ICMS

Aug 4, 2008 - Aug 8, 2008

William Robertson Building, University of Edinburgh

Organisers

Name	Institution
Diamantis, Nikolaos	University of Nottingham
Fesenko, Ivan	University of Nottingham
Goldfeld, Dorian	Columbia University
Hoffstein, Jeffrey	Brown 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)
Joseph Hundley (Southern Illinois University)
Ozlem Imamoglu (ETH Zurich)
David Kazhdan (Hebrew University, Jerusalem)
Alex Kontorovich (Brown University)
Kohji Matsumoto (Nagoya University)
Omer Offen (Humbold University Berlin)
S. J. Patterson (University of Gottingen)
Masatoshi Suzuki (Rikkyo University, Tokyo)

Recent achievements have led to a flurry of research in multiple Dirichlet series. This international workshop on multiple Dirichlet series will be an important milestone in putting into global perspective the achievements up to the present and formulating a program for the future development of the subject.

The most important objectives of the research workshop will be:

• to present mini courses on the foundations and recent research in the theory of multiple Dirichlet series.
• to formulate a general program for future development of the subject and present the main outstanding conjectures.
• to discuss and outline the most promising research directions for future breakthroughs in the subject.
• to stimulate active participation of the UK and EU number theorists in the workshop via further development of the subject.
• to forge connections between multiple Dirichlet Series and related fields.

Participation is by invitation only: those interested in attending should contact Nikos Diamantis (pmznd@maths.nott.ac.uk). There will be a contribution to local costs (including accommodation and subsistence) from awards to ICMS and the organisers. A Registration Fee of 30.00 GBP will be payable by all delegates.

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

Arrangements

Venue
The workshop will be held in the Lecture Theatre, 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 Hall, 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
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.

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

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.

Graduate students are strongly encouraged to apply for support to cover their local expenses. Please send a CV and one letter of recommendation to irene.moore@icms.org.uk before March 31, 2008.

 

Programme

Provisional 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 Zurich 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

9.00 - 10.00	Ben Brubaker MIT 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	Young-Ju Choie POSTECH TBA
11.45 - 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 Holy Cross Multiple Dirichlet series for Bn

 

Wednesday 06 August

9.00 - 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 Non-vanishing theorems
11.30 - 12.30	Kohji Matsumoto Nagoya University Multiple zeta-functions of root systems
	Free Afternoon

 

Thursday 07 August

9.00 - 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 University of 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
11.45 - 14.00	Lunch
14.00 - 15.00	Alina Bucur MIT Integral Moments III
15.00 - 15.30	Coffee/tea break
15.30 - 16.30	Joseph Hundley Southern Illinois University The descent method for GSpin groups

 

Friday 08 August

9.00 - 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
11.45 - 14.00	Lunch
14.00 - 15.00	TBA
15.00 - 15.30	Coffee/tea break
15.30 - 16.30	TBA

Presentations:

Presentation Details
Brubaker, Ben
WMDs and Crystals (mini-course)
View Abstract Hide 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 Hide 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 Hide 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 Hide 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 Hide Abstract
TBA
Diaconu, Adrian
Moments of automorphic L-functions (also Intergral Moments II)
View Abstract Hide Abstract
In this talk, I will present some new results about classical moments of automorphic L-functions. I will begin by recalling some history of the subject.
Frechette, Sharon
Multiple Dirichlet series for Bn
View Abstract Hide 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 Hide 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 Hide Abstract
All Rankin-Selberg integral representations for L-functions admit deformations into spectral relating second moments of the L-functions to triple integrals of eigenfunctions.
Goncharov, Alexander
Arithmetic analysis of the multiple zeta values
View Abstract Hide 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 Hide 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.
Hundley, Joseph
The Adjoint L function of SU(2,1)
View Abstract Hide 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 Hide 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
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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
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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.
Matsumoto, Kohji
Multiple zeta-functions of root systems
View Abstract Hide 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 Hide 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 Hide 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
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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
Aizenbud, Avraham	Weizmann Institute of Science
Alkauskas, Giedrius	Univeristy of Nottingham
Baruch, Ehud	Israel Institute of Technology
Beineke, Jennifer	Western New England College
Bian, Ce	University of Bristol
Boecherer, Siegfried	University of Mannheim
Booker, Andrew	University of Bristol
Brubaker, Ben	MIT
Bucur, Alina	Massachusetts Institute of Technology
Bump, Daniel	Stanford University
Chida, Masataka	Kyoto University
Chinta, Gautam	City College of New York
Choie, YoungJu	POSTECH
Cornelissen, Gunther	Universiteit Utrecht
Diaconu, Adrian	University of Minnesota
Diamantis, Nikolaos	University of Nottingham
Feigon, Brooke	University of Toronto
Ferapontov, Jenya	Loughborough University
Fesenko, Ivan	University of Nottingham
Frechette, Sharon	College of the Holy Cross
Friedberg, Solomon	Boston College
Fujii, Akio	Rikkyo University
Funke, Jens	Durham University
Garbin, Daniel	City University of New York
Garrett, Paul	University of Minnesota
Giovannopoulos, Fotios	University of Goettingen
Goldfeld, Dorian	Columbia University
Goncharov, Alexander	Brown University
Gourevitch, Dimitry	Weizmann Institute of Science
Gunnells, Paul	University of Massachusetts Amherst
Heim, Bernhard	Max Planck Institut für Mathematik
Hoffstein, Jeffrey	Brown University
Howell, Gareth	Cardiff University
Hughes, Christopher	University of York
Hundley, Joseph	Southern Illinois University
Huxley, Martin	University of Cardiff
Imamoglu, Özlem	ETH Zurich
Kazhdan, David	Hebrew University
Kontorovich, Alex	Brown University
Lagarias, Jeffrey C.	University of Michigan
Lapid, Erez	The Hebrew University of Jerusalem
Lee, Min	Columbia University
Li, Wen-Ching	Penn State University
Louvel, Benoit	Georg-August-Universitat
Luo, Wenzhi	The Ohio State University
Matsumoto, Kohji	Nagoya University
Mohler, Joel	Lehigh University
Moreno, Carlos	City University of New York
Morrow, Matthew	University of Nottingham
Munshi, Ritabrata	Rutgers University
O'Sullivan, Cormac	Bronx Community College
Offen, Omer	Humbold University Berlin
Pal, Ambrus	Imperial College London
Patterson, S. J.	Mathematisches Institut
Perelli, Alberto	Universita' di Genova
Petridis, Ioannis	University College London
Plunkett-Levin, Shaunna	Cardiff University
Pollington, Andrew	National Science Foundation
Raulf, Nicole	Université des Sciences et Technologies de Lille
Ricotta, Guillaume	Universite de Bordeaux 1
Risager, Morten	University of Aarhus
Sahi, Siddhartha	Rutgers University
Sczech, Robert	Rutgers University
Sengupta, Jyoti	Tata Institute of Fundamental Research
Stark, Harold	UCSD
Suzuki, Masatoshi	Rikkyo University
van Order, Jeanine	University of Cambridge
Whitehouse, David	Massachusetts Institute of Technology
Yamasaki, Yoshinori	Kyushu University
Zhang, Qiao	Texas Christian University