Chaire Aisenstadt 2005-2006 Aisenstadt Chair
Dr. Terence Tao

Conférence André-Aisenstadt Lecture

Photos de l'événement / Events' photos

(Cette conférence s’adresse à un large auditoire. /
Suitable for a general audience.)

Le lundi 3 avril 2006 / Monday, April 3, 2006

16 h 15 / 4:15 am

Pavillon André-Aisenstadt,
Université de Montréal
2920, chemin de la Tour
Salle / Room 1140

Long Arithmetic Progressions in the Primes

A famous and difficult theorem of Szemeredi asserts that every subset of the integers of positive density will contain arbitrarily long arithmetic progressions; this theorem has had four different proofs (graph-theoretic, ergodic, Fourier analytic, and hypergraph-theoretic), each of which has been enormously influential, important, and deep. It had been conjectured for some time that the same result held for the primes (which of course have zero density). I shall discuss recent work with Ben Green obtaining this conjecture, by viewing the primes as a subset of the almost primes (numbers with few prime factors) of positive relative density. The point is that the almost primes are much easier to control than the primes themselves, thanks to sieve theory techniques such as the recent work of Goldston and Yildirim. To "transfer" Szemeredi's theorem to this relative setting requires that one borrow techniques from all four known proofs of Szemeredi's theorem, and especially from the ergodic theory proof.

Une réception suivra la conférence au Salon Maurice-l'Abbé,
Pavillon André-Aisenstadt (Salle 6245).
There will be a reception after the lecture in Salon Maurice-l'Abbé,
Pavillon André-Aisenstadt (Room 6245).

Conférences dans le cadre de l’École CRM-Clay en combinatoire additive
Lectures at the CRM-Clay School on Additive Combinatorics

"Combinatorial and ergodic techniques for proving Szemeredi-type theorems "

This is a series of lectures exploring the graph theory, hypergraph theory, and ergodic theory approaches to Szemeredi's famous theorem on arithmetic progressions in sets of positive density, emphasizing the connections between the techniques.

Le vendredi 31 mars 2006 / Friday, March 31, 2006
15h45 / 3:45 pm

Le samedi 1er avril 2006 / Saturday, April 1, 2006
9h30 / 9:30 am

Le dimanche 2 avril 2006 / Sunday, April 2, 2006
9h30 / 9:30 am

Pavillon Roger-Gaudry, Université de Montréal
Salle / Room M-415

Le mardi 4 avril 2006 / Tuesday, April 4, 2006
11h00 / 11:00 am

Pavillon J.-Armand-Bombardier, Université de Montréal
Salle / Room 1035

Conférence dans le cadre de l’Atelier en combinatoire additive
Lectures at the Workshop on Additive Combinatorics

"An Infinitary Approach to (Hyper)graph Regularity and Removal"

The famous Szemeredi regularity lemma gives a structural theorem for very large (but finite) dense graphs, which then has many applications to such graphs, for instance in being able to efficiently eradicate all copies of a given subgraph by edge removal if the original number of such copies was small. These results have been extended to hypergraphs (leading for instance to another proof of Szemeredi's theorem on arithmetic progressions) but the proofs, while elementary and finitary, are somewhat messy and lengthy in nature. Here we present an alternate "infinitary" route to these results, by passing from a sequence of large finite dense deterministic graphs to an infinite dense random graph, and analysing the resulting object instead. The advantage of doing this is that many of the "epsilon" quantities present in the finitary theory go to zero in the infinite limit, and one can now bring techniques from infinitary probability theory (in particular, the theory of conditional independence) to bear on the subject.

Le jeudi 6 avril 2006 / Thursday, April 6, 2006
9h45 / 9:45 am

Pavillon Roger-Gaudry, Université de Montréal
Salle / Room M-415

March 22 mars 2006