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Host: Miami University
Homepage: http://www.math.cmu.edu/~eschimme/Appalachian/Index.html
Description:
This workshop will be in three approximately equal-sized parts: one part ultrafilters, one part cardinal characteristics of the continuum, and one part connections between ultrafilters and cardinal characteristics. My (present and still subject to change) plans for the three parts are as follows.
In the first part, I would like to decribe several ways of thinking about ultrafilters, coming from combinatorics, topology, and model theory. Then I want to describe some of the connections between ultrafilters and partition theorems. Third, I'll briefly describe some of the structure of the world of ultrafilters --- the Rudin-Keisler ordering, limits, and tensor products. Finally, I'll describe some ultrafilter-related axioms, independent of ZFC, that have been useful in various contexts.
In the second part, I'll describe a few specific cardinal characteristics and connections between them. I'll explain some of the ways that one thinks about these, including Vojtas's ``generalized Galois-Tukey connections'' and Zapletal's ``tame'' characteristics. I'll also introduce the groupwise density number, a non-tame characteristic with rich connections to ultrafilters.
In the last part, I'll explore some of those connections, as well as other results that relate cardinal characteristics of the continuum to cardinals associated with ultrafilters, for example the minimal number of generators of an ultrafilter or the cofinality of an ultrapower. Some of those connections also involve the partition properties and other structural notions discussed in the first part.
Speakers: Andreas Blass: Ultrafilters and cardinal characteristics of the continuum
Date received: November 17, 2009
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