Scientific

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.

I will discuss several recent results on the Turán density of long cycle-like hypergraphs. These results (due to Kamčev–Letzter–Pokrovskiy, Balogh–Luo, and myself) all follow a similar framework, and I will outline a general strategy to prove Turán-type results for tight cycles in larger uniformities or for other "cycle-like" hypergraphs.

One key ingredient in this framework, which I hope to prove in full, is a hypergraph analogue of the statement that a graph has no odd closed walks if and only if it is bipartite. More precisely, for various classes C of "cycle-like" r-uniform hypergraphs — including, for any k, the family of tight cycles of length k modulo r — we equiivalently characterize C-hom-free hypergraphs as those admitting a certain type of coloring of (r-1)-tuples of vertices. This provides a common generalization of results due to Kamčev–Letzter–Pokrovskiy and Balogh–Luo.

In this talk we will introduce the modular method, the approach followed by Wiles to prove Fermat’s Last Theorem. We will explain the role of elliptic curves, modular forms, and Galois representations in this framework, and discuss how the method has evolved in recent years.

‘Reef halos’ are rings of sand, barren of vegetation, encircling reefs. However, the extent to which various biotic (e.g., herbivory) and abiotic (e.g., temperature, nutrients) factors drive changes in halo prevalence and size remains unclear. The objective of this study was to explore the effects of herbivore biomass, primary productivity, temperature, and nutrients on reef halo presence and width. First, we conducted a field study using artificial reef structures and their surrounding halos, finding that halos were more likely to be observed with high herbivorous fish biomass, and halos were larger under high temperatures. There was a distinct interaction between herbivorous fish biomass and temperature, where at high fish biomass, halos were more likely to be observed under low temperatures. Second, we incorporated environmental drivers into a consumer-resource model of halo dynamics. Certain formulations of temperature-dependent vegetation growth caused halo width and fish density to change from a fixed to an oscillating system, supporting the idea that environmental drivers can cause temporal fluctuations in halo width. Our unique combination of field-based and mechanistic modeling approaches has enhanced our understanding of the role of environmental drivers in grazing patterns, which will be particularly important as climate change causes shifts in marine systems worldwide.