Number Theory

Understanding how network structure gives rise to spatiotemporal dynamics and computation is a central challenge in computational neuroscience and artificial intelligence. Despite increasingly detailed connectomic data in neuroscience and large-scale datasets in machine learning, establishing principled links between connectivity, dynamics, and function in nonlinear neural systems remains difficult. In this talk, I will present a mathematical framework that directly relates network architecture to emergent dynamical patterns and computational capabilities in analytically tractable models. Our approach focuses on networks of coupled oscillators, which are widely used to model interacting neural populations and have recently gained interest as computational substrates in artificial neural networks. With this approach, we can show how key structural features of these networks — including connectivity patterns and transmission delays — determine the emergence and stability of spatiotemporal activity, enabling analytical predictions of collective phenomena such as traveling waves. When applied to empirically derived brain networks, the framework provides a rigorous connection between large-scale anatomy, distance-dependent delays, and wave dynamics observed at mesoscopic and whole-brain scales. Building on these results, we introduce a new class of neural networks that leverage structured spatiotemporal dynamics for computation while remaining exactly solvable. Together, these results outline a general strategy for linking network structure, emergent dynamics, and computation, with implications for understanding neural activity and for developing interpretable dynamical models for neural computation.

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.