Groupe de travail en combinatoire arithmétique 2013 - 2014
L'objectif de ce groupe de travail est de traiter des travaux récents ainsi que des techniques classiques dans le domaine de la combinatoire arithmétique (ou dans le voisinage proche de ce domaine). Le groupe est ouvert à toute personne intéressée. Si vous souhaitez recevoir les annonces par mail, merci de m'envoyer un message.
Les exposés ont lieu à l'Institut Henri Poincaré, à 16h sauf avis contraire.
Entre 2011 et 2013, le groupe de travail était organisé par Julia Wolf.
La liste des séances précédentes peut être consultée ici.
| Sujet: |
An introduction to Freiman's theorem and its generalisations |
| Résumé: |
Given a finite set A of integers, define A+A := { x + y : x, y
in A }. A classical theorem of Freiman classifies the finite sets A of
integers that are "approximately closed", in the sense that A+A is "not
too much bigger than" A. More precisely, Freiman's theorem states that
if |A+A|< K|A| for some parameter K>1, then A must resemble something
like an arithmetic progression; moreover, how closely A "resembles" an
arithmetic progression is quantified in terms of K in a certain precise
sense.
It makes sense to ask an analogous question in a more general setting:
if A is a finite set in an arbitrary group, and we write AA := { xy : x,
y in A }, then what can we say about A under the assumption that |AA| <
K|A|? In recent years, there has been much progress on the answer to
this more general question, and many interesting applications.
|
| Orateur: |
Matthew Tointon |
| Lieu: |
IHP Salle 421 |
| Heure: |
16h |
lundi, 3 mars 2014
| Sujet: |
Zero-sum theory of abelian groups |
| Résumé: |
We give an overview of zero-sum theory in abelian groups, with an emphasis
on open questions and aspects of the theory related to the study of non-unique factorization in abelian cancellative monoids. |
| Orateur: |
Salvatore Tringali |
| Lieu: |
IHP Salle 421 |
| Heure: |
16h |
lundi, 10 février 2014
| Sujet: |
La méthode polynomiale - principe, sandwich au jambon et application |
| Résumé: |
A partir d'idées totalement élémentaires comme le principe de dichotomie
(l'ensemble des zéros d'un polynôme est soit très grand, soit très petit),
on peut démontrer de grandes choses, comme la conjecture d'Erdös sur le
problème des distances ou le théorème de Széméredi-Trotter. Enquête sur
une méthode en plein essor.
|
| Orateur: |
Pierre-Yves Bienvenu |
| Lieu: |
IHP Salle 314 |
| Heure: |
16h |
lundi, 20 janvier 2014
| Sujet: |
Extremal graphs and sets of integers that do not contain sumsets of
sets of prescribed size |
| Résumé: |
We study extremal problems on sets of positive integers that do not contain sumsets of sets of prescribed size and relate them with extremal
problems on r-partite graphs.
|
| Orateur: |
Javier Cilleruelo |
| Lieu: |
IHP Salle 421 |
| Heure: |
16h |
jeudi, 14 novembre 2013
| Sujet: |
On short intervals containing prime numbers |
| Résumé: |
Define a Goldbach number to be an even number which can be represented
as a sum of two primes. Following the procedure described in [1], we will present the proof of the following result:
For H in [N^{t+eps}, N], t=1/3, almost all even numbers in an interval [N, N+H] are Golbach numbers. (1)
There are now stronger improvements on the result above. However, the best is the combination of two important results [2] and [3], which
gives us the same result as (1) for t=21/800. Explaining the methods used in [2], we will firstly obtain the result of the type: for all x
large enough, the interval [x-x^{21/40}, x] contains prime numbers. After that we will show how to get the exponent factor for primes in
almost all short intervals (from [3]) by a short argument of Montgomery and Vaughan and finally obtain the best known improvement for the
result of type (1), with t=21/800.
[1] A. Perelli, J. Pintz, On the exceptional set for Goldbach's problem in short intervals,
J. London Math. Soc., (2) 47, 1993, pp. 41-49.
[2] R.C. Baker, G. Harman, J. Pintz, The difference between consecutive primes, II, Proc. London Math. Soc., (3) 83, 2001, pp. 532-562.
[3] C. Jia, Almost all short intervals containing prime numbers, Acta Arith., (76) 1, 1996, pp. 21-84.
|
| Orateur: |
Alisa Sedunova |
| Lieu: |
IHP Salle 421 |
| Heure: |
16h |
mardi, 15 octobre 2013
| Sujet: |
Norm forms as products of linear polynomials |
| Résumé: |
In this talk I will discuss how methods developed by Green and Tao can be
employed to prove the Hasse principle and weak approximation for certain
varieties defined by systems of linear equations involving norm forms.
This result allows us to show that the Brauer-Manin obstruction controls
weak approximation on smooth and projective models of varieties defined by
an equation of the form "norm form = P(t)", where P(t) is a product of
linear polynomials all defined over Q and where the norm form is defined
with respect to an arbitrary finite extension of Q.
(Joint work with Tim Browning.) |
| Orateur: |
Lilian Matthiesen |
| Lieu: |
IHP Salle 421 |
| Heure: |
16h |
mardi, 24 septembre 2013
| Sujet: |
Kakeya sets and directional maximal operators in the plane |
| Résumé: |
We discuss relatives of the two-dimensional Kakeya problem. This
leads us to new methods for construction of Besicovitch-type sets.
Time-permitting we may also discuss connections to the question of how
likely it is that a randomly dropped needle will land in a planar Cantor
set, and possibly also to the sum-product phenomenon. |
| Orateur: |
Michael Bateman |
| Lieu: |
IHP Salle 421 |
| Heure: |
16h |
|