# [Seminario] RECORDATORIO INVITACIÓN SEMINARIO Sistemas Dinámicos de Santiago, 01 DE ABRIL A CONTAR DE LAS 15:30 HRS.

Maria Ines Rivera mrivera en dim.uchile.cl
Lun Abr 1 08:05:12 -03 2019

Estimados Académicos(as) y Alumnos(as),

Se les recuerda que hoy  lunes 01 de abril a contar de las 15:30 hrs.
se realizarán 2 sesiones  del *Seminario Sistemas Dinámicos de
Santiago*, en la sala de Seminarios  John Von Neumann CMM, ubicada en la
Torre Norte, Piso 7 de Beauchef 851.

*TIME      (Mon 1st Apr) 3:30 pm - 4:20 pm**
**LOCATION  CMM**
**
**SPEAKER   Sebastián Donoso (CMM, Universidad de Chile)**
**
**TITLE     On subsets with no arithmetic progressions**
**
**ABSTRACT *

For $N\in \mathbb{N}$, let $\nu(N)$ be the maximal cardinality of a
subset of \{1,\ldots,N\} that contains no
arithmetic progression of length 3. Finding upper and lower bounds for
$\nu(N)$ has been a challenging problem for decades.

In this talk I will survey this problem and present a proof of a theorem
by Behrend in the 40's, that gave a surprising lower bound to $\nu(N)$.

*TIME      (Mon 1st Apr) 4:30 pm - 5:30 pm**
**
LOCATION  CMM**
**
SPEAKER   Angel Pardo (CMM, Universidad de Chile)**
**
TITLE     Counting problem on infinite periodic billiards and
translation surfaces**
**
ABSTRACT *

The Gauss circle problem consists in counting the number of integer
points of bounded length in the plane. This problem is equivalent to
counting the number of closed geodesics of bounded length on a flat two
dimensional torus or, periodic trajectories, in a square billiard table.
Many counting problems in dynamical systems have been inspired by this
problem. For 30 years, the experts try to understand the asymptotic
behavior of closed geodesics in translation surfaces and periodic
trajectories on rational billiards. (Polygonal billiards yield
translation surfaces naturally through an unfolding procedure.) H. Masur
proved that this number has quadratic growth rate.
In these talk, we will study the counting problem on infinite periodic
rational billiards and translation surfaces.
The first example and motivation is the wind-tree model, a Z^2-periodic
billiard model. In the classical setting, we place identical rectangular
obstacles in the plane at each integer point; we play billiard on the
complement.
I will first present some quite precise results that are only valid for
the wind-tree model (and some natural generalizations) and then, a
general result which is valid for a.e. infinite periodic translation
surfaces that uses completely different techniques: a dynamical
analogous, for the algebraic hull of a cocycle, to strong and
super-strong approximation on algebraic groups.

Esperando contar con su presencia, les saluda,

Ma. Inés Rivera
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