P-points were originally studied in the context of maximal ideals of rings of continuous functions on various topological spaces, but they have proven to be useful in various applications. Very early results showed that the P-points in the specific space $\beta \mathbb N\setminus \mathbb N$ have a very nice combinatorial characterization as a certain class of ultrafilters on $\mathbb N$. Further study has revealed various alternate characterizations, some of which will be examined in the first lecture. This introductory lecture will also discuss various methods of constructing P-points and related ultrafilters.
The second lecture will examine the forcing arguments needed to create models of set theory without any P-points, while the final lecture will discuss methods for killing some P-points while preserving others. Many questions in this are remain open and I will try to present some of them along with the context for asking them.
The first lecture will not assume any knowledge beyond elementary mathematics. The last two lectures will assume some more sophisticated knowledge of set theory, of the type that is usually covered in an introductory graduate course on set theory.
In the past decade, rapid progress has been made in the understanding of big Ramsey degrees, namely, how much infinite Ramsey theory do various countable, ultrahomogeneous structures satisfy? In particular, by work of Balko, Chodounský, Dobrinen, Hubička, Konečný, Vena, and myself, we now have a complete understanding of the situation for Fraïssé limits of finitely constrained binary free amalgamation classes, i.e. graph-like structures which forbid a specified finite list of clique-like substructures.
In the setting of small Ramsey degrees, works of Kechris--Pestov--Todorčević, Nguyen Van Thé, and myself connect this combinatorial property of a Fraïssé class to dynamical properties of the automorphism group of the Fraïssé limit. This tutorial will survey this connection, discuss recent progress in big Ramsey degrees, and explore the possible connections of big Ramsey degrees to dynamical phenomena.
|Jorge Antonio Cruz Chapital||On gaps, almost disjoint families and a ramsey ultrafilter.||abstract|
|Matheus Duzi Ferreira Costa||TBA|
|Jan Hubicka||Ramsey theorem for trees with sucessor operation|
|Jerzy Kąkol||Delta_1-spaces and Asplund spaces C_k(X) over Delta_1-spaces X|
|Maciej Korpalski||Straightening almost chains in P($\omega$)|
|Borisa Kuzeljevic||Orderings on P-point ultrafilters|
|Aleksandar Pavlović||Can an ideal give you a maximal space?|
|Tristan van der Vlugt|