Mathematics

In its December 2019 edition, the \textit{Notices of the American Mathematical Society} published an essay critical of the use of diversity statements in academic hiring. The publication of this essay prompted many responses, including three public letters circulated within the mathematical sciences community. Each letter was signed by hundreds of people and was published online, also by the American Mathematical Society. We report on a study of the signatories' demographics, which we infer using a crowdsourcing approach. Letter A highlights diversity and social justice. The pool of signatories contains relatively more individuals inferred to be women and/or members of underrepresented ethnic groups. Moreover, this pool is diverse with respect to the levels of professional security and types of academic institutions represented. Letter B does not comment on diversity, but rather, asks for discussion and debate. This letter was signed by a strong majority of individuals inferred to be white men in professionally secure positions at highly research intensive universities. Letter C speaks out specifically against diversity statements, calling them "a mistake," and claiming that their usage during early stages of faculty hiring "diminishes mathematical achievement." Individuals who signed both Letters B and C, that is, signatories who both privilege debate and oppose diversity statements, are overwhelmingly inferred to be tenured white men at highly research intensive universities. Our empirical results are consistent with theories of power drawn from the social sciences.

Computer-based testing is an expanding use of technology offering advantages to teachers and students. We studied Calculus II classes for STEM majors using different testing modes. Three sections with 324 students employed: Paper-and-pencil testing, computer-based testing, and both. Computer tests gave immediate feedback, allowed multiple submissions, and pooling. Paper-and-pencil tests required work and explanation allowing inspection of high cognitive demand tasks. Each test mode used the strength of its method. Students were given the same lecture by the same instructor on the same day and the same homework assignments and due dates. The design is quasi-experimental, but students were not aware of the testing mode at registration. Two basic questions examined were: (1) Do paper-and-pencil and computer-based tests measure knowledge and skill in STEM Calculus II in a consistent manner? (2) How does the knowledge and skill gained by students in a fully computer-based Calculus II class compare to students in a class requiring pencil-and-paper tests and hence some paper-and-pencil work. These results indicate that computer-based tests are as consistent with paper-and-pencil tests as computer-based tests are with themselves. Results are also consistent with classes using paper-and-pencil tests having slightly better outcomes than fully computer-based classes using only computer assessments.

Configuration spaces form a rich class of topological objects which are not usually presented to an undergraduate audience. Our aim is to present configuration spaces in a manner accessible to the advanced undergraduate. We begin with a slight introduction to the topic before giving necessary background on algebraic topology. We then discuss configuration spaces of the Euclidean plane and the braid groups they give rise to. Lastly, we discuss configuration spaces of graphs and the various techniques which have been developed to pursue their study.

There are n bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in different bags weigh 1, 2, 3, and so on to n grams exactly. There is a unique label from the set 1 through n attached to each bag that is supposed to correspond to the weight of the coins in that bag. The task is to confirm all the labels by using a balance scale once. We study weighings that we call downhill: they use the numbers of coins from the bags that are in a decreasing order. We show the importance of such weighings. We find the smallest possible total weight of coins in a downhill weighing that confirms the labels on the bags. We also find bounds on the smallest number of coins needed for such a weighing.

Scientists use a mathematical subject called 'topology' to study the shapes of objects. An important part of topology is counting the numbers of pieces and holes in objects, and people use this information to group objects into different types. For example, a doughnut has the same number of holes and the same number of pieces as a teacup with one handle, but it is different from a ball. In studies that resemble activities like "connect the dots", scientists use ideas from topology to study the shape of data. Data can take many possible forms: a picture made of dots, a large collection of numbers from a scientific experiment, or something else. The approach in these studies is called 'topological data analysis', and it has been used to study the branching structures of veins in leaves, how people vote in elections, flight patterns in models of bird flocking, and more. Scientists can take data on the way veins branch on leaves and use topological data analysis to divide the leaves into different groups and discover patterns that may otherwise be hard to find.

The regular hendecagon is the polygon with the smallest number of sides that cannot be constructed by single-fold operations of origami on a square sheet of paper. This article shows its construction by using an operation that requires two simultaneous folds.

Given fixed distinct points A,B,C,D , we examine properties of the locus of points X for which (XA,XC) , (XB,XD) are isogonal. This locus is a cubic curve circumscribing ABCD . We characterize all possible such cubics C∈ R 2 . These properties allow us to present constructions involving these cubics, such as intersections and tangent lines, using straightedge and compass.

In the study of the real projective plane, harmonic conjugates have an essential role, with applications to projectivities, involutions, and polarity. The construction of a harmonic conjugate requires the selection of auxiliary elements; it must be verified, with an invariance theorem, that the result is independent of the choice of these auxiliary elements. A constructive proof of the invariance theorem is given here; the methods used follow principles put forward by Errett Bishop.

The purpose of this paper is to provide a historical overview of some of the contemporary infinitesimalist alternatives to the Cantor-Dedekind theory of continua. Among the theories we will consider are those that emerge from nonstandard analysis, nilpotent infinitesimalist approaches to portions of differential geometry and the theory of surreal numbers. Since these theories have roots in the algebraic, geometric and analytic infinitesimalist theories of the late nineteenth and early twentieth centuries, we will also provide overviews of the latter theories and some of their relations to the contemporary ones.

In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his 1821 book Cours d'Analyse, may have plagiarized Bolzano's book Rein analytischer Beweis (RB), first published in 1817. That paper was subsequently discredited in several works, but some of its assumptions still prevail today. In particular, it is usually considered that Cauchy did not develop his notion of the continuity of a function before Bolzano developed his in RB, and that both notions are essentially the same. We argue that both assumptions are incorrect, and that it is implausible that Cauchy's initial insight into that notion, which eventually evolved to an approach using infinitesimals, could have been borrowed from Bolzano's work. Furthermore, we account for Bolzano's interest in that notion and focus on his discussion of a definition by Kästner (in Section 183 of his 1766 book), which the former seems to have misrepresented at least partially. Cauchy's treatment of continuity goes back at least to his 1817 course summaries, refuting a key component of Grattan-Guinness' plagiarism hypothesis (that Cauchy may have lifted continuity from RB after reading it in a Paris library in 1818). We explore antecedents of Cauchy and Bolzano continuity in the writings of Kästner and earlier authors.