Mary Ellen's conjectures

Abstract

For a special issue like this, it would seem to be very appropriate for us, the editors, to write an article summarizing Mary Ellen’s contributions to the field, and giving some account of her mathematical career. But there is an excellent article by Frank Tall [65] in the book The Work of Mary Ellen Rudin that does exactly this, covering her work and giving a sketch of her mathematical life up until shortly after her retirement in 1991. The article also includes a list of her students and their theses titles, her postdocs, and her publications. While she published a couple of papers on other topics after that time, she spent most of her post-retirement working on Nikiel’s conjecture, that the images of compact ordered spaces are exactly the compact monotonically normal spaces. She eventually solved it (positively) in 2001 [63]; it took one of the deepest and most difficult arguments of her career to do so. Instead of copying Frank’s article, we would like to do something a bit different. In the book Open Problems in Topology, published in 1990, Mary Ellen has an article [62] in which she states 17 “conjectures”. This is the way she chose to phrase some unsolved problems, most of them due originally to others, that she was particularly interested in. We quote from her article to give a sense of her reasons for choosing these particular problems: “The problems I list here have mostly been listed in Rudin [1988]; I have worked on all of them long enough to have some real respect for them; they have all stood the test of time and labor by various people; and I feel any solution to any of them would have applications at least to other problems exhibiting similar pathologies.” Here we go over Mary Ellen’s list of conjectures, giving a brief summary of her contributions and where each conjecture stands today. Many of these problems had or could be expected to have consistent answers. But Mary Ellen clearly had a bias for ZFC results, so instead of asking “Is it consistent that P holds?” she would ask “Is it true (in ZFC) that “not P” holds?”, or, as in this article, state “Conjecture: Not P holds.”.