

FIELDS MEDAL SYMPOSIUM 

SEPTEMBER 30  OCTOBER 3, 2013
The Fields Institute, Toronto, Canada

ABSTRACTS






Nalini
Anantharaman, Université ParisSud
Quantum ergodicity on large regular graphs (slides)
The usual Quantum Ergodicity Theorem (also known
as Shnirelman theorem) deals with Laplacian eigenfunctions
on compact manifolds. Assuming the geodesic flow
is ergodic, it says that “most” eigenfunctions
are uniformly distributed over phase space, in the
large frequency regime. We provide an analogous
statement for finite discrete graphs  in the regime
where the size of the graph goes to infinity : a
“typical” eigenfunction of the discrete
Laplacian on a “typical” large regular
graph is uniformly distributed over phase space.
This is joint work with Etienne Le Masson.






Yves Benoist, Université ParisSud
Dynamical system on the torus.
One of the aims of Ergodic Theory is to describe
the statistical behavior of the trajectories of
a transformation. Even for the action of unimodular
integral matrices on the 2torus this behavior can
be quite chaotic. With JF Quint, we focus on a very
concrete question. Let A and B be two such matrices
spanning a nonsolvable group. Let x be an irrational
point on the 2torus. We toss A or B, apply it to
x, get another irrational point y, do it again to
y, get a point z, and again. We check that this
random trajectory is equidistributed on the torus.
Moreover this phenomenon is quite general on any
finite volume homogeneous space.







Jean
Bourgain, Institute for Advanced Study
A spectral gap theorem in SL^2(R) and applications (slides)






Shimon Brooks, Bar Ilan University
Quasimodes and Quantum Unique Ergodicity (slides)
We investigate how equidistribution of eigenfunctions
is related to spectral multiplicities and arithmetic
structure, in the context of surfaces of constant
negative curvature. By studying quasimodes or
approximate eigenfunctions we highlight the role
that large "eigenspaces" can play in the
creation of scarring along geodesics, and how arithmetic
symmetries dramatically change the landscape.






Manfred
Einsiedler, ETH Zürich
Integer points on spheres and their orthogonal complement (slides)
Consider the collection of all primitive integer
vectors in $\mathbb{Z}^d$ of length $\sqrt{D}$.
After dividing by $\sqrt{D}$ one gets for $D\to\infty$
an equidistributing family of points on $\mathbb{S}^{d1}$
(if the set is nonempty and $d>2$.) The hardest
case of this statement is the case $d=3$ and was
obtained by Duke (after a breakthrough of Iwaniec
and initial progress by Linnik).
A related question is how the lattice in the orthogonal
complement of the vector looks like. W. Schmidt
showed that these lattices equidistribute if one
considers all vectors up to a particular length
(instead of just the vectors of a given length).
We will show by using homogeneous dynamics (under
mild conditions for $d=3,4,5$ and without additional
conditions for $d\geq 6$) that the pair consisting
of the normalized primitive vector of length $\sqrt{D}$
and the lattice in the orthogonal complement equidistribute
in $\mathbb{S}^{d1}$ times the moduli space of
lattices (up to rotations and scalar multiplication).
This is joint work with Menny Aka and Uri Shapira,
and is related to a theorem of Mozes and Shah, the
higher rank joining theorem by Lindenstrauss and
myself, and ongoing joint work with Margulis, Mohammadi,
and Venkatesh. As we will see, padic dynamics is
needed.






Alex Eskin, University of Chicago
The SL(2; R) action on Moduli space (slides)
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 affine 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.






Hillel
Furstenberg, Hebrew University
Complementary subactions for
higher rank actions
Let S' and S" be subsemigroups of a commutative
semigroup S with S=S'S", and suppose S acts on a
compact metric space X. For each x in X we associate
three dimensions  d'(x) = the Hausdorff dimension
of the orbit closure under S', similarly d"(x),
and d(x) = the hausdorff dimension for the orbit
closure under the full semigroup. In the simplest
situations one can verify that d'(x) + d"(x) is
not less than d(x). However there are counterexamples.
We have studied this question when S = N, the natural
numbers, X = R/Z and the action is by multiplication
mod 1. It is possible that the aforementioned inequality
holds for each x. We shall try to make plausible
the conjecture that the inequality holds but for
a set of exceptions having dimension 0.






Anatole Katok,
Pennsylvania State University
The Measure Rigidity Program
The original measure rigidity program concerns
with the description of all invariant measures for
various classes of algebraic (homogeneous and affine)
actions on locally symmetric spaces. While this
was completely successful in the parabolic (unipotent)
case, virtually all progress uptodate in the more
subtle normally hyperbolic and partially hyperbolic
cases relies on some sort of positive entropy assumption.
The most general results in that direction are due
to Einsiedler and Lindenstrauss. I will mention
certain ideas that may led to a progress in the
resistant zero entropy case.
Two new directions have developed during the recent
years. They use ideas developed in the measure rigidity
in combination with smooth ergodic theory (aka Pesin
theory) to study general smooth actions of several
classes of groups the the point of view of ergodic
theory, geometry, and topology. Most attention was
paid to actions of higher rank abelian groups. I
will discuss results in that direction results joint
with Federico Rodriguez Hertz obtained during the
last few years as well as remaining open problems.
A natural idea is use this sort of information to
advance Zimmer program , i.e classification of smooth
actions of higherrank Lie groups and their lattices.
An initial success in this direction was a 2010
paper with Rodriguez Hertz that proves rigidity
of realanalytic SL(n1,Z) actions on n
dimensional torus, n>2. Recently Aaron Brown,
Rodriguez Hertz and Zhiren Wang reached a remarkable
progress that involves both existence of invariant
measures and rigidity in a number of previuosly
unaccessible situations. An outstanding feature
of their approach that they avoid any use of Zimmer
cocycle superrigidity which always considered the
cornerstone of the program and use dynamical, entropybased
arguments instead. I
will discuss this work in the second part of my
talk.






Elon
Lindenstrauss, Hebrew University
Rigidity and flexibility for actions of diagonalizable groups
on homogenous spaces (slides)
A fundamental insight of Furstenberg is that the
action of higher rank abelian groups may have rigidity
properties that strongly contrast with the behavior
of each individual element. Surprisingly, these
rigidity properties turned out to be intimately
connected to ergodic theoretic entropy.
I will survey some of the research in this direction,
in particular regarding the study of invariant measures,
closed invariant sets, and periodic orbits of such
actions. The phenomenological description of this
behavior turns out to be quite intricate and intertwines
rigidity and flexibility. I will also discuss some
applications of these results to number theory and
(briefly) their relation to quantum ergodicity.






Gregory Margulis, Yale University
Effective results on distribution of values of quadratic
forms at integral points
Let Q be an irrational real quadratic form in n
variables where n>2, and let a<b. One of the
questions which will be discussed is the size of
the smallest integral solution of the inequality
a<Q(x)<b. Another problem is how to get an
effective error bound for the number of integral
solutions of the above inequality in the ball of
"large" radius centered at 0. The talk
is based on joint works with Elon Lindenstrauss
and other mathematicians.






Shahar Mozes, Hebrew University
Equidistribution of primitive rational points on
expanding horospheres
In a joint work with Manfred Einsiedler, Nimish
Shah and Uri Shapira we confirm a conjecture of
Jens Marklof regarding the equidistribution of certain
sparse collections of points on expanding horospheres.
These collections are obtained by intersecting the
expanded horosphere with a certain manifold of complementary
dimension and turns out to be of arithmetic nature.
This equidistribution result is then used along
the lines suggested by Marklof to give an analogue
of a result of W. Schmidt regarding the distribution
of shapes of lattices orthogonal to integer vectors.






Hee Oh, Yale
University
Thin groups and Dynamics






Kannan
Soundararajan, Stanford University
Moments of zeta and Lfunctions 





Masaki
Tsukamoto, Kyoto University
Brody curves and mean dimension (slides)
One of the fundamental questions in function theory
is the diophantine problem for entire holomorphic
functions. In this talk I will describe the dynamical
approach to this problem. Our approach is based
on the notion ``mean dimension". This is a
topological invariant of dynamical systems introduced
by Gromov, and further investigated by Elon Lindenstrauss
and Benjamin Weiss. In this talk, first I will explain
the background materials on holomorphic curves and
mean dimension theory. Next I will describe the
main result: the estimate on the mean dimension
of the system of ``Brody curves''. Our estimate
is sharp when the target is the Riemann sphere.
This talk is based on the joint works with Shinichiroh
Matsuo.






Benjamin
Weiss, Hebrew University of Jerusalem,
Entropy in ergodic theory
After defining's Shannon's entropy I will give
a brief survey of how it has been used in ergodic
theory. In the second part of the talk I will describe
what sofic groups are and a new approach to the
extension of entropy theory to actions of these
groups. This class includes all residually finite
groups and in particular all finitely generated
linear groups.






Tamar Ziegler, Technion  Israel
Institute of Technology
Patterns in Primes and Dynamics of Nilmanifolds (slides)
I will survey some of the ideas behind the recent
developments in additive number theory and ergodic
theory leading to the proof of HardyLittlewood
type estimates for the number of prime solutions
to systems of linear equations of finite complexity.






