L4

In their recent ICM address, Dan Halpern-Leistner and Jarod Alper posed the question: does the stack of canonically polarized curves admit a theta-stratification whose semistable locus is the moduli space of stable curves? In other, perhaps more familiar, language: is there some notion for curves that understands instability the way Harder-Narasimhan type does for vector bundles/coherent sheaves?

I will explain some joint work with Dave Swinarski that solves the analogous problem for certain rational curves, and then sketch out an argument that I think should solve the problem in general.

Z/2 harmonic 1-forms are generalizations of harmonic 1-forms that allow topological twisting around a subspace of codimension 2. These objects were introduced by Taubes to compactify the moduli spaces of solutions to generalized Seiberg-Witten equations, and they show up in many other gauge theoretical problems.

Donaldson showed there is a deformation theory for so-called non-degenerate Z/2-harmonic 1-forms. In this presentation we study the deformations of the remaining degenerate solutions. For a natural class of degenerate examples, we prove that after a suitable perturbation of the ambient Riemannian metric, the form can be deformed to a nearby non-degenerate Z/2-harmonic 1-form.

Hypertoric varieties are conical symplectic singularities equipped with a hamiltonian action of a torus of maximal possible dimension. They behave analogously to toric varieties in many ways, with the hamiltonian torus replacing the dense torus. Examples include: (crepant partial resolutions of) rational surface singularities of type A, the cotangent bundle of projective space, and Nakajima quiver varieties with the `all 1s' dimension vector.

Arbo and Proudfoot conjectured a combinatorial classification of hypertoric varieties by zonotopal tilings. In this talk I will discuss a proof of the conjecture.

TBA

In this talk we will discuss $G_2$-instantons and their relation to the heterotic $G_2$-system on 7-dimensional 2-step nilpotent Lie groups. We will first consider left-invariant coclosed $G_2$-structures and present necessary and sufficient conditions for their characteristic connection to be a $G_2$-instanton, expressed in terms of the torsion of the $G_2$-structure, the torsion of the connection, and the Lie group structure. We will then explain how these conditions imply that the corresponding left-invariant Riemannian metrics define naturally reductive structures.

As a continuation, we will consider solutions to the heterotic $G_2$-system on 2-step nilmanifolds. We will describe constructions of solutions with non-abelian gauge groups and discuss how the geometry of the underlying $G_2$-structure and the choice of gauge connection enter the equations. In particular, we will present families of solutions arising from the $G_2$-instantons described in the first part.

This is joint work with Andrew Clarke, Andrés J. Moreno, and Udhav Fowdar.

This talk is divided in three parts. First, I review structural properties and examples of geometries with parallel skew torsion. Second, I take a closer look at those geometries where the torsion is in addition closed; this equips every tangent space with the structure of a metric Lie algebra. These manifolds always locally split as a product into well understood factors, allowing a complete local classification. Third, I investigate various G-structures defining a geometry of this type, with a focus on almost Hermitian structures, extending recent results of Barbaro-Pediconi-Tardini. This is joint work with Andrei Moroianu.

The apicomedial actomyosin network is crucial for generating mechanical forces in epithelial cells. Pulsatile, oscillatory behaviour of this contractile network occurs in multiple organisms and contexts, for example during mouse embryo compaction or Xenopus neurulation. However, the emergence and control of such pulsed contractions are not fully understood. Here, we study pulsed contractions of larval epithelial cells (LECs) during development of the abdomen in fruit flies. Specifically, we combine in vivo 4D microscopy and simulations of an active elastomer to investigate subcellular spatial patterns of actomyosin dynamics. Our model quantitatively matches in vivo observations and identifies cell geometry and polarity as key contributors to in vivo actomyosin dynamics. Moreover, our findings support the notion that spatiotemporal oscillatory behaviour of the actomyosin network is an emergent property, rather than driven by upstream signalling.

I will additionally give a brief overview of other dynamic phenomena in cell biology that my collaborators and I have studied, such as ultradian (i.e. faster-than-circadian) oscillations in gene expression.

Throughout, I will highlight open challenges for data-driven approaches in cell biology.

Evolution underlies the transformation of a normal cell to a cancer, yet learning the parameters defining this dynamic process from single-timepoint bulk samples is an open challenge. To understand how cancer cells evolve in vivo, we must rely on naturally occurring, heritable lineage tracing markers that encode the evolutionary history of a population of cells. Here, I shall introduce our work on identifying selectively neutral “epigenetic barcodes” and employing them as a molecular clock. By coupling this process with mathematical modelling and Bayesian inference, we characterised the evolutionary history of almost 2000 lymphoid cancers (Gabbutt et al., 2025). Across a broad range of cancer types, we demonstrated that tumour growth rates and malignancy ages differed by orders of magnitude. In 2 independent cohorts of patients with chronic lymphocytic leukaemia (CLL), a typically indolent and slow growing cancer, the inferred growth rates were highly prognostic. I shall further discuss recent work applying this approach to resolve the clonal relationship between acute myeloid leukaemia (AML) blasts and differentiated neutrophils in patients without bone marrow failure. 

The acquisition and maintenance of the correct cellular pattern and tissue architecture is essential for organ function in multicellular organisms. Beyond generating the required cellular diversity, developing tissues need to attain the appropriate morphology. Tissue architecture is built through symmetry breaking instabilities such as folding, branching, buckling or budding. Studying single-cell responses alone is not enough to reveal the mesoscale physical and biochemical processes that regulate tissue organization and morphology over time and scale. In our lab we investigate how the interaction between the extracellular matrix (ECM) and tissues regulates processes during development and cancer, with a focus on ECM mechanical properties. While most research in this domain has concentrated on the ECM's elasticity as a primary determinant of cell and tissue behaviour, it is important to note that the ECM possesses both viscous and elastic properties. I will present our findings demonstrating that the passive viscoelastic properties of the ECM regulate tissue architecture and patterning both during development and cancer. Specifically, we show that ECM viscoelasticity influences the spatial and temporal organization of multicellular tissues in breast and intestinal organoids. Overall, our work highlights the critical role of viscoelasticity in driving morphological symmetry breaking instabilities, a fundamental process in morphogenesis and oncogenesis, and suggests ways of controlling tissue through ECM mechanics.