For November, the *Monthly* departs from its usual format to highlight undergraduate research in mathematics with a special issue, guest edited by Scott Chapman and Joe Gallian. The seven articles span a range of subjects: Paul-Jean Cahen, Jean Chabert, and Kiran Kedlaya describe 1995 research of 2014 Fields Medalist Manjul Bhargava on *P*-orderings and generalized factorials; Colin Adams surveys contributions by undergraduates to knot theory; Cesar Silva likewise describes the work of various groups of undergraduate researchers in ergodic theory; Yufei Zhao reports on decades of results on independent sets in regular graphs, describing his own undergraduate research as well as that of other undergraduates and several well-known combinatorialists; Rishi Nath surveys results from several papers written by undergraduates about particular types of partitions of integers; Nathan Kaplan reports on research by four undergraduates on numerical semigroups; and Carlos Castillo-Chavez, Christopher Kribs, and Benjamin Morin describe Arizona State University’s Mathematical Theoretical Biology Institute and how undergraduate participants have applied mathematics to the life and social sciences in several research projects.

I hope that you find this issue as exciting as I do. My sincere thanks to Joe and Scott for putting it all together.

— *Susan Jane Colley, Editor*

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## Table of Contents

### From the Guest Editors

p. 771.

Scott T. Chapman and Joseph A. Gallian

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.771

### About the Cover

p. 772.

### Bhargava’s Early Work: The Genesis of *P*-orderings

p. 773.

Paul-Jean Cahen, Jean-Luc Chabert, and Kiran S. Kedlaya

As an undergraduate student, Manjul Bhargava gave a full answer to a question on polynomial functions on the integers. He immediately generalized this study to finite principal ideal rings, thanks to the amazingly simple notion of *P*-ordering. This tool, together with a beautiful generalization of factorials, allowed him to generalize many classical theorems. It turned out to be extremely useful for the study of integer-valued polynomials on subsets.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.773

### Turning Knots into Flowers and Related Undergraduate Research

p. 791.

Colin Adams

Knot theory is an excellent area for research by students. There are pretty pictures, plentiful opportunities to experiment, and deep mathematics. One recent area that presents many possibilities for research is a generalization of the usual projections of knots where at each crossing, two strands cross. At each crossing in an *n*-crossing projection, *n* strands cross. It turns out that every knot has an *n*-crossing projection and therefore an *n*-crossing number. Moreover, every knot has a projection with just one *n*-crossing. In fact, there is always such a projection that looks like a flower. We will discuss student work on these ideas and further possibilities for research.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.791

### On Mixing-Like Notions in Infinite Measure

p. 807.

Cesar E. Silva

Measurable dynamical systems are defined on a measure space, such as the unit interval or the real line, with a transformation or map acting on the space. After discussing dynamical properties for probability spaces such as ergodicity, weak mixing, and mixing, we consider analogs of mixing and weak mixing in infinite measure, and present related examples and definitions that are the result of research with undergraduates. Rank-one transformations are introduced and used to construct the main examples.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.807

### Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

p. 827.

Yufei Zhao

This survey concerns regular graphs that are extremal with respect to the number of independent sets and, more generally, graph homomorphisms. More precisely, in the family of of *d*-regular graphs, which graph *G* maximizes/minimizes the quantity *i*(*G*)^{1/v(G)}, the number of independent sets in *G* normalized exponentially by the size of *G*? What if *i*(*G*) is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.827

### Advances in the Theory of Cores and Simultaneous Core Partitions

p. 844.

Rishi Nath

The theory of *s*-core partitions, integer partitions whose hook sets avoid hooks of length *s*, lies at the intersection of a surprising number of fields, including number theory, combinatorics, and representation theory. A more recent trend has been to study partitions whose hook sets avoid multiple lengths, known as simultaneous core partitions. This paper, divided into five sections, is a review of five recent papers in this area by undergraduates. All of the authors surveyed conducted their research while participating in the University of Minnesota Duluth REU.

In the first section, we introduce partitions, the abacus, s-core partitions, and their connections to several fields. In the second section, we turn to self-conjugate *s*-core partitions and discuss several theorems of L. Alpoge on their asymptotic behavior and their connection, for small *s*, with points on curves. In the third section, we discuss simultaneous (*s*, *t*)-core partitions and the work of A. Aggarwal and V. Wang on the Armstrong conjecture. The fourth section highlights results of A. Aggarwal, A. Berger, and V. Wang on the enumeration, weight, and containment properties of simultaneous (*s*, *t*, *u*)-core partitions. In the final section, we mention some areas of ongoing research connected to the work discussed here.

The techniques used across these papers, ranging from generating functions and modular forms to more combinatorial tools such as abaci, posets, and lattice paths, give a flavor of the richness of the subject. We provide illustrative examples when full proofs are too lengthy.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.844

### Counting Numerical Semigroups

p. 862.

Nathan Kaplan

A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to *g*? We outline Zhai’s proof of a conjecture of Bras-Amorós that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems.We highlight the many contributions made by undergraduates to problems in this area.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.862

### Student-Driven Research at the Mathematical and Theoretical Biology Institute

p. 876.

Carlos Castillo-Chavez, Christopher Kribs, and Benjamin Morin

Highlighting the role of applying mathematics within the life and social sciences through research produced by undergraduate participants of Arizona State University’s (ASU) Mathematical and Theoretical Biology Institute (MTBI), we offer a brief account of what this community has learned from a research model that *deliberately* relinquishes the scientific agenda to its undergraduate participants. Over the past 21 years MTBI’s summer research program has focused on training a diverse population of student researchers in the art of identifying and investigating questions *primarily* at the interface of the mathematical, life, and social sciences. At the heart of this paper are insights gained from the research leadership of twenty-one cohorts of undergraduate students. We highlight three selected projects that capture MTBI’s philosophy of student-driven research, a model that has motivated hundreds of students to enroll in quantitative graduate programs across the nation. The first models *collaborative active learning* and its role in building robust communities of learners; the second studies the effects of oil spills on the spatial dynamics of loggerhead sea turtles; and the third develops a new, student-created version of pair-approximation modeling which has led to new research and a wide variety of applications.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.124.9.876

## MathBits

### Dihedral Angle of the Regular *n*-Simplex via Menelaus’ Theorem

p. 826.

### 100 Years Ago This Month in *The American Mathematical Monthly*

p. 892.