Mathematics

Integer lattices play a central role in mathematics and computer science, with applications ranging from number theory and coding theory to combinatorial optimization. Over the past three decades, they have also become a cornerstone of modern cryptography.

In this talk, I will describe the evolution of lattices in cryptography: from the early use of lattices to break classical cryptosystems; to their application in designing new encryption and digital signature schemes with (conjectured) post-quantum security; and to their role in achieving long-standing cryptographic goals such as fully homomorphic encryption that allow us to compute directly on encrypted data.

The talk will not assume any prior background in cryptography.

This presentation will introduce a strategy to simplify certain undecidable problems. We will then demonstrate how quantum physics has been utilized to tackle potentially undecidable problems, specifically within the realm of low-dimensional topology. This intersection is known as quantum topology, a field in which I have been actively working, and should time allow, I will present some of my recent results in this area.

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.

How does one describe the structure of a graph? What is a good way to measure how complicated a given graph is? Tree decompositions are a powerful tool in structural graph theory, designed to address these questions. To obtain a tree decomposition of a graph G, we break G into parts that interact with each other in a simple ("tree-like") manner. But what properties do the parts need to have in order for the decomposition to be meaningful? Traditionally a parameter called the "width" of a decomposition was considered, that is simply the maximum size of a part. In recent years other ways of measuring the complexity of tree decompositions have been proposed, and their properties are being studied. In this talk we will discuss recent progress in this area, touching on the classical notion of bounded tree-width, concepts of more structural flavor, and the interactions between them.

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.