Interactions between dynamics of group actions and number theory
GAN
9 June 2014 to 4 July 2014
Scientific Advisory Committee: Ben Green (Cambridge), Dmitry Kleinbock (Brandeis), Philippe Michel (Lausanne)
Programme Theme
In the last decade there have been several important breakthroughs in Number Theory and Diophantine Geometry, where progress on long-standing open problems has been achieved by utilising ideas originated in the theory of dynamical systems on homogeneous spaces. Dynamical systems techniques are applicable to a wide range of number-theoretic objects that have many symmetries. In particular:
- various question in Diophantine approximation have been studied using recurrence properties of flows on the space of unimodular lattices
- diagonal flows have played an important role in recent advances on quantum chaos and in the proof of the quantum unique ergodicity conjecture for arithmetic surfaces
- flows on homogeneous spaces of nilpotent groups have been used to produce new estimates on exponential sums and to study prime solutions of systems of linear equations
- the distribution of periodic orbits is connected to behaviour of period integrals of automorphic forms and to the problem of establishing subconvexity bounds for L-functions
The aim of this programme is to bring together researchers working in Number Theory and Homogeneous Dynamics to discuss the recent developments and open problems that lie at the crossroads of these fields and to encourage more interaction among people working in these diverse areas.
Images produced by Alex Barnett and Holger Then
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Preprints
| Title | Year | Programme |
|---|---|---|
Best possible rates of distribution of dense lattice orbits in homogeneous spacesAuthors: Anish Ghosh, Alex Gorodnik, Amos Nevo |
2013 | GAN |
Multiparameter spherical averages of Siegel transforms and applicationsAuthors: Jayadev Athreya, Anish Ghosh, J Tseng |
2013 | GAN |
Simultaneous dense and nondense orbits and the space of latticesAuthors: R Shi, J Tseng |
2013 | GAN |
Workshops
Groups, numbers and dynamics
30 June 2014 to 4 July 2014
Seminars
| Monday 9th June 2014 | |||
|---|---|---|---|
| 14:30 to 15:30 | Room 2 | ||
| Tuesday 10th June 2014 | |||
|---|---|---|---|
| 14:30 to 15:30 |
On continued fraction expansion of potential counterexamples to mixed Littlewood conjecture. |
Room 2 |  |
| Wednesday 11th June 2014 | |||
|---|---|---|---|
| 11:30 to 12:30 | Room 2 | |
|
| Thursday 12th June 2014 | |||
|---|---|---|---|
| 11:30 to 12:30 |
Distribution of values of linear maps on quadratic surfaces. |
Room 2 | |
| 14:30 to 15:30 | Room 2 | |
|
| Friday 13th June 2014 | |||
|---|---|---|---|
| 11:30 to 12:30 |
On continued fraction expansion of potential counterexamples II |
Room 2 | |
| 14:30 to 15:30 | Room 2 | |
|
| Monday 16th June 2014 | |||
|---|---|---|---|
| 16:00 to 17:00 |
Spectral gap in simple Lie groups : Clay Mathematics Institute Senior Scholar Lecture. |
Room 1 | |
| Tuesday 17th June 2014 | |||
|---|---|---|---|
| 11:30 to 12:30 | Room 2 | |
|
| 14:30 to 15:30 | Room 2 | |
|
| Wednesday 18th June 2014 | |||
|---|---|---|---|
| 11:30 to 12:30 | Room 2 | |
|
| 14:30 to 15:30 |
Effective Ratner Theorem for ASL(2, R) and the gaps of the sequence \sqrt n modulo 1 |
Room 1 | |
| Thursday 19th June 2014 | |||
|---|---|---|---|
| 14:30 to 15:30 |
Fourier analysis of multiplicative functions and a combinatorial application. |
Room 2 | |
| Friday 20th June 2014 | |||
|---|---|---|---|
| 10:30 to 11:30 | Discussion Room | ||
| 11:30 to 12:30 | Room 1 | ||
| 14:30 to 15:30 |
Asymptotical behavior of piecewise contractions of the interval. |
Room 2 | |
| Monday 23rd June 2014 | |||
|---|---|---|---|
| 11:00 to 12:00 | Room 2 | |
|
| 13:30 to 14:30 | Room 2 | |
|
| Tuesday 24th June 2014 | |||
|---|---|---|---|
| 14:30 to 15:30 |
Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces. |
Room 2 | |
| 15:45 to 16:45 |
Stationary Measures and Invariant Subsets of Homogeneous Spaces: Part I |
Room 2 | |
| Thursday 26th June 2014 | |||
|---|---|---|---|
| 13:30 to 14:30 | Room 2 | |
|
| Friday 27th June 2014 | |||
|---|---|---|---|
| 11:30 to 12:30 | Room 2 | |
|
| 14:30 to 15:30 | Room 2 | |
|
| 15:45 to 16:45 |
Stationary Measures and Invariant Subsets of Homogeneous Spaces: Part II |
Room 2 | |
| Monday 30th June 2014 | |||
|---|---|---|---|
| 08:30 to 08:55 | No Room Required | ||
| 08:55 to 09:00 | Room 1 | ||
| 09:00 to 09:50 |
Gap Distributions, Homogeneous Dynamics, and Random Lattices |
Room 1 | |
| 10:00 to 10:50 | Room 1 | |
|
| 11:00 to 11:30 | No Room Required | ||
| 11:30 to 12:20 |
Quasirandom Groups, Minimally Almost Periodic Groups and Ergodic Ramsey Theory |
Room 1 | |
| 12:30 to 13:30 | No Room Required | ||
| 13:30 to 14:20 | Room 1 | |
|
| 14:30 to 15:00 | No Room Required | ||
| 15:00 to 15:50 |
Counting and equidistribution of common perpendiculars: arithmetic applications |
Room 1 | |
| 16:00 to 16:50 | Room 1 | |
|
| 17:00 to 18:00 | No Room Required | ||
| Tuesday 1st July 2014 | |||
|---|---|---|---|
| 09:00 to 09:50 |
The Hausdorff dimension of not uniquely ergodic 4-IETs has codimension 1/2. |
Room 1 | |
| 10:00 to 10:50 | Room 1 | |
|
| 11:00 to 11:30 | No Room Required | ||
| 11:30 to 12:20 |
Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings |
Room 1 | |
| 12:30 to 13:30 | No Room Required | ||
| 13:30 to 14:20 | Room 1 | |
|
| 14:30 to 15:00 | No Room Required | ||
| 15:00 to 15:50 | Room 1 | |
|
| 16:00 to 16:50 |
Effective equidistribution for twisted averages of the horocycle flow |
Room 1 | |
| 18:30 to 19:30 | No Room Required | ||
| Wednesday 2nd July 2014 | |||
|---|---|---|---|
| 09:00 to 09:50 |
Attaching shortest vectors to lattice points and applications |
Room 1 | |
| 10:00 to 10:50 |
Dimension of Self-similar Measures and Additive Combinatorics |
Room 1 | |
| 11:00 to 11:30 | No Room Required | ||
| 11:30 to 12:20 | Room 1 | |
|
| 12:30 to 13:30 | No Room Required | ||
| 13:30 to 14:20 |
Lattice points counting and Diophantine approximable on manifolds |
Room 1 | |
| 14:30 to 15:00 | No Room Required | ||
| 15:00 to 15:50 | Room 1 | |
|
| 16:00 to 16:50 | Room 1 | |
|
| 19:30 to 22:00 |
Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Co. |
No Room Required | |
| Thursday 3rd July 2014 | |||
|---|---|---|---|
| 09:00 to 09:50 | Room 1 | |
|
| 10:00 to 10:50 | Room 1 | |
|
| 11:00 to 11:30 | No Room Required | ||
| 11:30 to 12:20 |
Metric Diophantine approximation: the well approximable theory on manifolds |
Room 1 | |
| 12:30 to 13:30 | No Room Required | ||
| 13:30 to 14:20 |
Hausdorff dimension estimates for bounded orbits on homogeneous spaces of Lie groups |
Room 1 | |
| 14:30 to 15:00 | No Room Required | ||
| 15:00 to 15:50 |
Limiting distributions of translates of curves on homogeneous spaces |
Room 1 | |
| 16:00 to 16:50 |
Homogeneous dynamics, unitary representations, and Diophantine exponents |
Room 1 | |
| 17:00 to 18:00 | Centre for Mathematical Sciences | ||
| 18:00 to 19:15 | No Room Required | ||
| Friday 4th July 2014 | |||
|---|---|---|---|
| 09:00 to 09:50 | Room 1 | |
|
| 10:00 to 10:50 | Room 1 | |
|
| 11:00 to 11:30 | No Room Required | ||
| 11:30 to 12:20 |
Spectral gap properties for random walks on homogeneous spaces: examples and consequences |
Room 1 | |
| 12:30 to 13:30 | No Room Required | ||
| 13:30 to 14:20 | Room 1 | |
|
| 14:30 to 15:00 | No Room Required | ||
| 15:00 to 15:50 | Room 1 | |
|
| 16:00 to 16:50 | Room 1 | |
|
Organisers
- Anish Ghosh University of East Anglia
- Alex Gorodnik University of Bristol
- Barak Weiss
Participants
- Menny Aka EPFL - Ecole Polytechnique Fédérale de Lausanne
- Jinpeng An Peking University
- Nalini Anantharaman Université Paris Saclay
- Jayadev Athreya University of Illinois
- Dmitry Badziahin University of York
- Yves Benoist Athens University of Economics and Business
- Victor Beresnevich University of York
- Jon Chaika University of Utah
- Yitwah Cheung San Francisco State University
- S.G. Dani Tata Institute of Fundamental Research
- Alex Eskin University of Chicago
- Livio Flaminio
- Alexander Gamburd City College of New York
- Anish Ghosh University of East Anglia
- Mike Hochman Hebrew University of Jerusalem
- Roman Holowinsky Ohio State University
- Bernard Host Université de Marne-la-Vallée
- Dmitry Kleinbock
- Alex Kontorovich Stanford University
- Bryna Kra Northwestern University
- Han Li Stanford University
- Elon Lindenstrauss Hebrew University of Jerusalem
- Francois Maucourant Université de Rennes 1
- Philippe Michel EPFL - Ecole Polytechnique Fédérale de Lausanne
- Amir Mohammadi Japan Atomic Energy Agency & University of Texas at Austin
- Shahar Mozes Hebrew University of Jerusalem
- Paul Nelson EPFL - Ecole Polytechnique Fédérale de Lausanne
- Amos Nevo University of Chicago
- Arnaldo Nogueira Aix Marseille Université
- Hee Oh Princeton University
- Mark Pollicott University of Manchester
- Jean-François Quint Université Bordeaux 1
- Alireza Salehi Golsefidy University of California, San Diego
- Nimish Shah Tata Institute of Fundamental Research
- Uri Shapira Technion - Israel Institute of Technology
- David Samuel Simmons Ohio State University
- Andreas Strömbergsson Uppsala Universitet
- Jim Tseng University of Bristol
- Corinna Ulcigrai University of Bristol
- Sanju Velani
- Thomas Ward University of East Anglia
- Barak Weiss
- Trevor Wooley University of Bristol
- Tamar Ziegler Technion - Israel Institute of Technology
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