Groups, Numbers, and Dynamics
Monday 30th June 2014 to Friday 4th July 2014
08:30 to 08:55  Registration  
08:55 to 09:00  Welcome from Christie Marr (INI Deputy Director)  INI 1  
09:00 to 09:50 
Gap Distributions, Homogeneous Dynamics, and Random Lattices
We discuss the connections between cusp excursions of the horocycle flow, random lattices, and gap distributions between Farey fractions, and show how some aspects of this picture can be generalized to the setting of translation surfaces.

INI 1  
10:00 to 10:50 
Divisibility properties in higher rank lattices
In a joint work with Manfred Einsiedler we discuss a relationship between the dynamical properties of a maximal diagonalizable group $A$ on certain arithmetic quotients and arithmetic properties of the lattice. In particular, we consider the semigroup of all integer quaternions under multiplication. For this semigroup we use measure rigidity theorems to prove that the set of elements that are not divisible by a given reduced quaternion is very small:
We show that any quaternion that has a sufficiently divisible norm is also divisible by the given quaternion. Restricting to the quaternions that have norm equal to products of powers of primes from a given list (containing at least two) we show that the set of exceptions has subexponential growth.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:20 
Quasirandom Groups, Minimally Almost Periodic Groups and Ergodic Ramsey Theory
According to the definition introduced by T. Gowers in 2008, a finite group G is called Dquasirandom for some parameter D, if all nontrivial unitary representations of G have dimension greater or equal to D. For example, the group SL(2, F_p) is (p1)/2 quasirandom for any prime p. Informally, a finite group is quasirandom if it is Dquasirandom for a large value of D. Answering a question posed by L.
Babai and V. Sos, Gowers have shown that, in contrast with the more familiar "abelian" situation, qusirandom groups can not have large productfree subsets.
The goal of this lecture is to discuss the connection between the combinatorial phenomena observed in quasirandom groups and the ergodic properties of the minimally almost periodic groups (these were introduced in 1934 by J. von Neumann as groups which do not admit nonconstant almost periodic functions). This connection will allow us to give a simple explanation the dynamical underpinnings of some of the Gowers' results as well as of the more recent results obtained in joint work with T. Tao and in the work of T. Austin.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:20 
A Salehi Golsefidy (University of California, San Diego) Expansion properties of linear groups
Starting with finitely many matrices S in GL(n,Q), we will discuss when the Cayley graphs of congruence quotients of the group generated by S modulo a sequence of integers can form a family of expanders. Then we will focus on the case of powers of primes and show that such a property is dictated by the Zariskitopology.

INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 15:50 
Counting and equidistribution of common perpendiculars: arithmetic applications
I will survey several arithmetic applications, obtained in joint works with Parkkonen, of our counting (and simultaneous equidistribution of initial and terminal tangent vectors)
of the common perpendiculars between two closed locally convex subsets of a negatively curved manifold: generalisation of Mertens' formula for imaginary quadratic number fields, counting quadratic irrationals with bounded crossratios, equidistribution of rational points in the Heisenberg group, counting of arithmetic chains in the Heisenberg group with
Cygan diameter bounded from below.

INI 1  
16:00 to 16:50 
Every flat surface is Birkhoff and Osceledets generic in almost every direction: Clay Mathematics Institute Senior Scholar Lecture
This is a simple argument which also works in the homogeneous setting.

INI 1  
17:00 to 18:00  Welcome Wine Reception 
09:00 to 09:50 
J Chaika (University of Utah) The Hausdorff dimension of not uniquely ergodic 4IETs has codimension 1/2.
The main results of this talk are:
a) The Hausdorff dimension of notuniquely 4IETs is 2 1/2 as a subset of
the 3 dimensional simplex
b) The Hausdorff dimension of flat surfaces in H(2) whose vertical flow is
not uniquely ergodic is 7 1/2 as a subset of an 8 dimensional space
c) For almost every flat surface in H(2) the set of directions where the
flow is
not uniquely ergodic has Hausdorff dimension 1/2.
These results all say that the Hausdorff codimension of these exceptional
sets is 1/2. MasurSmillie showed that the Hausdorff codimension was less
than 1. It follows from work of Masur that the Hausdorff codimension is at
least 1/2. This is joint work with J. Athreya.

INI 1  
10:00 to 10:50 
M Einsiedler (ETH Zürich) Adelic equidistribution and property (tau)
We give a uniform equidistribution statement for maximal subgroups on adelic quotients. The result is effective and applies to cases where the results of Mozes and Shah may not be applicable. The argument can also be used to give an alternative proof of property (tau) (but with weaker exponents than established by Clozel).
We will also discuss an extension/application concerning integer points on spheres and their orthogonal lattice.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:20 
Inhomogeneous multiplicative Littlewood conjecture and logarithmic savings
We use the dynamics on SL(3,R)/SL(3,Z) to get logarithmic savings in the inhomogeneous multiplicative Littlewood setting. This is a joint work with Alex Gorodnik.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:20 
L Flaminio (Université Lille 1) On the GreenfieldWallach and Katok conjectures
In the early 70's, Greenfield and Wallach studied fields globally
hypoelliptic vectors fields on compact manifolds and made the following
conjecture :
``Let $ G $ be a Lie group and let $ H $ be a closed subgroup Such That $G/H
$ is compact. Let $ X ^ * $ be the vector field on $ G / H $ determined by
some element $X$ in the Lie algebra of $G$. Given by If $ X ^ * $ is globally
hypoelliptic, then $ G / H $ is a torus.''
In a work in collaboration with F.~RodriguezHertz and G. Forni,
we gave a positive solution to this problem.
In this talk I will recall thistory of the problem, wxplain its relation with Katok's conjecture
on cohomologically free vector fields and give some idea of the proof.

INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 15:50 
Localized representation theory
The talk will concern an aspect of the representation theory of Lie groups motivated by applications to bounds for period integrals.
I will introduce a notion of a vector in an irreducible unitary representation being "localized". I will then plead the case that this definition is interesting.
For example, recall that Schur's lemma says that the equivariant operators on an irreducible unitary representation are constant multiples of the identity. I will describe an asymptotic classification of "approximately equivariant" operators when restricted to localized vectors.
I will indicate some applications of this simplesounding result, particularly to establishing the invariance of certain limiting measures that arise in semiclassical analysis and number theory.

INI 1  
16:00 to 16:50 
G Forni (University of Maryland) Effective equidistribution for twisted averages of the horocycle flow
We describe an approach, based on scaling, to bounds on twisted ergodic averages
of the classical horocycle flow. As far as we know the best estimates available so far are due to
A. Venkatesh and are based on effective equidistribution and rate of mixing for the horocycle flow itself. Our bounds are significantly better, in particular they are independent of the spectral gap of the representation. The main motivations come from the general question of establishing effective rate of equidistribution for unipotent flows and from the particular case of effective equidistribution for horocycle maps (or for the horocycle flow along polynomial sequences).
We present joint work with L. Flaminio (Lille) and J. Tanis (Rice).

INI 1  
18:30 to 19:30  Wine Reception at the Cambridge University Press Bookshop 
09:00 to 09:50 
Attaching shortest vectors to lattice points and applications
We highlight a simple construction, appeared in the work of D. Badziahin, A. Pollington and S. Velani where they proved Schmidt's conjecture, which attaches to a lattice point an integral vector that is shortest in a certain sense. Such a construction turns out to be useful in studying badly approximable vectors and bounded orbits of unimodular lattices. It can be used to prove: (1) The set $\mathrm{Bad}(i,j)$ of twodimensional badly approximable vectors is winning for Schmidt's game; (2) $\mathrm{Bad}(i,j)$ is also winning on nondegenerate curves and certain straight lines; (3) Threedimensional unimodular lattices with bounded orbits under a diagonalizable oneparameter subgroup form a winning set (at least in a local sense).

INI 1  
10:00 to 10:50 
Dimension of Selfsimilar Measures and Additive Combinatorics
I will discuss recent progress on the problem of computing the dimension of a selfsimilar set or measure in $\mathbb{R}$ in the presence of nontrivial overlaps. It is thought that unless the overlaps are "exact" (an essentially algebraic condition), the dimension achieves the trivial upper bound. I will present a weakened version of this that confirms the conjecture in some special cases. A key ingredient is a theorem in additive combinatorics that describes in a statistical sense the structure of measures whose convolution has roughly the same entropy at small scales as the original measure. As time permits, I will also discuss the situation in $\mathbb{R}^d$.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:20 
A Kontorovich (Yale University) Applications of Thin Orbits
We will discuss various natural problems in arithmetic and dynamics, which at their core are questions about thin groups and orbits.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:20 
Lattice points counting and Diophantine approximable on manifolds
It has been known for at least about a century that there exist algebraic badly approximable points confined by algebraic relations. On the other hand, until very recently nothing was known regarding badly approximable points subject to generic smooth (nonalgebraic) relations. In this talk I will describe a technique in metric Diophantine approximation that has recently enabled to shed some light on this problem. Time permitting I will mention some other applications of the technique.

INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 15:50 
U Shapira (Technion  Israel Institute of Technology) Escape of mass in positive characteristics
I will describe some peculiar features of the geodesic flow in positive characteristics which occur in sharp contrast to analogue discussions in characteristics zero. In particular, situations where a natural limiting procedure leads to escape of mass and accumulation on singular measures. This is joint work with Frederic Paulin and Alex Kemarsky.

INI 1  
16:00 to 16:50 
Oppenheim conjecture and related problems
Oppenheim conjecture which I proved in mideighties says that the set of values at integral points of an irrational indefinite quadratic form in 3 or more variables is dense in R (this is a slightly stronger than the original formulation). I will discuss various approaches to this conjecture and its quantitative generalizations.

INI 1  
19:30 to 22:00  Conference Dinner at Cambridge Union Society hosted by Cambridge Dining Co. 
09:00 to 09:50 
Perturbations of Weyl sums, and uniform distribution mod 1
Significant advances in our understanding of mean values of exponential sums have resulted from recent progress in Vinogradov's mean value theorem, in particular as a consequence of efficient congruencing methods. These advances lead to improved pointwise estimates for Weyl sums. In this talk we will explore the consequences of such ideas for perturbed exponential sums in which lower degree coefficients are permitted to vary over a set of large measure. The conclusions we obtain are related to recent work of Livio Flaminio and Giovanni Forni.

INI 1  
10:00 to 10:50 
Gauss maps for simultaneous approximation
Levy's constant measures the exponential growth rate for the
sequence of denominators of the convergents of a real number.
Khintchine proved existence for almost every real number and
Levy computed the constant to be $\pi^2/12\ln2$. This result
is a standard exercise in modern textbooks on ergodic theory.
In this talk, we generalize it to higher dimensions with Levy's
constant defined using the sequence of best approximation
denominators. The main ingredient of the proof is constructing
the analog of the Gauss map for continued fractions. This
work is joint with Nicolas Chevallier.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:20 
Metric Diophantine approximation: the well approximable theory on manifolds
I will give an overview of recent development regarding the metric theory of well approximable sets restricted to manifold. In particular, I will focus on the strengthening of the fundamental theorems of Khintchine and Gallagher, and demonstrate a basic principle that enables one to establish inhomogeneous extremality from (homogeneous) extermality. The end result of the latter is the inhomogeneous strengthening of the KleinbockMargulis theorem that validates the BakerSprindzuk Conjecture.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:20 
Hausdorff dimension estimates for bounded orbits on homogeneous spaces of Lie groups
This work is motivated by studying badly approximable vectors, that is, ${\bf x}\in {\bf R}^n$ such that $\q{\bf x}  {\bf p}\ \ge cq^{1/n}$ for all ${\bf p}\in {\bf Z}^n$, $q\in {\bf N}$. Computing the Hausdorff dimension of the set of such ${\bf x}$ for fixed $c$ is an open problem. I will present some estimates, based on the interpretation of a badly approximable vector via a trajectory on the space of lattices, and then use exponential mixing to estimate from above the dimension of points whose trajectories stay in a fixed compact set. Joint work with Ryan Broderick.

INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 15:50 
Limiting distributions of translates of curves on homogeneous spaces
We show that in various cases, under natural conditions, limits of translates of pieces of curves on homogeneous spaces get equidistributed in the whole space. These type of results have applications to diophantine approximation.

INI 1  
16:00 to 16:50 
Homogeneous dynamics, unitary representations, and Diophantine exponents
We will describe an explicit quantitative form of the duality principle in homogeneous dynamics. This allows the reduction of a diverse set of quantitative equidistribution problems on homogeneous spaces G/H to the problem of giving explicit spectral bounds for the restriction of automorphic representations of G to the stability subgroup H. We will demonstrate this approach by deriving bounds for Diophantine approximation exponent on homogeneous varieties, a problem raised by Serge Lang already in 1965, but which have seen little progress since then. The Diophantine exponents we derive are in many cases best possible, a remarkable fact that follows from an important and useful representationtheoretic phenomenon which we will highlight. Based on Joint work with Alex Gorodnik (Bristol University) and on joint work with Anish Ghosh (TIFR Mumbai) and Alex Gorodnik.

INI 1  
17:00 to 18:00 
The SL(2,R) action on Moduli space: Mordell Lecture
We prove some ergodictheoretic rigidity properties of the action of SL(2; R) on the moduli space of compact Riemann surfaces. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2; R) is supported on an invariant a ffine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work. This is joint work with Maryam Mirzakhani and Amir Mohammadi.


18:00 to 19:15  Wine Reception at CMS 
09:00 to 09:50 
M Pollicott (University of Warwick) Higher Teichmuller Theory and Thermodynamical Formalism
One can consider cocompact Fuchsian groups from the point of view of representations of a surface group into SL(2, R). There has been recent interest in representations into SL(d, R) for d > 2 which we might call ``Higher Teichmuller'', particular with the recent work of Bridgeman, Canary, Labourie and Sambarino. This talk will describe the usefulness of classical ideas in thermodynamic formalism in the modern study of such representations.

INI 1  
10:00 to 10:50 
Diophantine property in groups
Let x, y be two real numbers and consider all numbers that can be expressed from them using addition and subtraction. Denote by B_l the set of numbers that can be obtained by using x and y at most l times. A simple BorelCantelliargument shows that the smallest positive element of B_l is at least cl^(1e) for almost all pairs x,y. In the lecture we will investigate the analogue of this property in noncommutative Lie groups.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:20 
Y Guivarch (Université de Rennes 1) Spectral gap properties for random walks on homogeneous spaces: examples and consequences
Coauthors: J.P Conze, B. Bekka, E .LePage
Let E be a homogeneous space of a Lie group G, p a finitely supported probability measure on G, such that supp(p) generates topologically G. We show that, in various situations, convolution by p has a spectral gap on some suitable functional space on E . We consider in particular Hilbert spaces and Holder spaces on E and actions by affine transformations. If G is the motion group of Euclidean space V ,we get equidistribution of the random walk on V. If G is the affine group of V,p has a stationary probability, and the projection of p on GL(V) satisfies "generic" conditions we get that the random walk satisfies Frechet's extreme law , and Sullivan's Logarithm law. 
INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
13:30 to 14:20 
S G Dani (Indian Institute of Technology) Multidimensional metric approximation by primitive points
We consider Diophantine inequalities of the form $ \Theta {\bf q} + {\bf p}  {\bf y} \leq \psi( {\bf q} )$, with $\Theta$ is a $n\times m$ matrix with real entries, ${\bf y} \in \mathbb R^n$, $m,n\in {\bf N}$, and $\psi$ is a function on ${\bf N}$ with positive real values, and seek integral solutions ${\bf v} =({\bf q}, {\bf p})^t$ for which the restriction of ${\bf v}$ to the components of a given partition $\pi$ are primitive integer points. In this setting, we shall discuss metrical results in the style of the KhintchineGroshev Theorem. Solutions for analogous doubly metrical inequalities will also be discussed.

INI 1  
14:30 to 15:00  Afternoon Tea  
15:00 to 15:50 
Unipotent flows on infinite volume manifolds
Unipotent flows on homogeneous spaces obtained as quotients of a Lie group by a lattice have been extensively studied over the past 40 years or so, with several fundamental applications. Starting from the late 80's there have been several results aiming at extending some of these results to certain infinite volume settings. However, our understanding of the subject, beyond the horospherical foliation in the rank one case, remains rather poor. In this talk we will discuss some, ongoing, attempts in this direction. This is a joint project with Hee Oh.

INI 1  
16:00 to 16:50 
Y Benoist ([Université ParisSud 11]) Random walk on padic flag varieties
According to a theorem of Furstenberg, a Zariski dense probability
measure on a real semisimple Lie group admits a unique stationary
probability measure on the flag variety. In this talk we will see that a
Zariski dense probability measure on a padic semisimple Lie group
admits finitely many stationary probability measures on the flag variety
and we will classify these measures. This is a joint work with JF. Quint.

INI 1 