There will be five research groups in 2023. Click on the titles to learn more about each project. Participants are only expected to have the listed prerequisite for the projects. In particular, it is not necessary to fully understand the project descriptions below to be eligible for our research experience. The first two weeks of the program are designed to introduce the background required to ensure participants' success in their projects.
Description
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.
Prerequisites
A course in Abstract Algebra or Number Theory.
References
Description
Phylogenetics is the study of the evolutionary relationships between organisms. One of the main challenges in the field is to infer an evolutionary tree from biological data for a given collection of organisms. In recent times, deeper biological knowledge about hybridization and gene flow has led to consider phylogenetic networks as an alternative representation of the evolutionary history of a collection of organisms. Developing effective methods for inferring phylogenetic trees or networks has lead to a number of interesting mathematical questions across a variety of fields including combinatorics and algebraic geometry.
In this project, we will introduce the basic concepts in phylogenetics from a mathematical point of view. Our goal is to understand the algebraic variety associated to a given phylogenetic network. Our particular project will center in understanding some algebraic invariants of these varieties (degree, dimension, Euclidean distance degree, maximum likelihood degree) for some concrete families of phylogenetic networks.
Prerequisites
A course in abstract algebra is preferred. Experience with proof-writing is expected.
References
Description
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, Charles Doran, Kevin Iga, Jordan Kostiuk, Greg Landweber and Stefan Méndez-Diez determined that Adinkras are a type of Dessin d'Enfant; they showed this by explicitly exhibiting a Belyi map as a composition \( \beta: S \to \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \). They computed the first arrow as a map from a certain compact connected Riemann surface \( S \) to the Riemann sphere \( \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R) \), and the second as a map which keeps track of the "coloring" of the edges.
Adinkras naturally have square faces. This keeps track of the non-commutative nature of the supersymmetric operators. While Dessin d'Enfants correspond to triangular tilings of Riemann surfaces, there is a similar construction -- called "origami" -- which correspond to square tilings. In this project, we attempt to discover how to express the construction of Doran, et al. as a composition \( \beta: S \to E(\mathbb C) \to \mathbb P^1(\mathbb C) \) for some elliptic curve elliptic curve \( E \) such that the map corresponds to an "origami", that is, a map which is branched over just one point. This project will not assume any prior knowledge of physics, group theory, or elliptic curves.
Prerequisites
Any proof-based course, such as Abstract Algebra, Discrete Math, or Number Theory; knowledge of Group Theory is preferred.
References
Description
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.
Prerequisites
A course in abstract algebra. Experience with proof-writing is expected.
References
Description
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.
Prerequisites
Linear algebra and a course in either number theory or abstract algebra.
References
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.