Multiple Dirichlet Series and Applications to Automorphic Forms

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Multiple Dirichlet Series and Applications to Automorphic Forms

 04 - 08 Aug 2008
 William Robertson Building, University of Edinburgh

Organiser

Name
Institution
Diamantis, NikolaosUniversity of Nottingham
Fesenko, IvanUniversity of Nottingham
Goldfeld, DorianColumbia University
Hoffstein, JeffreyBrown University

Speakers
Ben Brubaker (MIT)
Alina Bucur (MIT)
Daniel Bump (Stanford University)
Gautam Chinta (City College of New York)
Young-Ju Choie (Postech, Korea)
Adrian Diaconu (University of Minnesota)
Sharon Frechette (College of the Holy Cross)
Solomon Friedberg (Boston College)
Paul Garrett (University of Minnesota)
Alexander Goncharov (Brown University)
Paul Gunnells (University of Massachusetts, Amherst)
Bernhard Heim (Max Planck Institut für Mathematik)
Joseph Hundley (Southern Illinois University)
Ozlem Imamoglu (ETH Zurich)
David Kazhdan (Hebrew University, Jerusalem)
Alex Kontorovich (Brown University)
Wenzhi Luo (Ohio State University)
Kohji Matsumoto (Nagoya University)
Omer Offen (Humboldt University of Berlin)
S J Patterson (University of Gottingen)
Masatoshi Suzuki (Rikkyo University, Tokyo)

 

This workshop was partially supported by the EPSRC, the NSF and the LMS.

 

Short report

Zeta and L-functions are fundamental objects in Number Theory and their deeper understanding is crucial for most major programmes in the subject. The study of multiple Dirichlet series (MDS) began more recently than that of classical Dirichlet series but is rapidly developing and, in recent years, there has been a surge of important breakthroughs. The main achievements in the theory so far include the construction of Weyl group multiple Dirichlet series, applications of MDS to the problem of non-vanishing of twists of L-functions, to convexity breaking, to moments of zeta and L-functions and the connections to multiple zeta function values. The workshop brought together the main workers in the field of MDS and other leading international experts in allied subjects, as well as younger researchers and post-graduate students working in related areas. The workshop centred around excellent talks reporting on the most recent developments in MDS and in areas that have the potential of fruitful interaction with the subject. We expect that the questions during the talks and the one-to-one discussions during the meeting will stimulate further progress in the major problems and will point to further research directions in the broader area of MDS.

Download the PDF file of the full report here

Arrangements

Venue
The workshop will be held in the Lecture Theatre, G08, on the ground floor of the William Robertson Building, University of Edinburgh, George Square. The University of Edinburgh provides this useful interactive map of the area with the building marked in a yellow circle.


Registration
There will be a combined Registration and Welcome Buffet from 19.00 to 20.30 on Sunday 3 August, in the St Trinnean's Room, St Leonard's Hall, Pollock Halls. Food will be available throughout that period and you may register at any time between 19.00 and 20.30.

Those who cannot register on Sunday evening may do so at the workshop venue before the talks start on Monday morning.

Travel
Information about travel to the UK and Edinburgh is available here.

If you have requested accommodation (see below), travel information and directions to Pollock Halls can be found on the website for Edinburgh First, who co-ordinate the accommodation bookings, http://www.edinburghfirst.com/edinburghfirst/travel.asp . We recommend that you take a taxi to the Halls. A taxi should cost between 6.00 and 7.00 GBP from the main railway station (Waverley Station) and between 17.00 and 21.00 GBP from Edinburgh airport. Alternatively there are frequent buses from the airport to Waverley Station, from where you can get a taxi to Pollock Halls. For timetable and details see http://www.flybybus.com/index.php.

The second of these two maps of Pollock Halls shows the location of both Pollock Halls and William Robertson Building where the lectures will take place. It is approximately a 15 minute walk between the two sites.

Lothian buses charge £1.10 for a single, £2.50 for a day ticket. Please note that the exact fare is required and no change is given. There are several Lothian Buses from the centre of town to Pollock Halls: Service No. 30, 14, 33 or the X48 Park and Ride Bus. If you enter the Service No on the Lothian Buses webpage, you will be able to view the timetable for each service bus. Leave the bus at The Commonwealth Pool on Dalkeith Road. Pollock Halls is situated just behind The Commonwealth Pool.

The location of Pollock Halls (18 Holyrood Park Road) can be viewed by following this link to multimap.

Accommodation
ICMS has arranged rooms in university accommodation in Pollock Halls for participants who require it. 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. August is very busy in Edinburgh and early booking is recommended.

Pollock Halls of Residence
18 Holyrood Park Road
Edinburgh, EH16 5AY
Reception Contact Numbers
+44 (0)131 667 1971 (Tel)
+44 (0)131 668 3217 (Fax)

On arrival at Pollock Halls, please go to the Reception Building on your left, where you will pick up your key and some further information. You can get into your room after 14.00. If you arrive earlier you can leave your luggage at Reception. Pollock Halls has 24 hour reception.

Meals and Refreshments
There will be a Registration buffet in the St Trinneans Room in St Leonards, Pollock Halls on Sunday evening 3 August and a Workshop dinner on the evening of Thursday 7 August at the Playfair Library, Old College. Morning and afternoon refreshments will be provided at the workshop venue as well as lunch on Monday 4 August.

Computer access
At Registration you will be issued with a username, password and further information which will enable you to access either the wireless network or a public PC nearby. William Robertson Building is within a wireless zone as are parts of Pollock Halls.

Financial support available
The buffet supper on Sunday, a light lunch Monday, and the workshop dinner on Thursday evening will be provided free of charge to participants.

We hope that some participants may be able to cover their own travel expenses so we can offer some degree of financial assistance to more junior participants. Further information will be provided in individual invitation letters.

If ICMS has agreed to reimburse some of your travel costs, this will be stated in the 'Financial Arrangements' section of the 'Final Information' email. All reimbursement occurs after the workshop. You will be given a claim form at Registration and asked for your bank details to enable payment directly into your account. Please note that receipts are required for all expenses claimed. It would be useful if you can bring your bank details to the workshop.

US delegates can claim directly through an NSF grant in the USA. You will be issued with an NSF claim form at the workshop Registration. To claim through NSF you MUST use a US carrier. Please note that original receipts and boarding passes are required for all expenses claimed. All receipts and completed forms should be submitted within 10 business days after completion of travel. If ICMS has agreed to reimburse some of your travel costs through the NSF grant, this will be stated in the 'Financial Arrangements' section of the 'Final Information' email.


Programme

Monday 04 August

09.00 - 09.45

Registration

09.45 - 10.00

Introduction and welcome

10.00 - 11.00

Solomon Friedberg Boston College
WMDs and Crystals I

11.00 - 11.30

Coffee/tea break

11.30 - 12.30

Özlem Imamoglu ETH Zürich
On the cycle integrals of the j function

12.30 - 14.00

Lunch

14.00 - 15.00

Paul Garrett University of Minnesota
Integral Moments I

15.00 - 15.30

Coffee/tea break

15.30 - 16.30

Alexander Goncharov Brown University
Arithmetic analysis of the multiple zeta values

 

Tuesday 05 August

09.00 - 10.00

Ben Brubaker Massachusetts Institute of Technology
WMDs and Crystals II

10.00 - 10.15

Coffee/tea break

10.15 - 11.15

David Kazhdan Hebrew University of Jerusalem
The 2-dimensional Langlands correspondence

11.30 - 12.30

YoungJu Choie POSTECH
Quasi modular forms

12.30 - 14.00

Lunch

14.00 - 15.00

Adrian Diaconu University of Minnesota
Integral Moments II

15.00 - 15.30

Coffee/tea break

15.30 - 16.30

Sharon Frechette College of the Holy Cross
Multiple Dirichlet series for B_n

 

Wednesday 06 August

09.00 - 10.00

Gautam Chinta City College of New York
WMDs and Crystals III

10.00 - 10.15

Coffee/tea break

10.15 - 11.15

Alex Kontorovich Brown University
First nonvanishing theorems

11.30 - 12.30

Kohji Matsumoto Nagoya University
Multiple zeta-functions of root systems

 

Free Afternoon

 

Thursday 07 August

09.00 - 10.00

Paul Gunnells University of Massachusetts Amherst
WMDs and Crystals IV

10.00 - 10.15

Coffee/tea break

10.15 - 11.15

S J Patterson Georg-August-Universität Göttingen
Metaplectic multiple Dirichlet series - arithmetic aspects

11.30 - 12.30

Paul Garrett University of Minnesota
Automorphic spectral identities involving second moments of L-functions

12.30 - 14.00

Lunch

14.00 - 15.00

Alina Bucur Massachusetts Institute of Technology
Integral Moments III

15.00 - 15.30

Coffee/tea break

15.30 - 16.30

Wenzhi Luo Ohio State University
The closed geodesics on the modular surface and the Maass-Shintani lift

 

Friday 08 August

09.00 - 10.00

Daniel Bump Stanford University
WMDs and Crystals V

10.00 - 10.15

Coffee/tea break

10.15 - 11.15

Omer Offen Humbold University Berlin
Unitary periods

11.30 - 12.30

Masatoshi Suzuki Rikkyo University
Mean-periodicity and zeta functions

12.30 - 14.00

Lunch

14.00 - 15.00

Bernhard Heim Max Planck Institut für Mathematik
A trace formula of special values of automorphic L-functions

15.00 - 15.30

Coffee/tea break

15.30 - 16.30

Joseph Hundley Southern Illinois University
The adjoint L function of SU(2,1)


Presentations

Brubaker, Ben
WMDs and Crystals (mini-course)
View Abstract
Weyl group multiple Dirichlet series are Dirichlet series in several complex variables whose coefficients are constructed from n-th order Gauss sums, with groups of functional equations isomorphic to Weyl groups. These are usually finite reflection groups but infinite Kac-Moody Weyl groups are also potentially within the scope of the subject. The coefficients of an n-th order series have a multiplicativity that is twisted by n-th order power residue symbols, so that they are usually not Euler products, but their description mainly requires specification of their p-parts. These p-parts are fascinating combinatorial objects that resemble characters of representations of Lie groups, but with each weight multiplied by a product of n-th order Gauss sums. They can be described as metaplectic Whittaker coefficients, as sums over the Weyl group resembling the Weyl character formula, or as sums over crystal bases. Five lectures on this topic will be given by Friedberg, Brubaker, Chinta, Gunnells and Bump describing these different approaches and their interrelations.
Bucur, Alina
Integral Moments III
View Abstract
This is the third part of the mini-course on Integral Moments. I will give an outline of the proof of the analytic continuation of the fourth moment MDS over Fq(T). Then I will explain how to get the desired asymptotic formula once we have the analytic continuation. I will try to show the method of mutliple Dirichlet series associated to Coxeter groups at work in this concrete example. I will explain the classical approach and the new ideas and techniques we used to overcome the obstruction posed by the infinite group of functional equations.
Bump, Daniel
WMDs and Crystals (mini-course)
View Abstract
Weyl group multiple Dirichlet series are Dirichlet series in several complex variables whose coefficients are constructed from n-th order Gauss sums, with groups of functional equations isomorphic to Weyl groups. These are usually finite reflection groups but infinite Kac-Moody Weyl groups are also potentially within the scope of the subject. The coefficients of an n-th order series have a multiplicativity that is twisted by n-th order power residue symbols, so that they are usually not Euler products, but their description mainly requires specification of their p-parts. These p-parts are fascinating combinatorial objects that resemble characters of representations of Lie groups, but with each weight multiplied by a product of n-th order Gauss sums. They can be described as metaplectic Whittaker coefficients, as sums over the Weyl group resembling the Weyl character formula, or as sums over crystal bases. Five lectures on this topic will be given by Friedberg, Brubaker, Chinta, Gunnells and Bump describing these different approaches and their interrelations.
Chinta, Gautam
WMDs and Crystals (mini-course)
View Abstract
Weyl group multiple Dirichlet series are Dirichlet series in several complex variables whose coefficients are constructed from n-th order Gauss sums, with groups of functional equations isomorphic to Weyl groups. These are usually finite reflection groups but infinite Kac-Moody Weyl groups are also potentially within the scope of the subject. The coefficients of an n-th order series have a multiplicativity that is twisted by n-th order power residue symbols, so that they are usually not Euler products, but their description mainly requires specification of their p-parts. These p-parts are fascinating combinatorial objects that resemble characters of representations of Lie groups, but with each weight multiplied by a product of n-th order Gauss sums. They can be described as metaplectic Whittaker coefficients, as sums over the Weyl group resembling the Weyl character formula, or as sums over crystal bases. Five lectures on this topic will be given by Friedberg, Brubaker, Chinta, Gunnells and Bump describing these different approaches and their interrelations.
Choie, YoungJu
Quasi modular forms
View Abstract
Quasimodular forms generalize classical modular forms and were introduced by Kaneko and Zagier . Since then they have been studied in connection with problems not only in number theory but also in applied mathematics. Unlike modular forms, derivatives of quasimodular forms are also quasimodular forms. We describe quasimodular forms for a discrete subgroup $\G$ of $SL(2, \bR)$ and introduce quasimodular polynomials, which can be identified with quasimodular forms. We consider connections between Jacobi-like forms and quasimodular polynomials and study Hecke operators on quasimodular polynomials that are compatible with those on modular and Jacobi-like forms.
Diaconu, Adrian
Intergral Moments II
View Abstract
The first part of this talk will complement "Integral moments I" providing some details on how the subconvexity bound in "t"-aspect of GL_2 L-functions over number fields is obtained.
Then, in the second part, new developments in understanding the analytic continuation of certain multiple Dirichlet series associated to moments of quadratic Dirichlet L-functions over function fields will be presented.
Frechette, Sharon
Multiple Dirichlet series for Bn
View Abstract
We construct Weyl group multiple Dirichlet series associated to the root system B_n using Proctor patterns, close relatives of the Gelfand-Tsetlin patterns. We prove functional equations for these MDS when n=1, via the Casselman-Shalika formula. We also prove that our description matches the stable case, as described by Brubaker, Bump and Friedberg. This is joint work with Jennifer Beineke and Ben Brubaker.
Friedberg, Solomon
WMDs and Crystals (mini-course)
View Abstract
Weyl group multiple Dirichlet series are Dirichlet series in several complex variables whose coefficients are constructed from n-th order Gauss sums, with groups of functional equations isomorphic to Weyl groups. These are usually finite reflection groups but infinite Kac-Moody Weyl groups are also potentially within the scope of the subject. The coefficients of an n-th order series have a multiplicativity that is twisted by n-th order power residue symbols, so that they are usually not Euler products, but their description mainly requires specification of their p-parts. These p-parts are fascinating combinatorial objects that resemble characters of representations of Lie groups, but with each weight multiplied by a product of n-th order Gauss sums. They can be described as metaplectic Whittaker coefficients, as sums over the Weyl group resembling the Weyl character formula, or as sums over crystal bases. Five lectures on this topic will be given by Friedberg, Brubaker, Chinta, Gunnells and Bump describing these different approaches and their interrelations.
Garrett, Paul
Automorphic spectral identities involving second moments of L-functions (also Intergral Moments I)
View Abstract
Abstarct for Automorphic spectral identities involving second moments of L-functions
All Rankin-Selberg integral representations for L-functions admit deformations into spectral relating second moments of the L-functions to triple integrals of eigenfunctions.

Abstract for Integral moments I
Integral moments are weighted averages of powers of L-functions over families of automorphic forms. These arise in automorphic spectral decompositions. One application of asymptotics with errors terms for integral moments is to subconvex bounds, and this has been completely carried out for GL(2) in the t-aspect, over number fields. The general context and background will be discussed, and extensions to GL(n). A variety of further possibilities will be sketched.
Goncharov, Alexander
Arithmetic analysis of the multiple zeta values
View Abstract
Arithmetic analysis is supposed to be a brunch of mathematics investigating periods (=integrals of rational differential forms) by using arithmetic algebraic geometry. We will do arithmetic analysis of MZV's and, using this, relate MZV's to the geometry of modular varieties.
Gunnells, Paul
WMDs and Crystals (mini-course)
View Abstract
Weyl group multiple Dirichlet series are Dirichlet series in several complex variables whose coefficients are constructed from n-th order Gauss sums, with groups of functional equations isomorphic to Weyl groups. These are usually finite reflection groups but infinite Kac-Moody Weyl groups are also potentially within the scope of the subject. The coefficients of an n-th order series have a multiplicativity that is twisted by n-th order power residue symbols, so that they are usually not Euler products, but their description mainly requires specification of their p-parts. These p-parts are fascinating combinatorial objects that resemble characters of representations of Lie groups, but with each weight multiplied by a product of n-th order Gauss sums. They can be described as metaplectic Whittaker coefficients, as sums over the Weyl group resembling the Weyl character formula, or as sums over crystal bases. Five lectures on this topic will be given by Friedberg, Brubaker, Chinta, Gunnells and Bump describing these different approaches and their interrelations.
Heim, Bernhard
A trace formula of special values of automorphic L-functions
View Abstract
Deligne introduced the concept of critical values of automorphic L-functions.The arithmetic properties of L-functions and specical values play a fundamental role in modern number theory. In this talk we present a trace formula which relates special values of the Hecke, Rankin, and the central value of the Garrett triple L-function attached to primitive newforms.
This type of trace formula is new and involves special values in the convergent and non-convergent domain of the underlying L-functions. We indicate several applications.
Hundley, Joseph
The Adjoint L function of SU(2,1)
View Abstract
We describe a Rankin-Selberg integral construction for a certain Langlands L function attached to an automorphic form on "SU(2,1)"-- the quasisplit form of SL(3). In the split case, the construction was given by Ginzburg, and corresponds to the adjoint representation of the L group. Time permitting, we also discuss the possibility of extending the construction of Bump-Ginzburg for the adjoint L-function of SL(4) to the quasisplit case.
Imamoglu, Özlem
On the cycle integrals of the j function
View Abstract
Borcherds and Zagier established some connections between the traces of singular moduli, infinite products and weakly holomorphic modular forms of half-integral weight In this talk I will report on joint work with W. Duke and A. Toth where we consider some parallel problems for cycles integrals of $j$, which are real quadratic analogues of singular moduli.
Kazhdan, David
The 2-dimensional Langlands correspondence
View Abstract
Hecke operators for 2-dimensional fields. If G is a reductive group over a 1-dimensional field F=mF _q((t)) one can define a convolution on the space of mcH of complex-valued function on Kbackslash G/K where Ksubset G is the special maximal subgroup. The corresponding {it spherical Hecke algebra} is isomorphic to the ring of algebraic representations of the dual group ^LG. I'll discuss a definition and the description of the spherical Hecke algebra for 2-dimensional fields F.
Kontorovich, Alex
First nonvanishing theorems
View Abstract
We will discuss recent joint work with Jeff Hoffstein on the first non-vanishing quadratic twist of an automorphic L-series on GL(r), r=1,2,3.
Luo, Wenzhi
The closed geodesics on the modular surface and the Maass-Shintani lift
View Abstract
It is well-known that the closed geodesics on the modular surface, when collected according to the discriminants, are equidistributed with respect to the invariant hyperbolic measure, by the works of Duke and Iwaniec. We study and evaluate asymptotically the variance of this distribution on the unit tangent bundle, and show it is equal to the classic variance of the geodesic flow as introduced and studied by Ratner, but twisted by certain central L-value.

Our approach is via Weil representation and the theta correspondence, generalizing the works of Maass, Shintani and Katok-Sarnak. This is a joint work with Rudnick and Sarnak.
Matsumoto, Kohji
Multiple zeta-functions of root systems
View Abstract
Witten zeta-functions associated with semisimple Lie algebras were introduced by E. Witten in connection with some problem in quantum gauge theory. The original Witten zeta-functions are of one variable, but it is convenient to introduce the multi-variable version. From this viewpoint, Witten zeta-functions are special cases of more general multiple zeta-functions of several variables associated with root sets.
The family of zeta-functions of root sets has a recursive relation, which can be described explicitly by Mellin-Barnes integrals, and is corresponding to the cutting procedure of edges of Dynkin diagrams. By using this recursive relation, it is possible to obtain analytic properties (analytic continuation, location of poles, etc.) of zeta-functions of root sets. Moreover, we can find various functional relations among those zeta-functions, which can be explained in terms of symmetricity of associated Weyl groups. From this viewpoint we can prove a generalization of Witten's volume formulas.
Offen, Omer
Unitary Periods
View Abstract
I will discuss the study of period integrals over a unitary group, of automorphic forms on GL(n) over a quadratic extension. In particular, I will explain Jacquet's factorization of the period of a cusp form, a stabilization process in order to express the period of an Eisenstein series as a finite sum of factorisable linear forms and some local applications.
Patterson, S J
Metaplectic multiple Dirichlet series - arithmetic aspects
View Abstract
The simplest Dirichlet series are those first defined by Kubota where the coefficients are Gauss sums. One of the applications of these series is to study the distribution of Gauss sums. The multiple Dirichlet series have coefficients which are products of Gauss sums. The recent developments in this area mean that one can determine their analytic properties without recourse to the device of Eisenstein series. This approach is very natural from an arithmetic point of view. In the talk I shall explore the arithmetic aspects, especially what one could call the "Hecke theory", of these series. I shall also discuss what one might expect to learn from the residues, themselves interesting arithmetic functions.
Suzuki, Masatoshi
Mean-periodicity and zeta functions
View Abstract
In this talk we propose new bridges between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and the functional equation of the Hasse zeta function of arithmetic scheme with its expected analytic shape is shown to imply the mean-periodicity of a certain explicitly defined function associated to the zeta function. Conversely, the mean-periodicity of this function implies the meromorphic continuation and functional equation of the zeta function. This opens a new road to the study of zeta functions via the theory of mean-periodic functions which is a part of modern harmonic analysis. The background of this new bridges is the higher dimensional adelic analysis, which aims to study the Hasse zeta functions using integral representations on higher adelic spaces, harmonic analysis and analytic duality on higher adelic spaces.

Participants

Name
Institution
Avraham, AizenbudWeizmann Institute of Science
Giedrius, AlkauskasUniversity of Nottingham
Ehud, BaruchIsrael Institute of Technology
Jennifer, BeinekeWestern New England College
Ce, BianUniversity of Bristol
Siegfried, BoechererUniversity of Mannheim
Andrew, BookerUniversity of Bristol
Ben, BrubakerMassachusetts Institute of Technology
Alina, BucurMassachusetts Institute of Technology
Daniel, BumpStanford University
Masataka, ChidaKyoto University
Gautam, ChintaCity College of New York
YoungJu, ChoiePOSTECH
Gunther, CornelissenUniversiteit Utrecht
Adrian, DiaconuUniversity of Minnesota
Nikolaos, DiamantisUniversity of Nottingham
Brooke, FeigonUniversity of Toronto
Jenya, FerapontovLoughborough University
Ivan, FesenkoUniversity of Nottingham
Sharon, FrechetteCollege of the Holy Cross
Solomon, FriedbergBoston College
Akio, FujiiRikkyo University
Jens, FunkeDurham University
Daniel, GarbinCity University of New York
Paul, GarrettUniversity of Minnesota
Fotios, GiovannopoulosGeorg-August-Universität Göttingen
Dorian, GoldfeldColumbia University
Alexander, GoncharovBrown University
Dimitry, GourevitchWeizmann Institute of Science
Paul, GunnellsUniversity of Massachusetts Amherst
Bernhard, HeimMax Planck Institut für Mathematik
Jeffrey, HoffsteinBrown University
Gareth, HowellCardiff University
Christopher, HughesUniversity of York
Joseph, HundleySouthern Illinois University
Martin, HuxleyCardiff University
Özlem, ImamogluETH Zürich
David, KazhdanHebrew University of Jerusalem
Alex, KontorovichBrown University
Jeffrey C., LagariasUniversity of Michigan
Erez, LapidHebrew University of Jerusalem
Min, LeeColumbia University
Wen-Ching, LiPenn State University
Benoit, LouvelGeorg-August-Universität Göttingen
Wenzhi, LuoOhio State University
Kohji, MatsumotoNagoya University
Joel, MohlerLehigh University
Carlos, MorenoCity University of New York
Matthew, MorrowUniversity of Nottingham
Cormac, O'SullivanBronx Community College
Omer, OffenHumboldt University of Berlin
Ambrus, PálImperial College London
S J, PattersonGeorg-August-Universität Göttingen
Alberto, PerelliUniversita' di Genova
Ioannis, PetridisUniversity College London
Shaunna, Plunkett-LevinCardiff University
Andrew, PollingtonNational Science Foundation
Nicole, RaulfUniversité des Sciences et Technologies de Lille
Guillaume, RicottaUniversité de Bordeaux 1
Morten, RisagerUniversity of Aarhus
Siddhartha, SahiRutgers University
Robert, SczechRutgers University
Jyoti, SenguptaTata Institute of Fundamental Research
Harold, StarkUniversity of California, San Diego
Masatoshi, SuzukiRikkyo University
Jeanine, van OrderUniversity of Cambridge
David, WhitehouseMassachusetts Institute of Technology
Yoshinori, YamasakiKyushu University
Qiao, ZhangTexas Christian University