Understanding the mechanisms behind the spatial distribution, self-organisation and aggregation of organisms is a central issue in both ecology and cell biology. Since self-organisation at the population level is the cumulative effect of behaviours at the individual level, it requires a mathematical approach to be elucidated.
In nature, every individual, be it a cell or an animal, inspects its territory before moving. The process of acquiring information from the environment is typically non-local, i.e. individuals have the ability to inspect a portion of their territory. In recent years, a growing body of empirical research has shown that non-locality is a key aspect of movement processes, while mathematical models incorporating non-local interactions have received increasing attention for their ability to accurately describe how interactions between individuals and their environment can affect their movement, reproduction rate and well-being. In this talk, I will present a study of a class of advection-diffusion equations that model population movements generated by non-local species interactions. Using a combination of analytical and numerical tools, I will show that these models support a wide variety of spatio-temporal patterns that are able to reproduce segregation, aggregation and time-periodic behaviours commonly observed in real systems. I will also show the existence of parameter regions where multiple stable solutions coexist and hysteresis phenomena.
Overall, I will describe various methods for analysing bifurcations and pattern formation properties of these models, which represent an essential mathematical tool for addressing fundamental questions about the many aggregation phenomena observed in nature.
 

We consider a system of interacting particles as a model for a foraging ant colony, where each ant is represented as an active Brownian particle. The interactions among ants are mediated through chemotaxis, aligning their orientations with the upward gradient of a pheromone field. Unlike conventional models, our study introduces a parameter that enables the reproduction of two distinctive behaviours: the conventional Keller-Segel aggregation and the formation of travelling clusters without relying on external constraints such as food sources or nests. We consider the associated mean-field limit of this system and establish the analytical and numerical foundations for understanding these particle behaviours.

Cells must reliably coordinate responses to noisy external stimuli for proper functionality whether deciding where to move or initiate a response to threats. In this talk I will present a perspective on such cellular decision making problems with extreme statistics. The central premise is that when a single stochastic process exhibits large variability (unreliable), the extrema of multiple processes has a remarkably tight distribution (reliable). In this talk I will present some background on extreme statistics followed by two applications. The first regards antigen discrimination - the recognition by the T cell receptor of foreign antigen. The second concerns directional sensing - the process in which cells acquire a direction to move towards a target. In both cases, we find that extreme statistics explains how cells can make accurate and rapid decisions, and importantly, before any steady state is reached.

Turing patterns have long been proposed as a mechanism for spatial organization in biology, but their relevance remains controversial due to the stringent fine-tuning often required. In this talk, I will present recent efforts to engineer synthetic Turing systems in bacterial colonies, highlighting both successes and limitations. While our three-node gene circuit generates patterns, challenges remain in extending these results to broader contexts. Additionally, I will discuss our exploration of machine learning methods to address the inverse problem of pattern formation, helping the design process down the road. This work addresses the ongoing task in translating theory into robust biological applications, offering insights into both current capabilities and future directions.

Immunological health relies on a balance between the ability to mount an immune response against potential pathogens and tolerance to self. However, how we keep that balance in health and what goes wrong in disease is not well understood. Here, I will describe combination of novel experimental and computational approaches using multi-omics datasets, imaging and functional experiments to dissect the role and defects in immune cells across several disease areas in cancer and autoimmunity. We show how shared mechanisms that are disrupted across diseases, including cellular, migration, immuno-surveillance, regulation and activation, as well as the immunological features associated with better prognosis and immunomodulation.

Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.

Let G be the points of a reductive group over a p-adic field. According to Bernstein, the category of smooth complex representations of G decomposes as a product of indecomposable subcategories (blocks), each determined by inertial supercuspidal support. Moreover, each of these blocks is equivalent to the category of modules over a Hecke algebra, which is understood in many (most) cases. However, when the coefficients of the representations are now allowed to be in a more general ring (in which p is invertible), much of this fails in general. I will survey some of what is known, and not known.