PRiME (Pomona Research in Mathematics Experience)

Previous PRiME Projects

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PRiME 2023

The 8-week residential program ran from Sunday June 11 through Saturday August 5 through Pomona College as the "Pomona Research in Mathematics Experience". It was sponsored by the National Science Foundation (DMS-2113782).

PRiME 2023 Participants: Alex Abrams (Loyola Marymount University), Jewel Aho (University of St. Thomas), Tesfa Asmara (Pomona College), David Bonds (California State University at Los Angeles), Louis Burns (Pomona College), Arsh Chhabra (Pomona College), Elise Farr (Boston University), Galileo Fries (Colorado College), Akilah Goldson (Sarah Lawrence College), Xuehuai He (Pomona College), Julian Hutchins (Morehouse College), Stephen Hutchins (California State University at Stanislaus), Erik Imathiu-Jones (California Institute of Technology), Matilda LaFortune (Scripps College), Vuong Nguyen Hoang (Wingate University), Thea Nicholson (Xavier University), Elena O'Grady (Reed College), Eli Pregerson (Harvey Mudd College), Aniyah Stephen (Hartwick College), and Melinda Yang (Pomona College).

Group #1: Elliptic Curves with a Non-Trivial Isogeny

Led by Alex Barrios (University of St. Thomas) with Fabian Ramirez (University of California at Irvine) as research assistant.

• Abstract: Consider the implicit function \( E:y^2+ a_1 \, x \, y +a_3 \, y = x^3 + a_2 \, x^2 + a_4 \, x + a_6 \) where each \( a_i \in \mathbb{Q} \). We say that \( E \) is a rational elliptic curve if there is an additional point \( \mathcal{O} \) not on the graph of \( E \) and the discriminant \( \Delta \) of \( E \) is nonzero. We note that the condition that \( \Delta \neq 0 \) is equivalent to the existence of a unique tangent line at each point \( (x, y) \) on the graph of \( E \). Elliptic curves are rich with algebraic structure. For instance, the set of rational points over a field forms an abelian group. Elliptic curves that are separated by a surjective group homomorphism are known as isogenous elliptic curves. We call the group homomorphism an isogeny, and we say that the isogeny is cyclic of degree \( n \) if the kernel of the group homomorphism is cyclic of order \( n \).

  Elliptic curves that admit a cyclic isogeny of degree \( n \) are parameterizable. In this project, we will consider various parameterized families of isogenous elliptic curves - in particular, proving results for these families of elliptic curves yields with a non-trivial isogeny. In this project, we will consider elliptic curves over \( \mathbb{Q} \) and tackle the question of determining minimal models for elliptic curves with a non-trivial isogeny. These are equations of elliptic curves for which each \( a_i \in \mathbb{Z} \) and \( \left\vert \Delta\right\vert \) satisfies the property that it is the smallest over all \( \mathbb{Q} \)-isomorphic elliptic curves. Determining such models is a first step in calculating important information about an elliptic curve. This project builds on previous work by the PRiME 2019 and 2021 cohorts.

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

Group #2: The Geometry of Small Chemical Reaction Networks

Led by Luis David Garcia Puente (Colorado College) with Mark Curiel (University of Hawai'i at Manoa) as research assistant.

• Abstract: The study of chemical reaction networks forms a vital part of systems biology. In this project, we focus on chemical reaction networks with mass action kinetics for which the reaction dynamics are given by polynomial equations. This restriction allow us to apply methods from algebraic geometry to study several features of chemical reaction networks. In particular, we will analyze the algebraic complexity of the optimization problem which seeks to find the nearest point in the steady state algebraic variety to a given to a given data point.

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

Group #3: Adinkras as Origami

Led by Edray Goins (Pomona College) with Cameron Thomas (University of Georgia at Athens) as research assistant.

• Abstract: Around 20 years ago, physicists Michael Faux and Jim Gates invented Adinkras as a way to better understand Supersymmetry. These are bipartite graphs whose vertices represent bosons and fermions, and whose edges represent operators which relate the particles. Recently, Doran et al. determined that Adinkras are a type of Dessin d'Enfant by explicitly exhibiting a Belyi map as a composition \( \beta: S \to \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \). We are interested in exhibiting the same Belyi map as a different composition \( \beta: S \to E(\mathbb C) \to \mathbb P^1(\mathbb C) \).

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

Group #4: Trace Ideals over Semigroup Rings

Led by Haydee Lindo (Harvey Mudd College) with Olivia Del Guercio (Rice University) as research assistant.

• Abstract: The well-known trace map on matrices can be generalized to a map on any module over a commutative ring. The image of such a map is a trace ideal. In particular, given a ring \( R \) the trace ideal of an \( R \)-module \( M \) is the ideal generated by the homomorphic images of \( M \) in \( R \). There has been a recent uptick in the study of trace ideals within commutative algebra, often taking advantage of particular ring settings to aid in the calculations and applications of trace ideals. In this project we will investigate trace ideals in semigroup rings, highly structured rings with interesting number-theoretic and algebro-geometric properties.

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

Group #5: Abelian Extensions Arising from Torsion Points of Elliptic Curves

Led by Lori D. Watson (Trinity College) with Japheth Varlack (Wake Forest University) as research assistant.

• Abstract:If \(K\) is a number field and \(E\) is an elliptic curve defined over \(K\), we can obtain a Galois extension \(K(E[n])/K\) of \(K\) by adjoining all of the points on \(E\) of order dividing \(n\). In general these extensions are nonabelian; in fact, if \(E/K\) is an elliptic curve without complex multiplication, then for all sufficiently large primes \(p\), the Galois group of \(K(E[p])/K\) is isomorphic to \(GL_2( \mathbb Z/p \mathbb Z)\). Nevertheless, abelian extensions \(K(E[n])/K\) do occur. In 2015, Gonzalez-Jimenez and Lozano-Robledo determined when \( \mathbb Q(E[n])/\mathbb Q\) is an abelian extension. In this project, we will begin to investigate which abelian extensions \(K(E[n])/K\) can arise when \(K\) is a real quadratic field.

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

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PRiME 2022

The 8-week residential program ran from Sunday, June 12 through Saturday, August 6 through Pomona College as the "Pomona Research in Mathematics Experience". It was sponsored by the National Science Foundation (DMS-2113782).

PRiME 2022 Participants: Quincy Alston (University of Pennsylvania), Elise Alvarez-Salazar (University of California at Santa Barbara), Zoe Batterman (Pomona College), Ines Chung-Halpern (Yale University), Malike Conteh (Pomona College), Michaela Fitzgerald (Stonehill College), Ricardo Garcia (California Institute of Technology), Calvin "Barry" Henaku (Washington University of St. Louis), Mehek Mehra (Harvey Mudd College), Gesa Nestler (University of Tennessee at Knoxville), Joelle Ocheltree (Hartwick College), Ebenezer Semere (Pomona College), Jack Straus (College of William and Mary), and Sarah Szafranski (University of Redlands).

Group #1: Good Elliptic Curves

Led by Alex Barrios (Carleton College) with Summer Soller (University of Utah) as research assistant.

• Abstract: By an \( ABC \) triple, we mean a triple \( P=(a,b,c) \) such that \( a \), \(b \), and \( c \) are relatively prime positive integers with \( a+b=c \). The \( ABC \) Conjecture asserts that for each \( \epsilon>0 \), there are finitely many \( ABC \) triples \( P=(a,b,c) \) such that \( \text{rad}(abc)^{1+\epsilon} < c \). Here, \( \text{rad}(n) \) denotes the product of the distinct primes dividing \( n \). An \( ABC \) triple \( P=(a,b,c) \) is said to be good if \( \text{rad}(abc) < c \). For instance, the \( ABC \) triple \( (1,8,9) \) is good since \( \text{rad}(1\cdot 8\cdot9) = 6 < 9 \). More generally, we have that for each positive integer \( k \), the \( ABC \) triple \( \left( 1,9^{k}-1,9^{k}\right) \) is good. This showcases the necessity of \( \epsilon>0 \) in the statement of the \( ABC \) Conjecture.

  It turns out that the \( ABC \) Conjecture is equivalent to a statement about elliptic curves! This equivalence leads to a dictionary between the realms of \( ABC \) triples and elliptic curves. In particular, there is a notion of good elliptic curves. This project will investigate the interplay between good \( ABC \) triples and good elliptic curves, with the goal of constructing new infinite sequences of good elliptic curves.

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

Group #2: Galois Groups of Fields Generated by Points of Curves

Led by Renee Bell (University of Pennsylvania) with Kayla Gibson (Rutgers University) as research assistant.

• Abstract: There is a deep link between the geometry of a curve defined by a polynomial equation and solutions to the equation over the rational numbers, wonderfully illustrated by Faltings's theorem, which relates the amount of rational solutions to the genus of the curve. This result naturally invites one to consider the set of solutions over a finite extension of the rational numbers, and which field extensions give new points on the curve. Along these lines, Mazur and Rubin outlined an approach to the study of a curve by understanding the field extensions of \( \mathbb Q \) generated by a single point on that curve. We may ask which field extensions arise this way, what their Galois groups can be, how many there are up to bounded complexity, and how all these questions relate to the geometry of the curve. This project will build on results of Lemke Oliver and Thorne for elliptic curves, exploring these questions for different families of plane curves, starting with explicit computations of examples and using tools such as geometry of numbers, Hilbert irreducibility, Newton polygons, and linear optimization.

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

Group #3: Automorphism Groups of Partially Ordered Sets

Led by Cory Colbert (Washington and Lee University) with Rodrigo Smith (Clemson University) as research assistant.

• Abstract: In 2020, Johnathan Barmak proved that every finite group \(G\) is the automorphism group of a poset \(P\) with \(4 \, |G|\) points. In his paper, he also discusses the least positive number of points in a poset \(P\) needed to realize a group \(G\), denoted as \( \beta(G) \). For our research, we proved that every finitely generated abelian group is the automorphism group of some poset. We also investigated the \( \beta(S_n) \) and \( \beta(Z_{p^k}) \). Lastly, we developed code that determines the automorphism group of a given finite poset.

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

Group #4: Monodromy Groups of Belyi Lattes Maps

Led by Edray Goins (Pomona College) with John Michael Clark (University of Texas at Austin) as research assistant.

• Abstract: An elliptic curve \( E: y^2 + a_1 \, x \, y + a_3 \, y = x^3 + a_2 \, x^2 + a_1 \, x + a_6 \) is a cubic equation which has two curious properties: (1) the curve is nonsingular, so that we can draw tangent lines to every point \( P = (x,y) \) on the curve; and (2) the collection of complex points, namely \( E(\mathbb C) \), forms an abelian group under a certain binary operation \( \bigoplus: E(\mathbb C) \times E(\mathbb C) \to E(\mathbb C) \). In particular, for every positive integer \( n \), the map \( P \mapsto [n] P \) which adds a point \( P \in E(\mathbb C) \) to itself \( n \) times is a group homomorphism.
  A rational map \( f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \) from the Riemann Sphere to itself is said to be a Lattes Map if there are "well-behaved" maps \( \beta: E(\mathbb C) \to \mathbb P^1(\mathbb C) \) and \( L: E(\mathbb C) \to E(\mathbb C) \) such that \( f \circ \beta = \beta \circ L \). We are interested in those Lattes Maps \( f \) which are also Belyi Maps, that is, the only critical values are \( 0 \), \( 1 \), and \( \infty \). Work of Zeytin classifies all such maps: For example, if \( E: y^2 = x^3 + 1 \) then \( \beta: (x,y) \mapsto (y+1)/2 \) while \( L = [n] \) for some positive integer \( n \). In this project, we are interested in the corresponding Belyi Lattes Map \( f_n \). What can we say about such maps? What are their Dessin d'Enfants? Preliminary work shows that this should be a bipartite graph with \( 3 \, n^2 \) vertices. What are their monodromy groups? Preliminary work shows this should be a group of size \( 3 \, n^2 \).
  Is there something deeper going on here?

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

Group #5: Valuation Trees and Primitive Prime Divisors

Led by Bianca Thompson (Westminster College) with Jasmine Camero (Emory University) as research assistant.

• Abstract: A sequence is said to have primitive prime divisors if each term after a certain point always has a new prime divisor. Understanding primitive prime divisors is a step in understanding the backwards orbit of a function. Valuation trees help us visualize the prime structure for a sequence for specific primes. A valuation tree classifies integers via their prime divisibility and organizes the information to better understand how primes might be growing in the sequence. Is there a way to combine information about valuation trees over all primes \( p \) in order to understand the primitive prime divisors of a sequence? Understanding backwards orbits of rational maps are of interest in number theory, in particular rational maps where at least two critical points collide in their forward orbit. Thus, can we extend results on valuation trees created for polynomials to rational maps in order to understand their structure?

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

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PRiME 2021

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).

Group #1: Modular Curves and Elliptic Curves

Led by Alex Barrios (Carleton College).

• Abstract: TBA

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

Group #2: Critical Points of Toroidal Belyi Maps

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]

Group #3: Monodromy of Compositions of Toroidal Belyi Maps

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

• Abstract: Say that \( \beta: \mathbb{P}^1(\mathbb{C}) \rightarrow \mathbb{P}^1(\mathbb{C}) \) is a Dynamical Belyi map. Given any Toroidal Belyi map \( \gamma: E(\mathbb{C}) \rightarrow \mathbb{P}^1(\mathbb{C}) \), the composition \( \beta \circ \gamma: E(\mathbb{C}) \rightarrow \mathbb{P}^1(\mathbb{C}) \rightarrow \mathbb{P}^1(\mathbb{C})\) is also a Toroidal Belyi map. There is a group \( \text{Mon}(\beta) \), the monodromy group, which contains information about the symmetries of a Belyi map \( \beta \). It is well-known that, for any Toroidal Belyi map \( \gamma \), (i) there is always a surjective group homomorphism \( \text{Mon}(\beta \circ \gamma) \twoheadrightarrow \text{Mon}(\beta) \), and (ii) the monodromy group \( \text{Mon}(\beta \circ \gamma) \) is contained in the wreath product \( \text{Mon}(\gamma) \wr \text{Mon}(\beta) \). In this project, we study how the three groups \( \text{Mon}(\beta) \), \( \text{Mon}(\beta \circ \gamma) \), and \( \text{Mon}(\gamma) \wr \text{Mon}(\beta) \) compare as we vary over Dynamical Belyi maps \( \beta \).

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

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PRiME 2020

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).

Research Project: The Black Mathematician Chronicles: Our Quest to Update the MAD Pages

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"]

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PRiME 2019

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).

Group #1: Towards a Database of Belyĭ Maps

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!"]

Group #2: Minimal Discriminants of Rational Elliptic Curves

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 \(\mathbb Q\)-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] | [Final Presentation]

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PRiME 2018

Program did not run.

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PRiME 2017

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).

Group #1: Cartographic Groups of Regular Toroidal Graphs

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]

Group #2: On the Speed of an Excited Asymmetric Random Walk

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]

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PRiME 2016

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).

Group #1: Toroidal Belyĭ Pairs and their Monodromy Groups

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]

Group #2: On the Speed of an Excited Asymmetric Random Walk

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]

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PRiME 2015

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).

Research Project: Visualizing Dessins d'Enfants on the Torus

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]

[Speaker Series Poster]

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PRiME 2014

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).

Research Project: Associating Finite Groups with Dessins d'Enfants

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]

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PRiME 2013

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).

Group #1: Drawing Planar Graphs via Dessins d'Enfants

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]

Group #2: Associating Finite Groups to Dessins D'Enfants

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]

[Advertising Poster] | [Speaker Series Poster]

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PRiME 2012

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).

Research Project: Arithmetic Progressions of Squares and Cubes over Quadratic Fields

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]

[Speaker Series Poster]