Topology Workshop

Workshop Schedule:

Tuesday, September 18th 
9:30 -10 am Paul Gartside
10 -11 am  Roy Shalev
11 -11:15 am coffee break 
11:15 -11:45 am Paul Gartside
11:45 am -12:30 pm Problems
12:30 -1:30 pm Lunch
1:30 -2:30 pm Thomas Gilton
2:30 - 2:45 pm coffee break 
2:45 - 3:30 pm Problems
Workshop Dinner

Wednesday, September 19th
9:30 -10 am Paul Gartside
10- 11 am Todd Eisworth
11 - 11:15 am coffee break 
11:15 -11:45 am Paul Gartside
11:45 am -12:30 pm Problems
12:30 - 1:30 pm Lunch
1:30 - 2:30 pm James Cummings
2:30 - 2:45 pm coffee break 
2:45 - 3:30 pm Problems

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Abstracts: 

Roy Shalev

Title: Cofinal types below $\aleph_\omega$

Abstract: It is proved that for every positive integer $n$, the number of non-Tukey-equivalent directed sets of cardinality $\leq \aleph_n$ is at least $c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the Tukey class of directed sets of cardinality $\leq \aleph_n$ contains an isomorphic copy of the poset of Dyck $(n+2)$-paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

Todd Eisworth

Title: A compactum with no Bernstein decomposition

Abstract: A Bernstein decomposition of a space $X$ is a partition of the space into two pieces, neither of which contains a copy of the Cantor set.  Bernstein’s classic argument shows that the real line has a Bernstein decomposition, and results of Weiss and others show that very weak versions of square suffice to push an induction through all spaces of larger cardinality. Shelah proved the consistency (assuming large cardinals) of a regular space $X$ of cardinality $\aleph_{\omega+1}$ with the property that every partition of the space into countably many pieces contains a (closed) subspace homeomorphic to the Cantor space $2^{\omega}$.  His proof (once understood) has a lot of flexibility and we show how to obtain the consistency of a zero-dimensional example with even stronger partition properties that are inherited by any compactification of the space.   The proof is connected to pcf theory through its use of strongly almost disjoint families with strong freeness properties.

James Cummings

Title: Prikry forcing, scales, and linear orderings

Abstract: I will discuss some results and problems concerning compactness and minimality in some classes of linear orderings.

Paul Gartside

Title: The Tukey Order of Relations

 Abstract: I will discuss the origins of the Tukey order on directed sets and relations. Connect them to fundamental questions in topology. Connect them  to PCF theory and Shelah's colorings (Pr_0 etc). And ask a lot of questions...

Thomas Gilton

Title: The Cofinality of Generating Families

Abstract: In this talk, we'll discuss work of Paul Gartside and the speaker on cofinal sets related to how much information is needed to capture the topology of subspaces of $\R$. We'll focus on compact covers and on generating sets of sequences and of compact sets. In each case, by bringing Tukey theory to bear, we can characterize these generating families by using certain cardinal invariants. As such, they become amenable to set-theoretic (PCF in particular) techniques. We will close by discussing how to create models with plenty of spaces of a fixed compact covering number, but with varied values of the $k$-ness number (how many compact sets are needed to generate the topology).