Interactions between dynamics of group actions and number theory

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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

Click here to download the programme's final scientific report

Preprints

Title	Year	Programme
Best possible rates of distribution of dense lattice orbits in homogeneous spaces Authors: Anish Ghosh, Alex Gorodnik, Amos Nevo Download PDF	2013	GAN
Multiparameter spherical averages of Siegel transforms and applications Authors: Jayadev Athreya, Anish Ghosh, J Tseng Download PDF	2013	GAN
Simultaneous dense and nondense orbits and the space of lattices Authors: R Shi, J Tseng Download PDF	2013	GAN

Title

Year

Programme

Best possible rates of distribution of dense lattice orbits in homogeneous spaces

Authors: Anish Ghosh, Alex Gorodnik, Amos Nevo

Download PDF

2013

GAN

Multiparameter spherical averages of Siegel transforms and applications

Authors: Jayadev Athreya, Anish Ghosh, J Tseng

Download PDF

2013

GAN

Simultaneous dense and nondense orbits and the space of lattices

Authors: R Shi, J Tseng

Download PDF

2013

GAN

Workshops

GANW01

Groups, numbers and dynamics

30 June 2014 to 4 July 2014

Seminars

Monday 9th June 2014
14:30 to 15:30	Points on spheres and their orthogonal lattices	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	Simultaneous dense and nondense orbits for commuting maps	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	Rigid and flexible properties of group automorphisms	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	Some applications of Mahler's method	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	Partial sums of excursions along random geodesics.	Room 2
14:30 to 15:30	Periodicity, complexity and shifts.	Room 2

Wednesday 18th June 2014
11:30 to 12:30	Lower bounds for entropy of random walks.	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	Informal (Corinna Ulcigrai)	Discussion Room
11:30 to 12:30	Spectral gap in simple Lie groups II	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	Khintchine types of translated coordinate hyperplanes	Room 2
13:30 to 14:30	Hall rays for Lagrange spectra of Veech surfaces	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	Effective Ratner equidistribution for SL$(2,\mathbb R)\ltimes(\mathbb R^2)^{\oplus k}$ and applications to quadratic forms	Room 2

Friday 27th June 2014
11:30 to 12:30	Improvement of Dirichlet's Theorem and homogeneous dynamics	Room 2
14:30 to 15:30	On the p-adic Littlewood conjecture for quadratics	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	Registration	No Room Required
08:55 to 09:00	Welcome from Christie Marr (INI Deputy Director)	Room 1
09:00 to 09:50	Gap Distributions, Homogeneous Dynamics, and Random Lattices	Room 1
10:00 to 10:50	Divisibility properties in higher rank lattices	Room 1
11:00 to 11:30	Morning Coffee	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	Lunch at Wolfson Court	No Room Required
13:30 to 14:20	Expansion properties of linear groups	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
16:00 to 16:50	Every flat surface is Birkhoff and Osceledets generic in almost every direction: Clay Mathematics Institute Senior Scholar Lecture	Room 1
17:00 to 18:00	Welcome Wine Reception	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	Adelic equidistribution and property (tau)	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:20	Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings	Room 1
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:20	On the Greenfield-Wallach and Katok conjectures	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 15:50	Localized representation theory	Room 1
16:00 to 16:50	Effective equidistribution for twisted averages of the horocycle flow	Room 1
18:30 to 19:30	Wine Reception at the Cambridge University Press Bookshop	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	Morning Coffee	No Room Required
12:30 to 13:30	Lunch at Wolfson Court	No Room Required
13:30 to 14:20	Lattice points counting and Diophantine approximable on manifolds	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 15:50	Escape of mass in positive characteristics	Room 1
16:00 to 16:50	Oppenheim conjecture and related problems	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	Perturbations of Weyl sums, and uniform distribution mod 1	Room 1
10:00 to 10:50	Gauss maps for simultaneous approximation	Room 1
11:00 to 11:30	Morning Coffee	No Room Required
11:30 to 12:20	Metric Diophantine approximation: the well approximable theory on manifolds	Room 1
12:30 to 13:30	Lunch at Wolfson Court	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	Afternoon Tea	No Room Required
15:00 to 15:50	Limiting distributions of translates of curves on homogeneous spaces	Room 1
17:00 to 18:00	The SL(2,R) action on Moduli space: Mordell Lecture	Centre for Mathematical Sciences
18:00 to 19:15	Wine Reception at CMS	No Room Required

Friday 4th July 2014
10:00 to 10:50	Diophantine property in groups	Room 1
11:00 to 11:30	Morning Coffee	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	Lunch at Wolfson Court	No Room Required
13:30 to 14:20	Multi-dimensional metric approximation by primitive points	Room 1
14:30 to 15:00	Afternoon Tea	No Room Required
15:00 to 15:50	Unipotent flows on infinite volume manifolds	Room 1
16:00 to 16:50	Random walk on p-adic flag varieties	Room 1

Interactions between dynamics of group actions and number theory

Organisers

Anish Ghosh University of East Anglia
Alex Gorodnik University of Bristol
Barak Weiss

Interactions between dynamics of group actions and number theory

Programme theme

Preprints

Best possible rates of distribution of dense lattice orbits in homogeneous spaces

Multiparameter spherical averages of Siegel transforms and applications

Simultaneous dense and nondense orbits and the space of lattices

Workshops

Groups, numbers and dynamics

Seminars

Organisers

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