ICMS

May 17, 2010 - May 19, 2010

15 South College, Edinburgh

Organisers

Name	Institution
Marklof, Jens	University of Bristol

Principal Speakers:

Alex Eskin (University of Chicago)
Elon Lindenstrauss (Princeton University/Hebrew University of Jerusalem)
Philippe Michel (EPF Lausanne)
Mark Pollicott (Warwick University)
Sanju Velani (University of York)
Tamar Ziegler (Technion)

This workshop on the modern frontiers of Dynamical Systems and Number Theory is the first EU-Young and Mobile Workshop. It is a joint initiative of the European Research Councils, and will provide support for a limited number of young* European researchers. The deadline for applications has now closed.

* less than 10 years after PhD

Sponsors to date

Engineering and Physical Sciences Research Council
Deutsche Forschungsgemeinschaft
Nederlandse Organisatie voor Wetenschappelijk Onderzoek
Russian Academy of Sciences
Istituto Nazionale di alta Matematica
i-MATH

 

Arrangements

The workshop will comprise

• 60 minute invited principal lectures
• 30 minute contributed talks
• Workshop sessions on selected open problems
• Poster session
• Conference dinners
  

Participation is by invitation only.  This workshop is now full and applications are closed.

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 be held in two locations in Edinburgh - 15 South College Street and the Hugh Robson Building, George Square (within 5 - 10 minute walking distance of each other). All lectures will be held in The Robson Lecture Theatre on the ground floor of the Hugh Robson Building. Lunch, refreshments and parallel workshops will take place in 15 South College Street. To view details of the lecture theatre and a list of the audio-visual equipment available click here. Follow this link for a map showing the location of 15 South College Street and the Hugh Robson Building.  This map may also prove useful. 

Wireless Access
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. For those without laptops, there will also be a couple of computers available for you to check your emails.

Accommodation
ICMS will arrange en-suite rooms in hotels and university accommodation nearby for those who have requested this. Accommodation is typically about 10 to 30 minutes walk from 15 South College Street. Internet access is not guaranteed.

Meals and Refreshments
A light buffet lunch, morning and afternoon refreshments will be provided on each day of the workshop in the Chapterhouse on the ground floor of 15 South College Street.

Workshop dinners will take place on:

• Monday 17 May at 19.00 in Howie's Restaurant, 29 Waterloo Place, Edinburgh, EH1 3BQ

• Tuesday 18 May at 19.00 in The Magnum Restaurant, 1 Albany Street, Edinburgh, EH1 3PY

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

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 18.00 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 at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map. The second railway station is called Haymarket and is at the West End of the city centre.

Please note, it is your own responsibility to ensure that you have adequate travel insurance to cover your trip.

Conference Fee
There will be a conference fee of 200.00 GBP unless funded by EPSRC. This fee will cover the cost of your bed and breakfast accommodation up to 3 nights and the workshop dinner on Monday evening (see further details under "Financial Arrangements"). Lunch on Monday, Tuesday and Wednesday, refreshments each day and the workshop dinner on Tuesday night will be provided.

Registration
Registration will take place 08.45 - 09.35 on Monday 17 May 2010 in the Chapterhouse on the ground floor of 15 South College Street, Edinburgh.

Poster Session
There will be a poster session during the workshop in the Chapterhouse of 15 South College Street (see programme).  Anyone wishing to present a poster is invited to do so. We have up to 20 spaces available.  If you would like to present a POSTER, please let Johanna Kytöharju know. Ideally posters should be A1 size.

Financial Arrangements
Your individual financial arrangements will be laid out in your invitation and final email.

The workshop is sponsored by the following funding agencies:

Engineering and Physical Sciences Research Council
Deutsche Forschungsgemeinschaft
Nederlandse Organisatie voor Wetenschappelijk Onderzoek
Russian Academy of Sciences - Russian Foundation for Basic Research
Istituto Nazionale di alta Matematica
i-MATH

If you have any questions related to your funding, please contact Caterina Mora at EPSRC or the relevant agency above.

Unless otherwise stated in your invitation or further correspondence, participants will be asked to pay a conference fee of 200.00 GBP in advance. This fee will cover the cost of your bed and breakfast accommodation up to 3 nights and the workshop dinner on Monday evening. Additional nights will be charged at 55.00 GBP per night. Lunch on Monday, Tuesday and Wednesday, refreshments each day and the workshop dinner on Tuesday night will be provided. This can be paid by providing credit/debit card details. Please note we do not accept American Express or Discover Card. 

Please pay this fee in advance by using this credit or debit card payment form. This same form can be used to pay for extra nights' accommodation. 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 you are unable to provide credit/debit card details, please email Johanna Kytöharju.

Programme

The workshop will be held in two locations in Edinburgh - 15 South College Street and the Hugh Robson Building, George Square (within 5 - 10 minute walking distance of each other). All lectures will be held in The Robson Lecture Theatre on the ground floor of the Hugh Robson Building. Lunch, refreshments and parallel workshops will take place in 15 South College Street.

Registration will take place 08.45 - 09.35 on Monday 17 May 2010 in the Chapterhouse on the ground floor of 15 South College, Edinburgh. 

The workshop ends on Wednesday afternoon of 19 May 2010 at 16.10.

Please see further details below the programme.

Monday 17 May 2010

08.45-09.35	Registration & coffee, Ground Floor, 15 South College
09.45-10.00	Welcome & introduction, The Robson Lecture Theatre, Hugh Robson Building
10.00-11.00	Mark Pollicott (University of Warwick) Zeta functions in number theory, dynamical systems and geometry
11.15-12.15	Alex Eskin (University of Chicago) TBA
12.30-13.45	Lunch at 15 South College
14.00-14.30	Barak Weiss (Ben Gurion University) Minkowski's conjecture - the dynamical perspective view presentation
14.30-15.00	Igor Wigman (KTH) Fluctuations of the nodal length of random Laplace eigenfunctions
15.00-15.30	Charlene Kalle (University of Warwick) Beta-expansions, natural extensions and multiple tilings
15.40-16.20	Tea/Coffee & poster session at 15 South College
16.30-17.30	15 two-minute talks *
19.00	Workshop Dinner - Howies, 29 Waterloo Place, Edinburgh

Tuesday 18 May 2010

09.00-10.00	Elon Lindenstrauss (The Hebrew University of Jerusalem) Homogeneous and non-homogenous orbit closures for the action of the diagonal group
10.10-10.30	Tea/Coffee at 15 South College
10.40-11.40	Tamar Ziegler (Technion - Israel Institute of Technology) On and around the Green - Tao transference principle
11.50-12.20	Michael Björklund (The Hebrew University of Jerusalem) On the ergodic theory of sumsets
12.30-13.45	Lunch at 15 South College
14.00-14.30	Alejandro Luque (Universitat de Politècnica de Catalunya) Diophantine Arnold tongues of critical maps view presentation
14.30-15.00	Anke Pohl (ETH Zurich) Maas cusp forms for Hecke triangle groups, closed geodesics and invariant measures
15.00-15.30	Alexander Bufetov (Steklov Mathematical Institute & Rice University) Limit theorems for translation flows
15.40-16.20	Tea/Coffee & poster session at 15 South College
16.30-18.30	Parallel workshops at 15 South College
19.00	Workshop Dinner - The Magnum Restaurant, 1 Albany Street, Edinburgh

 

Wednesday 19 May 2010

09.00-10.00	Sanju Velani (University of York) Littlewood, Schmidt and multiplicatively badly approximable sets
10.10-10.30	Tea/Coffee at 15 South College
10.40-11.10	Tobias Mühlenbruch (Fern Universität in Hagen) Continued fractions and Hecke triangle groups view presentation
11.10-11.40	Luca Marchese (Université de Genève) Khinchin type conditions and Teichmuller flow
11.40-12.30	Tea/Coffee and discussion at 15 South College
12.30-13.45	Lunch at 15 South College

* PDFs of the two-minute talks
Christoph Aistleitner, Orestis Georgiou, Oleg German, Anish Ghosh, Alan Haynes, Jens Marklof, Anke Pohl, Anders Södergren, Arturo Vieiro, Xin Li

Principal Lectures
There will be six principal lectures, two each morning, to give an overview over current hot research areas at the interface of number theory and dynamical systems. Speakers will be asked to conclude their lectures with a discussion of future research directions and open problems in their field.

Research Workshops
There will be four parallel research workshops on Tuesday evening (see programme), led by the principal speakers, whose aim is to initiate interaction and possibly collaboration between participants on these open problems. The format of the workshops will be completely informal.

Contributed Talks
Thank you to all who have suggested topics for contributed talks. All the suggested talks are very interesting, but due to limited time we are unable to accommodate every talk. The main criterion for the final selection was to obtain a varied program that covers most areas related to the conference theme. The contributed talks will be 25 minutes and 5 minutes for discussion. Could all speakers please send their PDF or PowerPoint slides to Johanna Kytöharju by Tue 11 May, so we can upload them in advance.

Two Minute Short Talks
All participants who will not give a 25 minute talk are invited to present their research in a two minute short talk. The organiser does this regularly at Bristol and it is actually great fun. The time limit is strict therefore we recommend that you prepare at most three slides and have a rehearsal. There will be time for 1-2 short questions after each talk. Please send the PDF or PowerPoint slides to Johanna Kytöharju by Tue 11 May, so we can upload them in advance.

Poster Session
If you would like to present a poster, please let Johanna Kytöharju know. We have up to 20 spaces available. Ideally posters should be A1 size. The posters will be in the Chapter House or the Atrium of of 15 South College Street. Posters can be put up during lunch break on Monday.  This will be an ideal vehicle to present your research.

 

Presentations:

Presentation Details
Björklund, Michael
On the ergodic theory of sumsets
View Abstract Hide Abstract
Given two sets S and S' in a countable amenable group we define their sumset (or product set) as the set of all pairwise products of elements in S and S'. Typical questions in additive combinatorics concern the size of this new set. Using ergodic theory and Pontryagin theory, most notably some variants of the return time ergodic theorem, we prove direct and inverse sumset theorems if the sets are large (positive upper Banach density), generalizing the classical sumset theorem of Cauchy-Davenport to infinite sets.
Bufetov, Alexander
Limit theorems for translation flows
View Abstract Hide Abstract
Consider a compact oriented surface of genus at least two endowed with a holomorphic one-form. The real and the maginary parts of the one-form define two foliations on the surface, and each foliation defines an area-preserving translation flow. By a Theorem of H.Masur and W.Veech, for a generic surface these flows are ergodic. The talk will be devoted to the speed of convergence in the ergodic theorem for translation flows. The main result, which extends earlier work of A.Zorich and G.Forni, is a multiplicative asymptotic expansion for time averages of Lipschitz functions. The argument, close in spirit to that of G.Forni, proceeds by approximation of ergodic integrals by special holonomy-invariant Hoelder cocycles on trajectories of the flows. Generically, the dimension of the space of holonomy-invariant Hoelder cocycles is equal to the genus of the surface, and the ergodic integral of a Lipschitz function can be approximated by such a cocycle up to terms growing slower than any power of the time. The renormalization effectuated by the Teichmueller geodesic flow on the space of holonomy-invariant Hoelder cocycles allows one also to obtain limit theorems for translation flows: it is proved that along certain sequences of times ergodic integrals, normalized to have variance one, converge in distribution to a non-degenerate compactly supported measure. The argument uses a symbolic representation of translation flows as suspension flows over Vershik's automorphisms, a construction similar to one proposed by S.Ito.
Eskin, Alex
TBA
View Abstract Hide Abstract
TBA
Kalle, Charlene
Beta-expansions, natural extensions and multiple tilings
View Abstract Hide Abstract
We introduce a class of piecewise linear transformations that can be used to generate beta-expansions with arbitrary digits. Under some conditions on beta and the digit set, we can construct a natural extension for such a transformation, which allows us to get an invariant measure equivalent to Lebesgue for the original transformation. From the natural extension, we obtain a multiple tiling of a Euclidean space. For the classic greedy beta-transformation (Tx = beta x (mod 1)) the Pisot conjecture states that this construction gives a proper tiling for all Pisot numbers beta. We give an example of a double tiling, showing that this conjecture is no longer true in the more general setting.
Lindenstrauss, Elon
Homogeneous and non-homogenous orbit closures for the action of the diagonal group
View Abstract Hide Abstract
The pointwise ergodic theorem can be used in many cases to understand the behavior of a.e. point of a system. But this is of little help if one wants to know how the orbit of a specific point behaves. For unipotent flows, Ratner's Orbit Closure Theorem gives us a complete description of the orbit closures of individual orbits (and they are always homogeneous). For actions of higher rank diagonalizable groups, our understanding is not nearly complete, but in joint work with Uri Shapira we analyze orbit closures of some explicit classes of points, in some of which orbit closures are homogeneous, and in some not. I will also give Diophantine applications of this analysis.
Luque, Alejandro
Diophantine Arnold tongues of critical maps
View Abstract Hide Abstract
Given a two-parametric family of circle maps, the set of parameters for which the rotation number takes a prefixed value is called Arnold tongue. KAM theory shows that, for analytic families of analytic circle maps satisfying some mild non-degeneracy conditions, the Arnold tongue is an analytic curve when the rotation number satisfies certain arithmetic (Diophantine) properties. This result does not give any information if the family includes some subfamily for which the maps are analytic but have a critical value. The goal of this talk is to study numerically the differentiability at the boundary (critical value) of Arnold tongues corresponding to Diophantine rotation numbers and to present a renormalization group explanation of the phenomena encountered. In particular, we show that Arnold tongues are differentiable at the critical point and we also predict which is the order of differentiability by means of renormalization group arguments. In this talk we present a joint work with R. de la Llave and some others with J. Villanueva and T.M. Seara.
Marchese, Luca
Khinchin type conditions and Teichmuller flow
View Abstract Hide Abstract
We study a diophantine condition for interval exchange transformations and translation surfaces, which is inspired by the classical Khinchin theorem. We prove that we have the same dichotomy as in Khinchin result. Then we deduce a sharp estimation on how fast the typical Teichmuller geodesic wanders towards infinity in the moduli space of translation surfaces.
Michel, Philippe
A survey on automorphic periods
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In this lecture we discuss several classical problems related to the behaviour of Laplace eigenfunctions on a locally homogeneous space when the space is of "arithmetic" origin. In that case, the arithmetic provide us with a handful of additional "global" structures (Hecke operators, Langlands functoriality principles, L-functions) and techniques to analyse them (harmonic analysis on adelic spaces, analytic number theory...); the above mentionned problems are then closely related to the study of "automorphic periods". We will survey how these are defined and in simple examples, how they can be approached.
Mühlenbruch, Tobias
Continued fractions and Hecke triangle groups
View Abstract Hide Abstract
I present the Nakada continuous fraction expansion which is of the form lambda a_0 -1/ (lambda a_1 -1/ (lambda a_2 -1/...)), lambda = 2 cos(Pi/q), q =3,4,5,....The coefficients a_i are non-zero integers, satisfying certain conditions. We present some properties of these continued fractions and show that they can be used to code closed geodesics on the surface H/G_q, H being the upper complex plane and G_q the q-th Hecke triangle group. As an application we demonstrate the construction of a transfer operator. It turns out this transfer operator can be used to express the Selberg zeta function associated to H/G_q.
Pollicott, Mark
Zeta functions in number theory, dynamical systems and geometry
View Abstract Hide Abstract
In each of the three different areas of mathematics mentioned in the title, the term zeta function often refers to a particular type of complex function defined in terms on fundamental quantities, e.g., prime numbers, periodic orbits, closed geodesics or eigenvalues of the Laplacian. In each individual case, the underlying philosophy is that a greater understanding of the domain and values of the zeta function gives more general information about the original setting. Moreover, there is frequently a subtle interplay between these different zeta functions which can sometimes allow one to translate insights from one field into another. This lecture will set the scene by discussing examples of this circle of ideas ideas. We will then explain recent advances in understanding the dynamical zeta function, and its possible implications to other areas. Finally, we will discuss some naturally arising conjectures.
Velani, Sanju
Littlewood, Schmidt and multiplicatively badly approximable sets
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I will discuss the conjectures of Littlewood and Schmidt in the theory of simultaneous Diophantine approximation. This will include an attempt to address the question: What are the analogues of the classical set of badly approximable numbers within the multiplicative frameworks of Littlewood's conjecture and its mixed counterpart?
Weiss, Barak
Minkowski's conjecture - the dynamical perspective
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Minkowski (1904?): any z = (z_1, ..., z_n) in R^n and lattice Tsubset R^n of volume 1 satisfy inf_{t in T} prod_1^n |t_i-z_i| leq 2^{-n}. Minkowski's conjecture / hypothesis, although on shaky heuristic evidence, remains unrefuted. Research incorporating a natural "dynamical yoga" affords some more insight nowadays.
Wigman, Igor
Fluctuations of the nodal length of random Laplace eigenfunctions
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Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree $n$ having Laplace eigenvalue $E=n(n+1)$. We study the length distribution of the nodal lines of random spherical harmonics. It is known that the expected length is of order $n$. It is natural to conjecture that the variance should be of order $n$, due to the natural scaling. Our principal result is that, due to an unexpected cancelation, the variance of the nodal length of random spherical harmonics is of order $log{n}$. This behaviour is consistent to the one predicted by Berry for nodal lines on chaotic billiards (Random Wave Model).
Ziegler, Tamar
On and around the Green - Tao transference principle
View Abstract Hide Abstract
In their seminal paper in 2004, Green and Tao proved a transference principle, allowing them to transfer Szemeredis theorem on arithmetic progressions in dense subsets of the integers to dense subsets of the primes. We will discuss several variations of this transference principle, and mention a few related open questions.

Participants

Name	Institution
Aistleitner, Christoph	Montanuniversität Leoben
Badziahin, Dzmitry	The University of York
Beresnevich, Victor	The University of York
Björklund, Michael	The Hebrew University of Jerusalem
Bufetov, Alexander	Steklov Mathematical Institute & Rice University
Di Nasso, Mauro	University of Pisa
Egger, Sebastian	Universität Ulm
Eskin, Alex	University of Chicago
Georgiou, Orestis	University of Bristol
Ghosh, Anish	University of East Anglia
Gorodnik, Alex	University of Bristol
Haynes, Alan	University of York
Jordan, Thomas	University of Bristol
Kalle, Charlene	University of Warwick
Knight, Georgie	Queen Mary, University of London
Kurlberg, Per	KTH
Lee, Siu Lun Alan	University of Bristol
Li, Xin	University of Muenster
Lindenstrauss, Elon	The Hebrew University of Jerusalem
Luque, Alejandro	Universitat de Politècnica de Catalunya
Mahdi, Adam	Universitat Autònoma de Barcelona
Marchese, Luca	Université de Genève
Marklof, Jens	University of Bristol
Michel, Philippe	EPF Lausanne
Mühlenbruch, Tobias	Fern Universität in Hagen
Munday, Sara	University of St Andrews
Pohl, Anke	ETH Zurich
Pollicott, Mark	University of Warwick
Slipantschuk, Julia	ETH Zurich
Södergren, Anders	Uppsala University
Strömberg, Fredrik	TU Darmstadt
Taelman, Lenny	EPFL Lausanne & Mathematisch Instituut Leiden
Ulcigrai, Corinna	University of Bristol
Velani, Sanju	University of York
Vieiro, Arturo	Universitat de Barcelona
Weiss, Barak	Ben Gurion University
Weiss, Christian	Max Planck Institute for Mathematics
Wigman, Igor	KTH
Ziegler, Tamar	Technion - Israel Institute of Technology