Six-functor formalisms are ubiquitous in mathematics, and I will start this talk by giving a quick introduction to them. A three-functor formalism is, as the name suggests, (the better) half of a six-functor formalism. I will discuss what it means for such a three-functor formalism to be unitary, and why commutative Von Neumann algebras (and hence, by the Gelfand-Naimark theorem, measure spaces) admit a unitary three-functor formalism that can be viewed as mixing sheaf theory with functional analysis. Based on joint work with André Henriques.

Blood vessels are among the most vital structures in the human body, forming intricate networks that connect and support various organ systems. Remarkably, during early embryonic development—before any blood vessels are visible—their precursor cells are arranged in stereotypical patterns throughout the embryo. We hypothesize that these patterns guide the directional growth and fusion of precursor cells into hollow tubes formed from initially solid clusters. Further analysis of cells within these clusters reveals unique organization that may influence their differentiation into endothelial and blood cells. In this work, I revisit the problem of pattern formation through the lens of active matter physics, using both developing embryonic systems and in vitro cell culture models where similar patterns are observed during tissue budding. These different systems exhibit similar patterning behavior, driven by changes in cellular activity, adhesion and motility.

Microbial communities contain many evolving and interacting bacteria, which makes them complex systems that are difficult to understand and predict. We use theory – including game theory, agent-based modelling, ecological network theory and metabolic modelling - and combine this with experimental work to understand what it takes for bacteria to succeed in diverse communities. One way is to actively kill and inhibit competitors and we study the strategies that bacteria use in toxin-mediated warfare. We are now also using our approaches to understand the human gut microbiome and its key properties including ecological stability and the ability to resist invasion by pathogens (colonization resistance). Our ultimate goal is to both stabilise microbiome communities and remove problem species without the use of antibiotics.

The development of a complex functional multicellular organism from a single cell involves tightly regulated and coordinated cell behaviours coupled through short- and long-range biochemical and mechanical signals. To truly comprehend this complexity, alongside experimental approaches we need mathematical and computational models, which can link observations to mechanisms in a quantitative, predictive, and experimentally verifiable way. In this talk I will describe our efforts to model aspects of embryonic development, focusing in particular on the planar polarised behaviours of cells in epithelial tissues, and discuss the mathematical and computational challenges associated with this work. I will also highlight some of our work to improve the reproducibility and re-use of such models through the ongoing development of Chaste (https://github.com/chaste), an open-source C++ library for multiscale modelling of biological tissues and cell populations.

Cell and tissue movement during development, immune response, and cancer invasion depends on chemical or mechanical guidance cues. In many systems, this guidance arises not from long-range, pre-patterned cues but from self-generated gradients locally shaped by cells. However, how heterogeneous cell mixtures coordinate their migration by self-generated gradients remains largely unexplored. In this talk, I will first summarize our recent discovery that immune cells steer their long-range migration using self-generated chemotactic cues (Alanko et al., 2023). I will then introduce a multi-component Keller-Segel model that describes migration and patterning strategies of heterogeneous cell populations (Ucar et al., 2025). Our model predicts that the relative chemotactic sensitivities of different cell populations determine the shape and speed of traveling density waves, while boundary conditions such as external cell and attractant reservoirs substantially influence the migration dynamics. We quantitatively corroborate these predictions with in vitro experiments on co-migrating immune cell mixtures. Interestingly, immune cell co-migration occurs near the optimal parameter regime predicted by theory for coupled and colocalized migration. Finally, I will discuss the role of mechanical interactions, revealing a non-trivial interplay between chemotactic and mechanical non-reciprocity in driving collective migration.
 

Motivated by pattern formations and cell movements, many evolution equations incorporating spatial convolution with suitable integral kernel have been proposed. Numerical simulations of these nonlocal evolution equations can reproduce various patterns depending on the shape and form of integral kernel.The solutions to nonlocal evolution equations are similar to the patterns obtained by reaction-diffusion system and Keller-Segel system models. In this talk, we classify nonlocal interactions into two types, and investigate their relationship with reaction-diffusion systems and Keller-Segel systems, respectively. In these partial differential equation systems, we introduce multiple auxiliary diffusive substances and consider the singular limit of the quasi-steady state to approximate nonlocal interactions. In particular, we introduce how the parameters of the partial differential equation system are determined by the given integral kernel. These analyses demonstrate that, under certain conditions, nonlocal interactions and partial differential equation systems can be treated within a unified framework.  
This talk is based on collaborations with Hiroshi Ishii of Hokkaido University and Hideki Murakawa of Ryukoku University.