Project Leader: Aly Deines

Project Description: The conductor and minimal discriminant are two easily computable invariants that measure the bad reduction of an elliptic curve. The conductor of an elliptic curve over a number field is an arithmetic invariant, that measures the ramification in a certain extension field. The minimal discriminant is a geometric invariant. It counts the number of irreducible components of the elliptic curve over finite fields. When two elliptic curves have the same conductor and discriminant, we call them discriminant twins. This project explores when discriminant twins occur over number fields. Over the rationals there are only finitely many semistable isogenous discriminant twins. What happens over number fields? We will be looking at this problem over totally real class number one number fields.

Statement of Equity, Diversity, and Inclusion: Mathematics is a beautiful field that everyone deserves to have a part in. It requires creativity and unique viewpoints that we only get through diversity within the field. Fostering inclusion gives a voice to these different ideas and approaches. I am committed to creating and hosting a research environment in which all individuals can thrive. To this end my project has multiple entry points for different levels of math and computational skills. Before this, I have organized many Women in Sage workshops, sat on panels that promoted women in mathematics, participated in other such programs, and am excited to lead a project at a workshop committed to inclusion and diversity and give back to the community that has helped me so much.

Project Leader: Andrew Kobin

Professional Website: https://www.andrewkobin.com/

Project Description: Zeta functions are miracles of efficiency: they compress an infinite amount of information about an object into a single function whose analytic properties often encode facets of the original object in novel ways. In this project, we will explore connections between various number theoretic and algebro-geometric zeta functions and seek to fit them into a more general framework. Namely, most classical versions of zeta functions can be realized as the generating series for a distinguished 'zeta element' of the incidence algebra associated to a discrete decomposition space. This suggests a number of interesting directions for generalization.

Statement of Equity, Diversity and Inclusion: The possibilities for inclusivity in mathematics are infinite, but at the end of the day, we all have a duty to promote equity and uplift the voices of our fellow mathematicians. In this workshop, I will demonstrate how a mathematician can improve their community. However, my views on this are heavily informed by my privilege as a white male, so my primary objective will be to create space for each group member to shine. To this end, our project will be based around the strengths of each contributor as a researcher, creative mind and organizer. At a mathematical level, the project will accommodate various levels of experience, with the goal that everyone who is interested can get involved in a meaningful way.

Project Leader: Robert Lemke Oliver

Professional Website: https://math.tufts.edu/faculty/rlemkeoliver/

Project Description: A plane curve is one cut by a single polynomial equation in the xy-plane. A theorem of Faltings implies that a simple geometric invariant of the curve controls whether there may be infintely many rational points on the curve: if the genus of the curve is at least two, then it possesses only finitely many rational points. There are essentially only two interesting questions one may ask about a finite set -- How big is it? Can we find all of its elements? -- so in an effort to ask more interesting questions, we might consider a larger set of points on the curve, namely the set of points defined over a finite extension of the rationals. Unfortunately, if the genus of the curve is at least two, Faltings' theorem still applies, showing that there are only finitely many points over any given extension of the rationals. It turns out, however, that the set of fields over which there is a "new" point on the curve is infinite. This opens the door to a broader range of questions. What kinds of fields occur in this set? Are there some kinds of fields that don't occur? How many fields are there of a given flavor and bounded complexity? To what extent does this set determine the curve or its geometry? The project leader's take on these questions is from arithmetic statistics, but no experience with arithmetic statistics is assumed, and other perspectives and other directions are welcome.

Statement of Equity, Diversity and Inclusion: Mathematics is a field that should benefit from having as many viewpoints as possible, but as anyone with experience attending conferences or applying for grants can attest, it is maddeningly cliquish. "Good" math is a social construct. Engaging or interesting math, while personal, should not be.

A stated goal of this workshop is not to have participants cry, but it's also important that applicants don't cry. Cliquishness and elitism too often play a role even in the selection procedure for these project groups. To disrupt that norm, as well as my constructed social hierarchy, I would like to work exclusively with people who are not already in my clique and who are not already on a self-identified elite track. Everyone should have a safe space to make and explore math that may be outside their wheelhouse.

The project I am proposing has many different facets, can proceed in many different directions, and can be entered by participants with many different backgrounds. Everyone will have space to contribute and learn, and everyone will be heard.

Project Leader: Beth Malmskog

Professional Website: http://malmskog.wordpress.com

Project Description: On the most basic level, good error-correcting codes are able to both transmit data efficiently and to correct a large number of errors relative to their length. Algebraic geometry codes based on algebraic curves with many points over some finite field are able to do very well in both of these categories, while providing the extra perk of computational speed that comes from working over a relatively small field. Innovations in modern computing continue to spur demand for codes with new features. Cloud computing has created a desire for locally recoverable codes, i.e. codes where any missing symbol can be recovered by consulting only a small set of "helper symbols", called the recovery set for the symbol. One way to obtain such codes is to use coverings of curves over finite fields. By combining many covering maps to create a fiber product of curves, we can obtain codes where each symbol has many recovery sets. In this project, we will seek to understand the trade-offs and possibilities of this construction, pursue bounds on the parameters of these codes, and generate new, potentially optimal, examples.

Statement of Equity, Diversity, and Inclusion: My best experiences in mathematics, including the work this project builds on, have involved the power of bringing together people from different viewpoints and backgrounds to work on a problem in a positive, humane, and respectful setting, where everyone is included and everyone’s contributions and personality are valued. That is what I will strive for in this project group. In my mathematical life, my primary goal is to use mathematics to bring people together and lift people up. Mathematics is for everyone. We all need to work to make sure that the diversity of the mathematical community reflects that fact, to know and value each other so that everyone feels included, and to make sure that opportunities for success and advancement are spread across the community in an equitable way. A workshop like this one, which is explicitly built around that premise, is an opportunity to help build the mathematical world we want to live in, and I’m so excited to have this chance to share a project that I love and learn alongside all of you.

Project Leader: Travis Morrison

Professional Website: travismo.github.io

Project Description: An efficient algorithm for computing the endomorphism ring of a supersingular elliptic curve would break several isogeny-based cryptosystems. SIKE, for example, is an alternate candidate in round 3 of NIST's post-quantum cryptography standardization effort. The security of SIKE relies on the hardness of pathfinding in isogeny graphs, and the pathfinding problem is equivalent to the endomorphism ring problem for supersingular elliptic curves. In this project, we will improve on state-of-the-art algorithms for computing endomorphism rings. Along the way, each participant will learn about the future of elliptic curve cryptography and gain familiarity with computer algebra systems like Sage and Magma, where we will implement our algorithms.

Statement of Equity, Diversity, and Inclusion: Mathematics is better with you in it. This project group will be better with you in it, too. We will create a welcoming and inclusive environment for each member: everyone will be involved and kept up to speed on the status and the direction of the project. There are plenty of footholds in this project for students with a variety of backgrounds. If you aren't familiar with elliptic curves, isogenies, or quaternion algebras, you will be by the end of this project! Your energy will be well-spent and your contributions will be appreciated. Finally, I acknowledge that as a white, heterosexual, cisgender male, I am afforded many privileges in the mathematical community. I'm committed to making this community a just and equitable one, and I'm excited to work toward that at RNT 2020.

Project Leader: Aaron Pollack

Professional Website: https://www.math.ucsd.edu/~apollack/

Project Description: A classical holomorphic cuspidal modular form has a completed standard L-function, that has finitely many (in fact, zero) poles and satisfies a functional equation. One corollary of the exact functional equation is the proof of the existence of trivial zeros of the L-function at negative integers. One can formulate a completely analogous question for certain automorphic forms on the exceptional group G_2: can one obtain sufficiently precise an understanding of their standard L-function so as to prove a precise functional equation and obtain its trivial zeros? While a lot of the theory is in place, due to interesting technical difficulties, the final desired result is not achieved. The goal of this project is to complete some of the theory of the standard L-function on G_2 so as to obtain its precise functional equation and the existence of trivial zeros. The main technique involved should be the Rankin-Selberg method. Also likely necessary are the arithmetic invariant theory of cubic rings, p-adic calculations for small p, and archimedean calculations.

Statement of Equity, Diversity, and Inclusion: Mathematics is a human endeavor. It requires a large collective amount of time and effort for progress to be made, and the results belong to us all. Correspondingly, the knowledge and techniques should be shared among all who are interested. My goal as a project leader is to share the specific knowledge and techniques that I have learned to everyone who is involved in our project. I am committed to ensuring that all participants in our project can meaningfully contribute and learn. Moreover, I hope everyone involved with our project will lend their own knowledge and experience to our common goal, so that I may learn from you.

Throughout my career, I have benefited from a lot of personal and professional privilege. However, too frequently, the knowledge of mathematics and the privilege to work on it are divided in non-equitable ways. As a participant in this workshop, I am delighted to be involved with “Rethinking Number Theory” so that I may be in a position to discuss mathematics and our shared community with my fellow mathematicians.