ARCOÍRIS
A SEMINAR ON PCF THEORY AND ITS APPLICATIONS
TALKS


what?
“Arcoíris” is a seminar on PCF theory and its applications to other areas of mathematics, especially topology. We're based in Sao Paulo, Brazil.
PCF theory is Saharon Shelah’s study of ultraproducts of small sets of cardinals, that he first used to give unexpectedly sharp bounds to singular exponents and later on exploited to solve the most disparate set of problems, from the construction of a small Dowker space in Topology (with Kojman) to the almost free vs. free problem in group theory.
PCF theory provides every mathematician interested in infinite sets with a powerful tools to analyse problems involving singular cardinals, often within the usual axioms of set theory.
Menachem Kojman has written a 4 page teaser on PCF theory for the Topology Atlas.
The name of our seminar is the Portuguese word for "rainbow" and is a pun on the Hebrew name of PCF theory: תורת הקשתות, meaning "theory of arches" or "theory of rainbows".
PCF theory is Saharon Shelah’s study of ultraproducts of small sets of cardinals, that he first used to give unexpectedly sharp bounds to singular exponents and later on exploited to solve the most disparate set of problems, from the construction of a small Dowker space in Topology (with Kojman) to the almost free vs. free problem in group theory.
PCF theory provides every mathematician interested in infinite sets with a powerful tools to analyse problems involving singular cardinals, often within the usual axioms of set theory.
Menachem Kojman has written a 4 page teaser on PCF theory for the Topology Atlas.
The name of our seminar is the Portuguese word for "rainbow" and is a pun on the Hebrew name of PCF theory: תורת הקשתות, meaning "theory of arches" or "theory of rainbows".
WHERE AND WHEN?
We meet every Monday at 2 PM in room 266A of Building A (Bloco A) of the Institute of Mathematics and Statistics (IME), of the University of Sao Paulo, in Sao Paulo, Brazil.
Address: Rua do Matão, 1010  Cidade Universitária, São Paulo  SP, 05508090 Brazil 
WHY?
Besides in its intrinsic beauty, we're interested in the impact of PCF theory and singular cardinal arithmetic on other areas of mathematics. Here are a few examples of that (if you're interested in presenting any of these papers (or parts of them) at the seminar please drop us a line at: arcoirisPCF AT gmail.com).
 Shelah's proof of the existence of Jónsson Algebras on aleph_{omega+1}.
 Shelah's solution of the almost free vs. free problem in Group Theory.
 Kojman and Shelah's construction of a small Dowker space in ZFC.
 Shelah and Todorcevic's ZFC construction of a space whose cellularity increases in the square.
 Juhasz and Szentmiklossy's interpolation result for kappacompactness.
 Kojman and Shelah's ZFC determination of the Baire number of the space of all uniform ultrafilters over a singular cardinal of countable cofinality.
 Kojman and Lubitch's construction of small linearly Lindelof nonLindelof spaces.
 Andreas Blass's solution to a problem of Irwin involving a generalisation of free groups.
 Lajos Soukup's characterisation of the additivity spectrum of ideals on countable sets.
 Assaf Rinot's equivalence between Shelah's Strong Hypothesis and a topological reflection principle.
 Alberto Levi's reflection theorems for the Lindelof number.
 Kojman, Milovich and Spadaro's ZFC bounds for Noetherian type.
 Jindra Zapletal's cute proof of Kunen's inconsistency theorem using PCF scales.
 A result by Shelah on the number of open sets in a topological space.
 Lower and upper bounds on the Noetherian type by Spadaro.
NOTES AND INTRODUCTIONS to PCF THEORY
 Notes by Don Monk.
 Shelah's PCF theory and its applications, by Maxim Burke and Menachem Magidor.
 "A short proof of the PCF theorem", by Menachem Kojman.
 A historical survey on singular cardinals by Menachem Kojman.
 A video lecture on singular cardinal arithmetic by Menachem Kojman at the Institute for Advanced Study in Princeton.