the mapping class groups on spaces of conformal blocks allow to classify boundary conditions of two-dimensional rational conformal field theories.
Skoruppa, Nils-Peter
Weil representations of finite quadratic modules and automorphic forms of critical weight
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We shall show that Jacobi forms of critical weight can be identified with invariants of suitable Weil representations of finite quadratic modules. This fact can be used to study orthogonal and Siegel modular forms of critical weight. A recent result which was obtained along these lines is for example the nonexistence of Siegel modular forms of weight 1 and degree 2 on the groups Gamma_0(l).
Tuite, Michael
Vertex operator superalgebras on a genus two Riemann surface
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In this talk we describe the partition function (and some n-point correlation functions) for a fermionic vertex operator superalgebra on a genus two Riemann surface constructed from punctured tori. Comparing to a bosonized version we obtain a Jacobi triple like formula for the genus two Riemann theta function. This talk is based on collaborations with Geoff Mason and Sasha Zuevsky.
Yamauchi, Hiroshi
n-transposition groups and vertex operator algebras
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We give a general construction of n-transposition groups as automorphism groups of vertex operator algebras. Our argument can be applied to the Monster, the Babymonster and the largest Fischer group.

Participants

Name	Institution
Allcock, Daniel	University of Texas at Austin
Bantay, Peter	Eötvös Loránd University
Barraclough, Richard	Goldsmiths' College, University of London
Basak, Tathagata	University of Chicago
Carnahan, Scott	Massachusetts Institute of Technology
Cheng, Miranda C.N.	Harvard University
Cooney, Nicholas	University of Oxford
Cummins, Chris	Concordia University
Docherty, Pamela	University of Edinburgh
Dong, Chongying	University of California, Santa Cruz
Duncan, John	Harvard University
Gannon, Terry	University of Alberta
Griess, Robert	University of Michigan
Gritsenko, Valery	Universite Lille 1
Höhn, Gerald	Kansas State University
Hung Lam, Ching	Academia Sinica, Taipei
Hurley, Donny	National University of Ireland, Galway
Kac, Victor	MIT
Kohnen, Winfried	Universität Heidelberg
Kondo, Shigeyuki	Nagoya University
Konechny, Anatoly	Heriot-Watt University
Krauel, Matthew	University of California, Santa Cruz
Kühn, Ulf	Universität Hamburg
Kumar, Shrawan	University of North Carolina
Lentner, Simon	Ludwig-Maximillians-Universität München
Mason, Geoffrey	University of California, Santa Cruz
McKay, John	Concordia University
Milas, Antun	University at Albany, SUNY
Miyamoto, Masahiko	University of Tsukuba
Nikulin, Viacheslav	University of Liverpool
Norton, Simon	Cambridge University
Raulf, Nicole	Université des Sciences et Technologies de Lille
Ray, Urmie	Université de Reims
Scharlau, Rudolf	Technische Universität Dortmund
Scheithauer, Nils	Technische Universität Darmstadt
Schulze-Pillot, Rainer	Universität des Saarlandes
Schweigert, Christoph	Universität Hamburg
Skoruppa, Nils-Peter	Universität Siegen
Spencer, Christopher	University of Edinburgh
Tuite, Michael	National University of Ireland, Galway
Yamauchi, Hiroshi	Aichi University of Education
Zuevsky, Alexander	National University of Ireland, Galway