Number Theory

Drinfeld modules are the analogues of elliptic curves in positive characteristic. They are essential objects in number theory for studying function fields. They do not have points, in the traditional sense—we're going to count them anyway! The first methods achieving this were inspired by classical elliptic curve results; we will instead explore an algorithm based on so-called Anderson motives that achieves greater generality. Joint work with Xavier Caruso.

In number theory, we often consider a generalization of integers called algebraic numbers. Their definition is rather elementary, but their classification is nothing but. Algebraic numbers come in families, and we can attach each family an invariant measuring its size: the castle. Kronecker proved that an algebraic integer with castle strictly less than one is zero, and that an algebraic integer with castle exactly one is a root of unity. The classification of algebraic numbers with castle less than a prescribed constant is technical, but we managed to derive it for cyclotomic integers (a subclass of algebraic numbers) with castle less than 5.01, solving a conjecture of R. M. Robinson opened in 1965.

I will state our result, and rather than focus on the technical details, present the methodology that lead us to it. Indeed, this collaboration was initiated at the Rethinking Number Theory workshop: members from various career stages work in groups under the guidance of a project leader. The workshop organizers make it so that participants work with joy, autonomy and open-mindness. This allowed each of us to contribute to what we were best at. Joint work with J. Bajpai, S. Das, K. S. Kedlaya, N. H. Le, M. Lee and J. Mello; https://arxiv.org/abs/2510.20435.

A vanishing sum of roots of unity (VSRU) is a finite list $z_1,\ldots,z_K$ of $N$-th complex roots of unity whose sum is zero. While there are many simple examples—including the famous "beautiful equation" of Euler, $e^{i \pi} + 1 = 0$—such sums become extremely complex as the parameter $N$ attains more complex prime power divisors (and we will see several classical examples illustrating this idea, as well as new examples from my work).

One fruitful line of inquiry is to seek a quantitative relationship between the prime divisors of $N$, their associated exponents, and the cardinality parameter $K$. A theorem of T.Y. Lam and K.H. Leung from the early '90's states: $K$ must always be (at least) as large as the smallest prime dividing $N$. This generalizes the well known observation that that sum of all $p$-th roots of unity (where $p$ is any prime number) must vanish; and, one notices that Euler's equation is one example of this fact.

In this talk, we will discuss two significant strengthenings of this result (one due to myself and I. Łaba, another due to myself, G. Kiss, I. Łaba and G. Somlai), which are derived from complexity measurements for polynomials with integer coefficients which have many cyclotomic polynomial divisors. As applications, we give connections in two other areas of mathematics. The first is in the study of integer tilings: additive decompositions of the integers $Z = A+B$ as a sum set, where each integer is represented uniquely. The second application is to the Favard length problem in fractal geometry, which asks for bounds upon the average length of the projections of certain dynamically-defined fractals onto lines.

This talk is based upon my individual work, as well as my joint work with I. Łaba, as well as my joint work with G. Kiss, I. Łaba and G. Somlai. All are welcome, and the first 15-20 minutes will include introductory ideas and examples for all results discussed in the latter portion of the talk.

We will give an exposition on the recent progress in the study of unimodal sequences, beginning with the work of Isaac Newton and then to the contemporary papers of June Huh. We will also relate this topic to the Riemann hypothesis. In the process, we will connect many areas of mathematics ranging from number theory, commutative algebra, algebraic geometry and combinatorics.

A set {𝑎1,𝑎2,…,𝑎𝑚}of distinct positive integers is a Diophantine 𝑚-tuple if the product of any two distinct elements in the set is one less than a square. In this talk, I will discuss some recent results related to Diophantine tuples and their generalizations. Joint work with Ernie Croot, Seoyoung Kim, and Semin Yoo.