Colloquium: Peter Nyikos

PLEASE NOTE: LOCATION--Parker Hall 249

Speaker: Peter Nyikos, University of South Carolina

ABSTRACT: A quarter of a century after Peano caused a sensation with his space filling curves, the Hahn-Mazurkiewicz theorem gave a beautiful characterization of which spaces are a continuous image of the closed unit interval. They are precisely the compact, connected, locally connected, metrizable spaces. A problem that remained open for about six decades was to find a characterization of continuous images of orderable continua that was a natural generalization of this theorem. It was finally solved by Mary Ellen Rudin in 2000: all one needs is to change "metrizable" to "monotonically normal"; however, the proof was arguably the most difficult proof in all of point set topology, hence the long lag time.  Moreover, Rudin needed some exceptionally deep theorems of Treybig and Nikiel to complement her proof, which actually showed  that every compact monotonically normal space is the continuous image of an orderable compact space. This talk will review the highlights of this epic mathematical journey, and consider one minor improvement and one problem which, if answered affirmatively, would replace monotone normality with the much simpler and more general condition that every subspace is normal. However, this would require powerful axioms that are independent of the usual (ZFC) axioms of set theory.

Faculty host: Gary Gruenhage

Brief Description of the Speaker’s Academic and Professional Achievements/Credentials:

Peter Nyikos has been a leader in the field of set theoretic topology ever since he obtained his Ph.D. in 1971 at Carnegie-Mellon.  He has approximately 100 publications in refereed journals and many invited conference addresses.  He is on the editorial board of Topology and its Applications and has organized several AMS Special Sessions as well as a Spring Topology and Dynamics Conference.   A long-standing interest of his is the theory of non-metrizable manifolds.