Project Leader: Barinder S. Banwait

Project Description:  With the development of the L-functions and modular forms database (🔗 LMFDB) has come the ability to apply machine learning (ML) techniques to large datasets of objects arising in number theory. The first instance of this was as recent as 4 years ago, when He, Lee, and Oliver [🔗 HLO20a] used ML to predict Sato-Tate groups of jacobians of genus 2 curves with impressive accuracy. In 2022, these authors together with Pozdynakov [🔗 HLOP22]  discovered a new phenomenon, now known as Murmurations of elliptic curves, that remains to be explained.

This project proposes another application of ML techniques to the dataset of rational elliptic curves in the LMFDB, with the concrete question: Can an ML classifier be trained to predict the order of the Shafarevich-Tate group of an elliptic curve over Q? Note that this was already attempted - without success - in a different paper of He, Lee, and Oliver [🔗 HLO20b].

This project is strongly related to a heuristic due to Park, Poonen, Voight, and Matchett-Wood [🔗 PPVMW18] that models the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and has the very striking prediction that, not only are ranks of elliptic curves uniformly bounded, but moreover that there are only finitely many elliptic curves of rank greater than 21. Another aspect of this project will therefore be to investigate whether the elliptic curve data in the LMFDB supports the PPVMW heuristics, as well as to consider ways in which ML could be applied to ranks and Shafarevich-Tate groups simultaneously.

Statement of Equity, Diversity and Inclusion: See Barinder's Diversity Statement 🔗here.

Project Leader: María Chara

Project Description: In the paper 🔗 'On cyclic algebraic-geometry codes', published in 2022, we initiated the study of cyclic algebraic geometry codes. We provided conditions for constructing cyclic algebraic geometry codes within the framework of algebraic function fields over a finite field, utilizing their group of automorphisms. Additionally, we demonstrated that cyclic algebraic geometry codes constructed in this manner were closely associated with cyclic extensions. Furthermore, we conducted a detailed examination of the monomial equivalence of cyclic algebraic geometry codes constructed using our approach, particularly in the context of rational function fields.

The aim of this project is to further explore the classification of cyclic AG codes based on equivalences. Investigating the number of inequivalent cyclic AG codes that can be constructed, in terms of monomial equivalence of linear codes, may yield interesting insights into the construction of sequences of cyclic AG codes. It is worth noting that the question of whether the family of cyclic codes is asymptotically good remains unresolved.

Statement of Equity, Diversity and Inclusion: I have participated in two previous editions of RNT, and now I'm both excited and a bit terrified to be the project leader in a new edition of RNT. This will be my first time as a project leader, and I am happy that it's here where I feel safe to be myself, and I hope that every participant will feel the same. I firmly believe that everyone, regardless of gender or background, should have equal opportunities to be part of a group and to feel free to ask questions or provide ideas. It is imperative that we create an inclusive environment where all individuals feel welcome and empowered to pursue their passion for mathematics.

Having said this, I have to admit that I don't have a strong idea of the problem I want to develop with the group. I think that depending on the participants, we can explore different (all great) paths, and that is an excellent way to do math.

Project Leader: Karol Koziol

Project Description:  The Iwasawa algebra of a profinite group is a (usually non-commutative) ring which plays a role similar to the group ring of a finite group.  Moreover, when we take our profinite group to be GL₃(Zp) (or a certain subgroup thereof), this ring acts on representations of the p-adic group GL₃(Qp), and therefore plays an important role in the Langlands program.  However, despite its straightforward definition, obtaining an explicit description that one can actually work with can be quite nontrivial.  The goal of this project is to find a clean, usable description of Fp⟦I₁⟧, the (mod p) Iwasawa algebra of I₁, where I₁ denotes the pro-p-Iwahori subgroup of GL₃(Zp).  In particular, we will try to explicitly compute a certain graded algebra associated to Fp⟦I₁⟧, and investigate certain ideals of particular importance to the mod p Langlands program.  

Statement of Equity, Diversity and Inclusion:  My goal is to create an environment where all individuals feel empowered to contribute, and more importantly, where the same opportunities for joyful collaboration are afforded to everyone.  Mathematics functions best when ideas can flow freely from person to person, without any obstructions based on biases.  Everyone has different backgrounds and experiences, and putting them together broadens everyone's knowledge.  Upholding these values of diversity, equity, and inclusion requires active effort and advocacy.  I feel very grateful to be able to contribute to the mission of RNT  and to help make a step in a positive direction.

Project Leader: Anand Patel

Project Description: Ernst Steinitz (1871 - 1928) first discovered that an extension of number fields L/k produces a canonical element St(L)  (now known as its Steinitz class) in the ideal class group Cl(k).  It is defined quite naturally: the ring of integers OL is a torsion-free finitely generated module over the ring of integers Ok, and all such modules over a Dedekind domain like Ok are the direct sum of an ideal in Ok with a bunch of copies of Ok.  The ideal's isomorphism class is St(L). 

Despite being natural,  many basic questions about Steinitz classes remain open.   For instance, once d ≥ 6 it is not known whether every element of Cl(k) is realizable as St(L) for some degree d extension L/k-- we will call this the Steinitz Realization Problem.  (Using geometry of numbers techniques available for d ≤ 5, Bhargava, Shankar, and Wang showed in 2015 that every  ideal class is realized – in fact each class is realized equally often.)

The objective of this RNT project is to (1) gain a solid understanding of the objects showing up in the problem and (2) to address its direct analog in the geometric setting where k is replaced by the function field of a nonsingular affine algebraic curve.  The geometric setting affords useful techniques not available in the number field setting,  and so this project is perfect for people who want more experience (or even a first experience) with algebraic geometry.  

Statement of Equity, Diversity and Inclusion: To me, the enterprise of mathematics is more about the communication and sharing of ideas among human beings, over and beyond the theorems themselves. (I'm channeling Thurston here.) From this point of view,  anyone who wants to join the conversation must be welcomed and included, and should be valued for their unique perspective.  I strive to foster this spirit in my interactions, and will make sure it comes across in this RNT project!

Project Leaders: Sameera Vemulapalli and Erik Holmes

Project Description:  Given a number field K/Q, one may consider the unit group U of the ring of integers of K. The group U is a finitely generated abelian group, and modulo its torsion subgroup, it can naturally be endowed with the structure of a lattice. The rank of this lattice is determined by the signature (the number of real and complex embeddings) of the number field. Very little is known about unit lattices, both theoretically and computationally. In this project, we focus on the following questions:

1. Which lattices arise from unit groups of number fields? (Bounds)

2. Suppose we enumerated number fields of fixed degree and fixed signature in some way; how are the corresponding unit lattices distributed? Does this distribution depend on the way we enumerate number fields? How does the distribution change when we restrict to special subfamilies of number fields? (Distributions)

In this RNT project, we will use the computer algebra systems Magma and Pari to collect data on unit lattices, plot them, and see what the distribution of unit lattices looks like. We will do a lot of computation, data analysis, and examples.

Sameera's Statement of Equity, Diversity and Inclusion: Discovering new mathematics, in an empowering atmosphere with wonderful collaborators, is a true joy. Unfortunately, many people are not always afforded this opportunity, and I know how much damage a toxic mathematical environment can cause. I strive to create an environment in this project where:

1. everyone's ideas, questions, and contributions are valued;

2. everyone is comfortable being themselves and speaking up; 

3. and everyone is learning something.

Erik's Statement of Equity, Diversity and Inclusion: Mathematics can be a vibrant and exciting field with endless opportunities for healthy and fruitful collaborations. It can also be an elitist, isolating, unfriendly environment which can drive people away for math. Creating and supporting healthy and welcoming communities is one of my primary goals in mathematics so I am thrilled to be a part of this iteration of RNT and the community as a whole.

Our goal for this project - which has many entry points, motivations, and directions - is to be accessible to all participants. Whether you have an extensive number theory background, or this is your first time exploring about the area, there will be something in it for you. As a project leader I aim to ensure a safe and productive collaboration where every participant feels comfortable asking questions, and sharing their thoughts, opinions and perspectives. I also hope that this project excites the participants as much as it does us, and that it leads to lasting collaborations.

Mentor: Deewang Bhamidipati

Statement of Equity, Diversity and Inclusion: Mathematics is as rich as the people who contribute to it; its community is as rich as those who are a part of it. No one should be denied the opportunity to find, define and practice what math is and means to them, and everyone must have the platform to express this. We all need to work towards expanding opportunities, engage in conversations, and actively debunk and work against popular notions propagated by gatekeepers. Inclusivity, diversity, and equity are indispensable and indisputable components of a program that must sweep the mathematics community to make it truly a human expression. As a member of this year's RNT program, I aim to channel my beliefs to help my fellow graduate students who participate in this program know that they belong and are welcome in this space.