Sets that contain their circle centers

Say that a subset S of the plane is a "circle-center set" if S is not a subset of a line, and whenever we choose three noncollinear points from S, the center of the unique circle through those three points is also an element of S. A problem appearing on the Macalester College Problem of the Week website was to prove that a finite set of points in the plane, no three lying on a common line, cannot be a circle-center set. Various solutions to this problem that did not use the full strength of the hypotheses appeared, and the conjecture was subsequently made that every circle-center set is unbounded. In this article, we prove a stronger assertion, namely that every circle-center set is dense in the plane, or equivalently that the only closed circle-center set is the entire plane. Along the way we show connections between our geometrical method of proof and number theory, real analysis, and topology.Comment: 12 pages, 4 figure

Addressing the underrepresentation of women in mathematics conferences

Despite significant improvements over the last few generations, the discipline of mathematics still counts a disproportionately small number of women among its practitioners. These women are underrepresented as conference speakers, even more so than the underrepresentation of women among PhD-earners as a whole. This underrepresentation is the result of implicit biases present within all of us, which cause us (on average) to perceive and treat women and men differently and unfairly. These mutually reinforcing biases begin in primary school, remain active through university study, and continue to oppose women's careers through their effects on hiring, evaluation, awarding of prizes, and inclusion in journal editorial boards and conference organization committees. Underrepresentation of women as conference speakers is a symptom of these biases, but it also serves to perpetuate them; therefore, addressing the inequity at conferences is valuable and necessary for countering this underrepresentation. We describe in detail the biases against women in mathematics, knowing that greater awareness of them leads to a better ability to mitigate them. Finally, we make explicit suggestions for organizing conferences in ways that are equitable for female mathematicians.Comment: 26 pages. See also "An annotated bibliography of work related to gender in science" (arXiv:1412.4104