Project Leaders: Adrián Barquero Sánchez and Nicolás Sirolli

Project Description:  The study of arithmetical functions has been a very active area of research in number theory for centuries.

Functions like the Euler function φ(n), the number of divisors function τ(n), the totient multiplicity function t(n), the Möbius function μ(n), just to cite a few examples, have been the object of intensive and varied investigations from countless researchers.

Following the norm in number theory, it is natural to seek for generalizations of their definitions and results both to particular and general number fields. For example, in 1874 Mertens gave an asymptotic formula for the averages of the generalization of the Euler function to the ring of Gaussian integers. Only much more recently, in 2020 Arango-Piñeros and Rojas gave a definition of the Euler function to arbitrary global fields; and using a tauberian theorem, they obtained an asymptotic formula for the averages of the corresponding totient multiplicity function.

The goal of this project is to seek for further generalizations for definitions and results for arithmetic functions over particular and general number fields, starting with the most elementary and accessible, and then moving to deeper results using recent advances in analytic number theory.

We intend to complement our work by taking advantage of the possibility of carrying extensive numerical computations with SageMath, in order to formulate conjectures and test results.

Statement of Equity, Diversity and Inclusion: We believe that mathematics is inherently diverse and thrives when shared across different communities. Its beauty and power grow through collaboration and varied perspectives.

Our goal is to facilitate a supportive and inclusive environment by providing a friendly working atmosphere in which everyone willing to participate will have its place. 

We recognize that equity is not guaranteed by simply giving the same chances to everyone. Instead, special attention must be put on those of us coming from disadvantageous contexts and we will take steps to ensure meaningful inclusion.

Project Leaders: Mathilde Gerbelli-Gauthier

Project Description:  Locally symmetric spaces S are smooth manifolds built from a lattice Γ in a Lie group G: they are the higher-dimensional analogues of modular curves. In this project, we will be interested in the singular cohomology H^*(S,C) when S is compact. These cohomology groups are very hard to compute, but their structure can nevertheless be investigated and they admit many symmetries. Following work of Matsushima and Vogan-Zuckerman, H^*(S,C) can be computed in terms of representations of G, which can be themselves studied combinatorially. Beginning with locally symmetric spaces associated to the group SLn, the goal of the project is to spell out explicitly the relationship between these combinatorics and the decompositions of the cohomology, with a further view towards understanding how these decompositions interact with the cup product.

Statement of Equity, Diversity and Inclusion: This project is a collective endeavor: it will be the outcome of our work, ideas, tastes, questions, and relationships with one another. Establishing these relationships of mathematical solidarity and comradeship is as important to me as the mathematics that we will produce.

It is essential that everyone can show up as their whole selves as a participant this project: this means making space so we all have the possibility to share our personal, cultural and mathematical backgrounds and tastes; express openly our mathematical and non-mathematical ideas, thoughts, and questionings; and set our boundaries and name tensions when they arise. I want the space to be joyful and enriching for all participants, while especially foregrounding the experiences and needs of those of us who are made to feel the least at home in the mathematical community at large. 

I commit to openness, curiosity, self-reflection, care, and enthusiasm. I invite everyone in my project group to do the same. 

Project Leaders: Kiran Kedlaya

Project Description:  It is an old theorem of Kronecker that an algebraic integer whose conjugates in the complex numbers all have absolute value at most 1 is a root of unity. In 1968, Cassels proved a classification theorem for cyclotomic algebraic integers (i.e., sums of roots of unity) whose conjugates in the complex numbers all have absolute value at most √5, aside from an effectively computable but unspecified finite exceptional set. Building on work of Robinson-Wurtz from 2013, we will try to finish the computation of the exceptional set.

Statement of Equity, Diversity and Inclusion: Just as a biological population needs genetic diversity in order to survive and benefit from the process of natural selection, the mathematical enterprise needs a diversity of perspectives in order to generate the solutions to our most challenging problems. Indeed, for most of recorded history mathematics was propelled forward by exchanges of ideas across times, places, and cultures. Unfortunately, in recent centuries this plurality has too often been replaced by a monoculture in which many practicing and prospective mathematicians do not see themselves represented. I see rolling back this monoculture as an urgent act of decolonization for the sake of maintaining progress in our field to the benefit of all of humanity.

Project Leaders: Kevin McGown

Project Description: Let R be an integral domain with field of fractions K. The associated ring of integer-valued polynomials Int(R) is the collection of polynomials f in K[x] such that f(R) is contained in R. In the case of R=Z, one knows that the polynomials f_n(x) defined by "x choose n" form a so-called regular basis for Int(Z). In the spirit of arithmetic statistics, one can ask the following question: Given a family of number fields, what proportion of fields K, with ring of integers R, have the property that Int(R) has a regular basis? This is related to the Polya-Ostrowski group of K. There are other similar or related problems that we could consider as well.

Statement of Equity, Diversity and Inclusion: Mathematics is a discipline in constant need of new and innovative ideas.  As such, bringing together groups across cultural and other boundaries is essential for mathematics to thrive.  All too often, ego and bias get in the way of collaboration.  My goal is to create a welcoming atmosphere where anyone with the interest can contribute to the project and have an enjoyable and meaningful mathematical experience.

Project Leaders: Holly Swisher and Stephanie Treneer

Project Description:  The partition function p(n) is known to have many interesting properties which can be studied from a variety of perspectives including combinatorics, power series, and automorphic forms. A 2013 conjecture of Sun posits that p(n) is not a perfect power of an integer for any n>=2. Motivated by Sun's conjecture as well as recent related investigations of Merca, Ono and Tsai, we will explore relationships between p(n) and perfect powers. This will involve computational experimentation, formulation of conjectures, and rigorous proof using techniques from analysis, combinatorics, or automorphic forms.

Statement of Equity, Diversity and Inclusion: We believe that mathematics belongs to all of us, and everyone who has a desire to engage with it belongs in mathematics. We have seen and experienced mathematical spaces that are not welcoming to all voices, and we reject the gatekeeping that has been a part of our chosen field for too long. We have selected a project topic that we hope will inspire creative exploration where people with varied mathematical backgrounds can engage and make contributions, and we can't wait to hear from all of you!

Project Leaders: Allison Beemer and Jessalyn Bolkema

Project Description:  A protocol for Private Information Retrieval (PIR) allows a user to download information from a server, or a set of servers, without revealing which pieces of data they are specifically interested in. Suppose, for example, you don’t want your library to learn anything about your reading habits, so you download 12 randomly selected books in addition to the one you really want to read. Of course, to be fully private, you should download everything your library has to offer. But that is a prohibitively large amount of data! A good PIR strategy aims to balance efficiency with privacy, maximizing the amount of needed information (books you want to read) retrieved per download (this is called the download rate) while successfully obscuring the user’s intentions (actual reading list). Extensions of PIR include allowing the user side information (e.g. already owning certain books on the reading list), multiple servers (e.g. belonging to multiple libraries), server collusion (e.g. library consortiums), and/or coded databases (alas, no good book analogy).

More technically, we wish to design information download schemes while ensuring that the mutual information between download requests and desired information is zero, and minimizing the download rate: the ratio of number of desired messages to total downloaded information. We assume stored and retrieved messages are vectors over a finite field, so ensuring privacy requires controlling discrete probability distributions over that field.

Our proposed project is to generalize and adapt existing schemes for PIR with side information to the coded and colluding server case, with the possibility of developing novel, low-download rate schemes for this setting. 

Statement of Equity, Diversity and Inclusion: We are committed to supportive and empathetic collaboration that values the people doing math more highly than the mathematics being done – in part, because we know that this is where the really good work happens. We know that mathematical ability and mathematical interest are not concentrated in any one group; every person is capable of learning, doing, and enjoying mathematics. We also acknowledge that mathematical culture is just one part of human culture, and as such, is loaded with intersecting systems of oppression that have explicitly and implicitly contradicted this message of inclusion. We want to contribute to the dismantling of these oppressive forces. Thus, we have twin aims for this collaboration: (1) key making: supporting anyone who wants to be a part of the mathematical community, and (2) gate destroying: continuing to remove barriers to full participation in mathematics.

Mentor: Sandra Nair

Statement of Equity, Diversity and Inclusion: I firmly believe that scientific potential and talent do not see factors that have traditionally divided society, such as race, gender, ethnicity, sexual orientation, caste, religion, language etc. However, the fostering, nurturing and active development of these talents in a welcoming environment is crucial for individuals to fully realize their potential, and thus contribute to the edifice of mankind's knowledge and progress. For centuries, such growth opportunities have been restricted to the privileged few by sheer luck of birth and enforced institutional biases informed by colonialism, resulting in some communities being more marginalized than others in the modern world. We cannot change the past, but that is no excuse to pretend it never happened, for doing so is a guarantee that the worst parts of human history will repeat.