Department of Mathematics
The L space problem, the oscillation function,
and its applications.
Justin Moore
Boise State University
This talk will introduce the oscillation function and some of its applications in mathematics. A recent development in this area has been the use of this function to define a nonseparable topological space with no uncountable discrete subspaces (an L space), answering a problem asked by Hajnal and Juhasz in 1968. Examples of such spaces have long been known to be consistent with the usual axioms of mathematics (a Souslin Line is an L space), but this construction requires no additional axiomatic assumptions.
The L space also has an interpretation as a coloring a bipartite graph. In particular, there is an edge coloring of an uncountable complete bipartite graph with infinitely many colors so that all colors appear on any uncountable bipartite subgraph (here uncountable bipartite means that both "camps" of vertices in the graph are uncountable).
The oscillation function has also had application to the continuum problem. Forcing Axioms -- certain strengthenings of Baire's Category Theorem -- often impose considerable restrictions on the cardinality of the real line and frequently imply that it is the second uncountable cardinal. Two of these proofs make crucial use of the oscillation function and its properties.
All interested persons are welcome.