The 8-week virtual program ran from Monday, June 13 through Saturday, August 7 through Pomona College as the "Pomona Research in Mathematics Experience". It was sponsored by the National Security Agency (H98230-21-1-0015).

**PRiME 2021 Participants:** Edmond Anderson (Morehouse College), Tesfa Asmara (Pomona College), Alyssa Brasse (Hunter College), Nevin Etter (Washington and Lee University), Gustavo Flores (Carleton College), Aurora Hiveley (Macalester College), Erik Imathiu-Jones (California Institute of Technology), Maria Maalouf (California State University at Long Beach), Drew Miller (University of California at Santa Barbara), Cyna Nguyen (California State University at Long Beach), Isaac Robinson (Harvard University), Summer Soller (University of Utah), Sharon Spaulding (University of Connecticut), and Daniel Tedeschi (Grinnell College).

Led by Alex Barrios (Carleton College).

**Abstract:**TBA- [Project Summary] | [Final Poster] | [Final Presentation]

Led by Edray Goins (Pomona College).

**Abstract:**A Belyi map \( \beta: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C}) \) is a rational function with at most three critical values; we may assume these values are \( \{ 0, \, 1, \, \infty \} \). Replacing \( \mathbb{P}^1 \) with an elliptic curve \( E: \ y^2 = x^3 + A \, x + B \), there is a similar definition of a Belyi map \( \beta: E(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C}) \). Since \( E(\mathbb{C}) \simeq \mathbb T^2(\mathbb {R}) \) is a torus, we call \( (E, \beta) \) a Toroidal Belyi pair.There are many examples of Belyi maps \( \beta: E(\mathbb{C}) \to \mathbb P^1(\mathbb{C}) \) associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyi map of degree \( N \), the inverse image \( G = \beta^{-1} \bigl( \{ 0, \, 1, \, \infty \} \bigr) \) is a set of \( N \) elements which contains the critical points of the Belyi map. In this project, we investigate when \( G \) is contained in \( E(\mathbb{C})_{\text{tors}} \).

- [Project Summary] | [Final Poster] | [Final Presentation]

Led by Rachel Davis (University of Wisconsin at Madison).

**Abstract:**TBA- [Project Summary] | [Final Poster] | [Final Presentation]

The 6-week virtual program ran from Monday, June 15 through Friday, July 24 through Pomona College as the "Pomona Research in Mathematics Experience". It was sponsored by the National Science Foundation (DMS-1850909).

**PRiME 2020 Participants:** Malike Conteh (Pomona College), Nevin Etter (Washington and Lee University), Jonalyn A. Fair (Southern University and A&M College), Ramoi Hammond (University of Connecticut), Charles Hardnett (Vanderbilt University), Jasmine James (Central Washington University), Alexis Kelley (University of California at Merced), and Jayla Langford (Purdue University).

Led by Edray Goins (Pomona College) with Amy Oden (Pomona College) and Zakiya Jones (Pomona College) as research assistants.

**Abstract:**In 1997, Scott Williams (SUNY Buffalo) founded the website "Mathematicians of the African Diaspora," which has since become widely known as the MAD Pages. Williams built the site over the course of 11 years, creating over 1,000 pages by himself as a personal labor of love. The site features more than 700 African Americans in mathematics, computer science, and physics as a way to showcase the intellectual prowess of those from the Diaspora.Soon after Williams retired in 2008, Edray Goins (Pomona College), Donald King (Northeastern University), Asamoah Nkwanta (Morgan State University), and Weaver (Varsity Software) have been working since 2015 to update the Pages. Edray Goins led an REU of eight undergraduates during the summer of 2020 to write more biographies for the new MAD Pages.

In this talk, we discuss the results from Pomona Research in Mathematics Experience (PRiME), recalling some stories of the various biographies of previously unknown African American mathematical scientists, and reflecting on some of the challenges of running a math history REU.

- [Final Presentation] | [YouTube Video "Final Presentation"]

The 8-week residential program ran from Monday, June 9 through Friday, August 4 at Pomona College as the "Pomona Research in Mathematics Experience". It was sponsored by the National Science Foundation (DMS-1850909).

**PRiME 2019 Participants:** Myles Ashitey (Pomona College), Brian Bishop (Pomona College), Kendall Bowens (Tuskegee University), Alvaro Cornejo (University of California at Santa Barbara), Owen Ekblad (University of Michigan at Dearborn), Gabriel Flores (Wheaton College); Marietta Geist (Carleton College), Kayla Gibson (University of Iowa), Kayla Harrison (Eckerd College), Abby Loe (Carleton College), Tayler Fernandes Nuñez (Northeastern University), and Cameron Thomas (Morehouse College).

Led by Edray Goins (Pomona College) with Alia Curtis (Scripps College) as research assistant.

**Abstract:**A Belyĭ map \(\beta : \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C}) \) is a rational function with at most three critical values; we may assume these are \(\{0, 1, \infty\}\). A Dessin d'Enfant is a planar bipartite graph on the sphere obtained by considering the preimage of a path between two of these critical values, usually taken to be the line segment from 0 to 1. Such graphs can be drawn on the sphere by composing with stereographic projection: \(\beta^{-1} ([0,1]) \subseteq \mathbb{P}^1(\mathbb{C}) \simeq S^2(\mathbb{R})\). This project sought to either create or expand on a database of such Belyĭ pairs, their corresponding Dessins d'Enfant, and their monodromy groups. We did so for up to degree \(N = 5\) in the hopes of generating an algorithm to generate Dessins from monodromy triples.Arbitrarily choose loops \(\gamma\) around 0 and 1 in \(\mathbb{P}^1(\mathbb{C})\) that start and end at \(x_0\). Compute the paths that start at \(P_k\), where \(P_k\) is the \(k^{th}\) point that corresponds to the inverse image of \(x_0\). We refer to these paths as \(\widetilde{\gamma}\). Monodromy describes the movement of \(\widetilde{\gamma}\) and \(\gamma\) in correspondence to a Belyĭ map such that the endpoints of our path correspond to a \(\sigma_0\) and \(\sigma_1 \in S_N\) where \(\sigma_{\infty}\) such that \(\sigma_0 \circ \sigma_1 \circ \sigma_{\infty} = \mathbb{1}\) This project sought to simplify the concept of monodromy for a general audience in the form of an eight minute video. Our movie not only provides visualizations of monodromy on the Riemann Sphere but highlights monodromy's connection to Belyĭ maps and Dessin d'Enfant through real world examples.

- [Final Paper] | [Final Poster 1]| [Final Poster 2] | [YouTube Video "PrimeTime!"]

Led by Alex Barrios (Carleton College) with Tim McEldowney (University of California at Riverside) as research assistant.

**Abstract:**By a rational elliptic curve, we mean a projective variety of genus 1 that admits a Weierstrass model of the form \(y^2 = x^2+Ax+B\) where \(A\) and \(B\) are integers. For a rational elliptic curve \(E\), there is a unique quantity known as the minimal discriminant which has the property that it is the smallest integer (in absolute value) occurring in the \(Q\)-isomorphism class of \(E\). In 1975, Hellegouarch showed that for relatively prime integers \(a\) and \(b\) the elliptic curve \(y^2 = x(x+a)(x-b)\) comes equipped with an easily computable minimal discriminant. Recently, Barrios extended this result to all rational elliptic curves with non-trivial torsion subgroups. This project gives a classification of minimal discriminant for rational elliptic curves that admit an isogeny of degree \(N = 5, 6, 7, 8, 9, 13 \).- [Final Poster]

Program did not run.

The 8-week residential program ran from Monday, June 5 through Friday, July 28 at Purdue University as the "Purdue Research in Mathematics Experience". It was sponsored by the National Science Foundation (DMS-1560394).

**PRiME 2017 Participants:** Chineze Egbunike Christopher (Purdue University), Michael Cinkoske (Purdue University), Robert Julian Dicks (Emory University), Gina Marie Ferolito (Wellesley College), Joseph Jackson (Swarthmore College), Jacob Menix (Western Kentucky University), Joseph M. Sauder (Pontifical Catholic University of Puerto Rico), and Danika Keala Van Niel (Syracuse University).

Led by Edray Goins (Purdue University) with Abhishek Parab (Purdue University) as a research assistant.

**Abstract:**The research group led by Professor Goins studied "Belyĭ maps" (which are functions from the torus to the complex projective plane with three critical points) and their connection with bipartite graphs which are called "Dessin d'Enfant." The research in the summer of 2017 was a continuation of a research program that Professor Goins has been directing with undergraduate students for the past several years. The goals of the project during the 2017 REU were to (i) complete a database of Belyĭ pairs and the corresponding monodromy groups (or cartographic group) associated to the Dessin d'Enfant graph, (ii) compute the monodromy groups of compositions of Belyĭ maps, and (iii) characterize the monodromy groups for Dessin d'Enfants which are toroidal graphs.- [Project Summary] | [Final Poster (Christopher)]| [Final Poster (Dicks)] | [Final Poster (Ferolito)] | [Final Poster (Sauder)] | [Final Poster (Van Niel)] | [Final Presentation]

Led by Jonathan Peterson (Purdue University) with Zachary Letterhos (Purdue University) as a research assistant.

**Abstract:**An excited random walk is a non-Markovian extension of the simple random walk, in which the walk's behavior at time \( n \) is impacted by the path it has taken up to time \(n\). The properties of an excited random walk are more difficult to investigate than those of a simple random walk. For example, the limiting speed of an excited random walk is either zero or unknown depending on its initial conditions. While its limiting speed is unknown in most cases, the qualitative behavior of an excited random walk is largely determined by a parameter \(\delta\) which can be computed explicitly. Despite this, it is known that the limiting speed cannot be written as a function of \(\delta\). We offer a new proof of this fact, and use techniques from this proof to further investigate the relationship between \(\delta\) and limiting speed. We also generalize the standard excited random walk by introducing a "bias" to the right, and call this generalization an excited asymmetric random walk. Under certain initial conditions we are able to compute an explicit formula for the limiting speed of an excited asymmetric random walk.- [Final Paper] | [Final Poster]

The 8-week residential program ran from Monday, June 13 through Friday, August 5 at Purdue University as the "Purdue Research in Mathematics Experience". It was sponsored by the National Science Foundation (DMS-1560394) as well as generous gifts from Ruth and Joel Spira (BS '48, Physics).

**PRiME 2016 Participants:** Erin Bossen (Eastern Illinois University), Ivan Gonzalez (Florida International University), Jaime Dionel (University of Rochester), Brian Kidd (Purdue University), Owen Levin (University of Minnesota), Caitlin Lienkaemper (Harvey Mudd College), Gabriel Ngwe (Williams College), Baiming Qiao (Purdue University), Jacob Smith (Franklin College), and Kevin Stangl (UCLA).

Led by Edray Goins (Pomona College) with Mark Pengitore (Purdue University) as a research assistant.

**Abstract:**The research objective for the group led by Professor Goins was to study functions on the torus known as "Belyĭ maps" which are functions with at most three critical points taking values in the projective complex plane. Belyĭ maps can be visualized by drawing planar bipartite graphs which are called "Dessin d'Enfants." Since every elliptic curve can be identified with the torus, one calls the elliptic curve together with a Belyĭ map on the torus a "Belyĭ pair." The goals of the project were threefold: (1) to create a database of Belyĭ pairs, (2) to create a database of the corresponding Dessins d'Enfant, and (3) to create a database of the monodromy groups of Belyĭ pairs.- [Final Paper] | [Final Poster] | [Final Presentation]

Led by Jonathan Peterson (Purdue University) with Sung Won Ahn (Purdue University) as a research assistant.

**Abstract:**The research group led by Professor Peterson studied a self-interacting random walk model which is called an "excited random walk." Since this model of random motion is non-Markovian it is much more difficult to study than a classical random walk. Previous results in excited random walks had proved that such walks have a limiting speed, but unfortunately there is no explicit formula for the limiting speed of the walk. The primary objective of the research during the summer was to obtain the best possible upper and lower bounds to approximate the speed of the walk.- [Final Paper] | [Final Poster] | [Final Presentation]

The 8-week residential program ran from Monday, June 15 through Friday, August 7 at Purdue University as the "Purdue Research in Mathematics Experience". It was sponsored by the National Science Foundation (NSF) as well as generous gifts from Ruth and Joel Spira (BS '48, Physics).

**PRiME 2015 Participants:** Leonardo Azopardo (Purdue University), Sofia Lyrintzis (Purdue University), Bronz McDaniels (Purdue University), Maxim Millan (Purdue University), Yesid Sánchez (University of Puerto Rico at Rio Piedras), Danny Sweeney (Purdue University), and Sarah Thomaz (Purdue University).

Led by Edray Goins (Purdue University) with Hongshan Li (Purdue University) and Avi Steiner (Purdue University) as research assistants.

**Abstract:**With given Belyĭ maps and their corresponding elliptic curves, we can give a general description of their Dessins d'Enfants in 2 dimensions. We don't know, however, what these Dessins will look like when embedded on the torus, in 3 dimensions. Our goal is to create a program that will allow us to visualize these Dessins on the torus.- [Final Poster] | [Final Presentation]

The 8-week residential program ran from Monday, June 16 through Friday, August 8 at Purdue University as the "Purdue Research in Mathematics Experience". It was sponsored by the National Science Foundation (NSF) as well as generous gifts from Ruth and Joel Spira (BS '48, Physics).

**PRiME 2014 Participants:** Edwin Baeza (Purdue University), Luis Baeza (Purdue University), Conner Lawrence (Purdue University), and Chenkai Wang (Purdue University).

Led by Edray Goins (Purdue University) with Kevin Mugo (Purdue University) as a research assistant.

**Abstract:**Each finite, connected planar graph has an automorphism group \(G\); such permutations can be extended to automorphisms of the Riemann sphere \(S^2(\mathbb{R}) \simeq \mathbb{P}^1(\mathbb{C})\). In 1984, Alexander Grothendieck, inspired by a result of Gennadiĭ Belyĭ from 1979, constructed a finite, connected planar graph \(\Delta_\beta\) via certain rational functions \(\beta(z) = p(z)/q(z)\) by looking at the inverse image of the interval from 0 to 1. The automorphisms of such a graph can be identified with the Galois group \( \text{Aut}(\beta) \) of the associated rational function \(\beta:\mathbb{P}^1(\mathbb{C}) \rightarrow \mathbb{P}^1(\mathbb{C})\). In this project, we investigate how restrictive Grothendieck's concept of a Dessin d'Enfant is in generating all automorphisms of planar graphs. We discuss the rigid rotations of the Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron), the Archimedean solids, the Catalan solids, and the Johnson solids via explicit Belyĭ maps.- [Final Poster] | [Final Presentation]

The 8-week residential program ran from Monday, June 10 through Friday, August 2 at Purdue University as the "Purdue Research in Mathematics Experience". It was sponsored by the National Science Foundation (NSF) as well as generous gifts from Ruth and Joel Spira (BS '48, Physics) and Andris "Andy" Zoltners (MS '69, Mathematics).

**PRiME 2013 Participants:** Katrina Biele (University of Colorado at Colorado Springs), Kevin Bowman (Morehouse College), Sheena Chandra (Purdue University), Yuan Feng (University of Illinois at Urbana-Champaign), David Heras (College of William and Mary), Anji Li (Purdue University), Amanda Llewellyn (Harvey Mudd College), and Ahmed Tadde (Howard University).

Led by Edray Goins (Purdue University) with Andres Figuerola (Purdue University) as a research assistant.

**Abstract:**There are many examples of finite, connected planar graphs \( \Gamma \): for instance, there are paths, trees, cycles, webs, and prisms to name a few. In 1984, Alexander Grothendieck, inspired by a result of Gennadiĭ Belyĭ from 1979, constructed a finite, connected planar graph \( \Delta_\beta \) via certain rational functions \( \beta(z) = p(z) / q(z) \) by looking at the inverse image of the interval from 0 to 1. In this project, we investigate how restrictive Grothendieck's concept of a Dessin d'Enfant is in generating all planar graphs. We show that certain trees (such as stars, paths, and caterpillars), certain webs (such as cycles, dipoles, wheels, and prisms), and each of the Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron) can all be generated as Dessins d'Enfants by exhibiting explicit Belyĭ maps \( \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \).- [Final Presentation]

Led by Edray Goins (Purdue University) with Matt Toeniskoetter (Purdue University) as a research assistant.

**Abstract:**Each finite, connected planar graph has an automorphism group \( G \); such permutations can be extended to automorphisms of the Riemann sphere \( S^2(\mathbb R) \simeq \mathbb P^1(\mathbb C) \). In 1984, Alexander Grothendieck, inspired by a result of Gennadiĭ Belyĭ from 1979, constructed a finite, connected planar graph \( \Delta_\beta \) via certain rational functions \( \beta(z) = p(z) / q(z) \) by looking at the inverse image of the interval from 0 to 1. The automorphisms of such a graph can be identified with the Galois group \( \text{Aut}(\beta) \) of the associated rational function \( \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \). In this project, we investigate how restrictive Grothendieck's concept of a Dessin d'Enfant is in generating all automorphisms of planar graphs. We discuss the rigid rotations of the Platonic solids (the tetrahedron, cube, octahedron, icosahedron, and dodecahedron), the Archimedean solids, and the Catalan solids via explicit Belyĭ maps. Conversely, we enumerate groups of small order and discuss which groups can -- and cannot -- be realized as Galois groups of Belyĭ maps.- [Final Poster] | [Final Presentation]

The 8-week residential program ran from Monday, June 4 through Friday, August 3 at Purdue University as the "Purdue Research in Mathematics Experience". It was sponsored by the National Science Foundation (NSF) as well as generous gifts from Ruth and Joel Spira (BS '48, Physics) and Andris "Andy" Zoltners (MS '69, Mathematics).

**PRiME 2012 Participants:** Ronald Archer (Purdue University), Sergio García Currás (University of Puerto Rico at Rio Piedras), Han Liu (Purdue University), Benito Martínez (Purdue University), Stephen Mussmann (Purdue University), and Lirong Yuan (Purdue University).

Led by Edray Goins (Purdue University) with James Weigandt (Purdue University) as a research assistant.

**Abstract:**In 1640, Pierre de Fermat sent a letter to Bernard Frénicle de Bessy claiming that that there are no four or more rational squares in a nontrivial arithmetic progression; this statement was shown in a posthumous work by Leonhard Euler in 1780. In 1823, Adrien-Marie Legendre showed that there are no three or more rational cubes in a nontrivial arithmetic progression. A modern proof of either claim reduces to showing that certain elliptic curves have no rational points other than torsion.A 2009 paper by Enrique González-Jiménez and Jörn Steuding, extended by a 2010 paper by Alexander Diaz, Zachary Flores, and Markus Vasquez, discussed a generalization by looking at four squares in an arithmetic progression over quadratic extensions of the rational numbers. Similarly, a 2010 paper by Enrique González-Jiménez discussed a generalization by looking at three cubes in an arithmetic progression over quadratic extensions of the rational numbers. In this project, we give explicit examples of four squares and three cubes in arithmetic progressions, and recast many ideas by performing a complete 2-descent of quadratic twists of certain elliptic curves.

- [Final Paper] | [Final Poster] | [Final Presentation]

This material is based on work supported by the National Science Foundation under Grant No. DMS-2113782. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Contact edray.goins@pomona.edu for more information.