Date:

Wed, 27/12/201711:00-13:00

Location:

Ross 63

Abstract: The set theoretic generalizations of algebras have been

introduced in the 1960s to give a set theoretic interpretation of usual

algebraic structures. The shift in perspective from algebra to set

theory is that in set theory the focus is on the collection of possible

algebras and sub-algebras on specific cardinals rather than on

particular algebraic structures. The study of collections of algebras

and sub-algebras has generated many well-known problems in combinatorial

set theory (e.g., Chang’s conjecture and the existence of small singular

Jonsson cardinals).

In the 1990s Foreman and Magidor used algebras to initiate a study

of Singular Stationarity, i.e., a study of alternative notions of

stationarity for subsets of singular cardinals. They introduced and

developed two notions of singular stationarity called Mutual

Stationarity and Tight Stationarity, and used their findings to prove a

fundamental result concerning generalized nonstationary ideals.

The two notions of singular stationarity have been studied in the last

two decades, and the main purpose of the talk to describe the related

known and recent results.

I will start by giving some background material on algebras and

stationary sets, and describe the history of a well-known problem by

Jonsson. We will then proceed to describe the work done on the notion of

Mutually Stationary Sequences and sketch a recent proof which is based

on the existence of special types of strong filters.

In the second part of the talk, we will connect the notions of Singular

Stationarity to results from PCF theory and Extender-based forcing

methods.

introduced in the 1960s to give a set theoretic interpretation of usual

algebraic structures. The shift in perspective from algebra to set

theory is that in set theory the focus is on the collection of possible

algebras and sub-algebras on specific cardinals rather than on

particular algebraic structures. The study of collections of algebras

and sub-algebras has generated many well-known problems in combinatorial

set theory (e.g., Chang’s conjecture and the existence of small singular

Jonsson cardinals).

In the 1990s Foreman and Magidor used algebras to initiate a study

of Singular Stationarity, i.e., a study of alternative notions of

stationarity for subsets of singular cardinals. They introduced and

developed two notions of singular stationarity called Mutual

Stationarity and Tight Stationarity, and used their findings to prove a

fundamental result concerning generalized nonstationary ideals.

The two notions of singular stationarity have been studied in the last

two decades, and the main purpose of the talk to describe the related

known and recent results.

I will start by giving some background material on algebras and

stationary sets, and describe the history of a well-known problem by

Jonsson. We will then proceed to describe the work done on the notion of

Mutually Stationary Sequences and sketch a recent proof which is based

on the existence of special types of strong filters.

In the second part of the talk, we will connect the notions of Singular

Stationarity to results from PCF theory and Extender-based forcing

methods.