Project Leader: Brandon Alberts 

Project Description:  "Number Field Counting" refers to the area of arithmetic statistics that asks for the distributions of number fields belonging to certain families, which under the Galois correspondence is equivalent to asking for the distribution of surjective homomorphisms. Here "distribution" refers to an asymptotic question - count how many members of this family have discriminant bounded above by X, then ask how quickly that count grows as X is taken to infinity. A twisted version of this question has arisen as a new approach to number field counting -  let T a group with a Galois action. The twisted question asks for the distribution of crossed homomorphisms from the absolute Galois group to T.

Thus far, the twisted question has only been solved when T is abstractly given by an abelian group (that is, T is a Galois module) using cohomology. When T is abstractly given by a nonabelian group a lot of cohomological tools break down, requiring us to look at alternative directions. The goal of this project is to explore the twisted question in the nonabelian setting, to understand the players involved in the twisted question, understand the goal of studying this twisted question, and to work towards understanding small nonabelian examples. No cohomological background is required for this project.

Statement of Equity, Diversity and Inclusion: The most fun I have in mathematics is when I'm working with others. Mathematics has often been compared to a language, which speaks to its communal aspect - if you only ever talk to yourself, you miss out on fully experiencing the language. I am privileged to have found a community that welcomed me into mathematics, and therefore welcomed me to a deeper, shared experience of the subject we enjoy. Not everyone can say the same, because our communities developed amongst a long line of exclusionary practices. It takes intentional work to make our communities into places that all mathematicians can access, participate in, and enjoy. More important than some final product or paper, I am committed to making this project group and workshop into a community that welcomes you, welcomes your mathematics, and can be a place that you might find some of the joy that drew you to this subject.

Project Leader: Nathan Chen

Project Description:  There has been recent interest in studying "measures of irrationality" for complex projective varieties. These are geometric invariants which are designed to measure how far a variety is from complex projective space. In this project, the goal would be to investigate arithmetic analogues of these invariants, specifically for curves, products of curves, and Jacobians. A related question would be to construct examples of abelian varieties which happen to contain a "large amount" of low degree points, although it is unclear at the moment what this means. Do these low degree points have to geometrically come from somewhere?

Statement of Equity, Diversity and Inclusion:  I believe that the potential to carry out mathematical research is distributed equally among different groups. This is why having a space such as Rethinking Number Theory is so important. Promoting diversity and establishing a positive and welcoming atmosphere, especially one in which everyone is respectful and mindful of others' points of views, is an essential part of any discussions. Although the initial steps of the learning process may be a struggle, the goal will always be to explore new topics and learn from each other as much as possible. With this in mind, I am excited to participate in this community!

Project Leader: Henry Chimal-Dzul

Project Description: A main requirement of all modern digital communications systems is reliability. Reliability is achieved by the implementation of error correcting codes, which are the main object of study in Coding Theory. Linear codes are subspaces of the finite dimensional space F_q^n, where F_q is a finite field. These subspaces are abundant and various algebraic constructions have been proposed in the literature. Recently a novel algebraic construction has been introduced using faithful representations of finite dimensional group algebras. The aim of this research project is to further explore this novel construction from various directions and to derive properties of the codes obtained from them. Special emphasis will be devoted to the constructions of quasi-cyclic low density parity check codes and quasi-cyclic moderate density parity check codes. These types of codes are one of the most important classes of codes for error correction in modern digital communications and for practical use in post-quantum cryptosystems. The project does not require previous knowledge on error-correcting codes or representation theory, but we will asume some background on linear algebra.

Statement of Equity, Diversity and Inclusion:  I believe that research in mathematics is a collaborative and creative endeavor, independent of race, religion and mathematical background. My most memorable experiences have been collaborating with multicultural groups. Because of this, I support a diverse and inclusive community and I recognize the efforts of an space like Rethinking Number Theory where I can promote diversity and a welcoming and respectful atmosphere for doing research together. 

Project Leader: Jamie Juul and Bella Tobin

Project Description:  Dynamical irreducibility (also called stability) is a central concept in arithmetic dynamics. A polynomial is said to be dynamically irreducible if every iterate of the polynomial is irreducible. Recently, there has been significant progress on understanding dynamical irreducibility of polynomials over fields of odd characteristic. Much of this work was motivated by Boston and Jones, who, in 2012, determined necessary and sufficient conditions for a quadratic polynomial to be dynamically irreducible over a finite field. In particular, they show a quadratic polynomial is dynamically irreducible over Fq for q odd if and only if its adjusted critical orbit contains no squares. In 2014, Gómez-Pérez, Nicolás, Ostafe, and Sadornil extended these results to find necessary conditions for stability of any polynomials in any degree over fields of odd characteristic. In this project we will investigate dynamical irreducibility over finite fields for families of polynomials of degree larger than 2.

Statement of Equity, Diversity and Inclusion:  We are committed to providing an inclusive environment where all members of the community, regardless of mathematical background, are welcome and appreciated. We believe that all team members are valuable and each person's unique perspective is equally essential to the success of the group.

We will specifically work to facilitate opportunities for every member to contribute. We do not assume any background in arithmetic dynamics and will ensure that this is an opportunity for group members to learn through the research process.

We believe these project ideas will likely lead the group to a publication and potential ongoing collaboration within the arithmetic dynamics community.

Statement of Equity, Diversity and Inclusion:  I am excited to be a part of RNT-4. I have had the opportunity to work with mathematicians at different career stages and from varied backgrounds. I truly believe that mathematical talent/potential is equally distributed among all groups; what is most important to succeed, is the hunger to learn more! I believe in fairness, and I believe that diversity brings more ideas, more perspectives, and as a result leads to better mathematics. I look forward to being a part of this workshop and I hope that everyone participating in this event will become a stronger and happier mathematician.

Project Leader: Adriana Salerno and Ursula Whitcher

Project Description:  The zeta function of a variety encapsulates information about the number of points on that variety over fields of prime characteristic. Physics experiments on the zeta functions of certain threefolds have shown surprising relationships between zeta function structure, geometry, and the physics of black holes. The corresponding story for surfaces involves links to classical objects such as elliptic curves with complex multiplication. We'll seek special structure in zeta functions of quartic surface pencils. Possible techniques may be drawn from the mathematics of modular curves, hypergeometric functions, or K3 surfaces, as well as extensive computer experiments. No prior experience with K3 surfaces or black holes is required!

Statement of Equity, Diversity and Inclusion:  We are committed to doing mathematics in a way that respects the distinct experiences and identities of each group member and seeks a path toward justice. We recognize that engaging in research mathematics is both intellectually and emotionally demanding, and we hope to develop new resources together.