Set Theory Seminar, Fall 2009

 

Our seminars are on Tuesdays in the Math/Geo building, room 120, 2:40 - 3:30, unless otherwise indicated.

September 8
Masaru Kada
Galois-Tukey connection involving sets of metrics
        (joint work with Yasuo Yoshinobu)

Abstract: Woods proved that the Stone-Cech compactification of a metrizable space X is approximated by the collection of Smirnov compactifications for all

compatible metrics on X.  As an improvement of Woods' Theorem, Kada, Tomoyasu and Yoshinobu studied, for a metrizable space X, how many metrics we actually need to approximate the Stone-Cech compactification by Smirnov compactifications.  After that, Todorcevic told us that the answers to such cardinality questions should be naturally deduced from structural information of the set of compatible metrics on the space, and so we should investigate the structure, not just cardinalities, of the set of metrics.
I will present some recent results on this topic.  I will show that the structure of the set of compatible metrics of a separable metrizable space X is nicely characterized using the notion of generalized Galois-Tukey connection.


September 15 -  22
Masaru Kada
How to amplify a metric, and how to gauge the amplitude of a metric

Abstract: I will present a part of the proof of our theorem, which provides a sequence of morphisms (in the sense of Galois-Tukey connections) involving the order structure of the set of metrics on a locally compact separable metrizable space. The key idea of the proof is a method to "amplify" a fixed metric by an "amplitude" indicated by a strictly increasing function, and to "gauge the amplitude" of a given metric using the fixed metric as a base. I will try to explain the effectiveness of our assumption that our space is locally compact and separable, and give some idea how to apply our result for a separable (but not necessarily locally compact) metrizable space.