**Abstract.**
A famous theorem of Roth asserts that any dense subset of the integers
{1, ..., N} must contain a three-term arithmetic progression provided
N is large enough in terms of the density of the set. This turns out
to be equivalent to the statement that a subset of {1 ,..., N} of
positive density delta must actually contain a lot of three-term
progressions: at least c(delta)N^2 of them, in fact, where c(delta) is
some positive constant depending only on the density delta. Similar
statements exist in Z/pZ, the integers modulo a prime p, and I shall
discuss the analogous problem in this setting: how many three-term
progressions must A contain if A is a subset of Z/pZ of density delta?
In particular, I shall outline how one can obtain an exact answer for
very large densities using some analytically-inspired ideas. Based on
joint work Ben Green.

**Abstract.**
We prove the existence of a non uniformly hyperbolic attractor for a positive set of
parameters a in the family:
\[(x,y)\mapsto (x^2+a+2y,0)+B(x,y)\]
Where $B$ is fixed a $C^2$ small function.
The proof uses the formalism of Yoccoz puzzle and analytical ideas of
Benedicks-Carleson.

**Abstract.** (given as a pdf-file)

**Abstract.**
We prove a structure theorem for a sumset of two sets A and B of
positive upper Banach density in any countable amenable group. More
precisely, we prove that AB is "piecewise syndetic" which means that there
exists a finite set K such that for any finite set F in G
("configuration") there exists an element g in G such that Fg is a subset
of ABK.
For abelian groups we prove even more, namely, if A and B have positive
upper Banach density then there exists a finite set K in G such that A+B+K
is a piecewise Bohr set (large pieces of almost periodic set -- contains a
lot of structure). The latter implies that there exist C, D, E sets of
positive upper Banach density such that C+D+E is a subset of A+B.
(joint work with M.Beiglbock and V.Bergelson)

**Abstract.** (given as a pdf-file)

**Abstract.** (given as a pdf-file)

**Abstract.**
We will introduce the Kloosterman sums
S(m,n;c)
and discuss about some of their applications in analytic number theory, in particular
applications on exponential sums and
on the fourth power moment of the Riemann zeta-function (following Heath-Brown,
Kuznetsov, Iwaniec and Motohashi).

**Abstract.**
In this talk I will give an overview of the renormalization theory for
unimodal maps. The focus will be on Marco Martens' proof of the existence of
renormalization fixed points and how it naturally leads to an algorithm
for constructing such fixed points (of any combinatorial type and
critical exponent). Finally, I will outline a computer implementation
of this algorithm.

**Abstract.**
We consider quasi-periodic perturbations of a
quadratic map exhibiting an attracting period-3 point. We will
rigorously show that such a perturbation can create so-called Strange
Nonchaotic Attractors, an object which lies between regularity and chaos.

**Abstract.**
Maps from double standard map family
f_a(x)=2x+a+(1/pi) sin(2 pi x) (mod~1),
have the property that they are double covers of the circle onto
itself with a unique inflexion point. They have been investigated most
recently by M. Misiurewicz and A. Rodrigues. In particular one can say
that they are hybrids between circle homeomorphims with inflexions and
quadratic maps of the interval.
The aim of the talk is to develop symbolic dynamics and kneading theory
for these maps and discuss the behaviour in parameter space (chaotic
behaviour, stable periodic orbits) comparing the situation to
the more standard cases of circle homeomorphisms and quadratic interval maps. This is joint work with
A. Rodrigues.

**Abstract.**
We study L^p-L^q boundedness and compactness of the operator
f -> w(x) int_{a(x)}^{b(x)}k(x,y)f(y)v(y)dy with given weight
functions w(x),v(y), differentiable strictly increasing border
functions a(x),b(x) and a kernel k(x,y) satisfying some growth
conditions. The results are applied for weighted L^p-L^q
boundedness of geometric mean operator f ->
exp[(b(x)-a(x))^(-1) int_a(x)^b(x) log f(y) dy]
and other related problems.
The talk is based on U.U.D.M. Reports 2008:30 and 2008:46.

**Abstract.**
This talk will present new developments in understanding the
analytic continuation of certain Dirichlet series in several complex
variables associated to moments of quadratic Dirichlet L-functions.

**Abstract.**
The question of equidistribution of Gamma orbits on a
homogeneous space X has been thoroughly studied in recent years from many
perspectives. In this talk I will tackle this question for Gamma a
nilpotent group and X a nilpotent Lie group and consider two types of
averages: the word length average and the random walk average. Using
unique ergodicity and precise geometric information on the shape of
nilpotent balls I will show how to answer the equidistribution problem in
that setting.

**Abstract.**
I will present a recent joint work with Manfred Einsiedler and
Lior Fishman in which we use rigidity results in dynamics to prove results
in Diophantine approximations. We study how certain fractals intersect
certain Diophantine classes. In particular I plan to concentrate on the
following theorem regarding the intersection of the middle third Cantor
set and the set of "Well Approximable" numbers:
Theorem: Let a_n be a random sequence of the digits 0 and 2 (each digit
appears with probability 1/2) and let x be the number in the unit interval
having this sequence as its base three expansion. Then with probability one
the coefficients in the continued fraction expantion of x, are unbounded.

**Abstract.**
In this talk, I will give a brief review of classical spectral
geometry and the study of the geodesic length spectrum on a Riemannian
manifold. I will then discuss some generalizations of the length spectrum
and some results on how much of the geometry is encoded in other geometric
spectra. This is joint work with Alan Reid.

**Abstract.**
A beautiful theorem of K. F. Roth from the 50's asserts that any
subset of the integers containing no three-term arithmetic
progressions with non-zero common difference has density zero.
In the 80's and 90's a beautiful model problem was considered: suppose
that A \subset (\Z/3\Z)^n contains no affine line. Then
|A|=O(3^n/n). A proof of this result (due to Meshulam) can be seen as a
finite field version of Roth's proof of his aforementioned
theorem, and in this setting the argument becomes much
simpler. Despite this no improvement is known and any bound of the shape
o(3^n/n) would be of considerable interest. In this talk we shall
consider the analogous problem for (\Z/4\Z)^n where an
improvement over Roth's argument is possible.

**Abstract.**
In this talk, we establish stable ergodicity for
diffeomorphisms with partially hyperbolic attractors whose Lyapunov
exponents along the center direction are all positive with respect to
the physical measures.

**Abstract.**
Let S be a group or semigroup acting on a variety V, let x
be a point on V, and let W be a subvariety of V. What can be said
about the structure of the intersection of the S-orbit of x with W?
Does it have the structure of a union of cosets of subgroups of S?
The Mordell-Lang theorem of Laurent, Faltings, and Vojta shows that
this is the case for certain groups of translations (the Mordell
conjecture is a consequence of this). On the other hand, Pell's
equation shows that it is not true for additive translations of the
Cartesian plane. We will see that this question relates to issues in
complex dynamics, simple questions from linear algebra, and techniques
from the study of linear recurrence sequences.

Roman Schubert (University of Bristol): Semiclassics and long time evolution: how compatible?

Oscar Marmon (Chalmers/GU): On the density of solutions to Diophantine equations

Håkan Hedenmalm (KTH): Heisenberg uniqueness pairs and the Klein-Gordon equation

Daniel Schnellmann (KTH): Almost Sure Equidistribution in Expansive Families

Tsachik Gelander (Hebrew University): Property (T) and rigidity for actions on Banach spaces

Richard Miles (KTH): Dirichlet series for finite combinatorial rank dynamics

Rikard Olofsson (KTH): Large Hecke eigenfunctions of quantized cat maps

Michael Björklund (KTH): Entropy of Algebraic Actions of the Discrete Heisenberg Group