C4

Asymptotically Locally Flat (ALF) Ricci-flat metrics are expected to model certain long-time singularities in four-dimensional Ricci flow, so understanding their stability is essential. In this talk, I will discuss that conformally Kähler, non-hyperkähler Ricci-flat ALF metrics are dynamically unstable under Ricci flow. Our work establishes three key tools in this setting: a Fredholm theory for the Laplacian on ALF metrics, the preservation of the ALF structure along the Ricci flow, and an extension of Perelman’s λ-functional to ALF metrics. This is joint work with Tristan Ozuch.

Topological data analysis (TDA) deals with quantifying the "shape of data" using tools from algebraic topology and computational geometry. In many contexts, data comes equipped with a labelling (for example, cell type annotations in spatial biology), and one is interested in quantifying not just the global structure of the data but the spatial relationships between labelled subsets of the data. I will give a brief introduction to TDA and then talk about chromatic Delaunay filtrations, a recently developed family of computational methods in TDA that can address the problem of quantifying spatial relationships in labelled point cloud datasets.

As temperatures are increasing, so is the presence of meltwater lakes sitting on the surface of the Greenland Ice Sheet. Such lakes have the possibility of draining through cracks in the ice to the bedrock. Observed discharge rates have found that these lakes can drain at three times the flow rate of Niagara Falls. Current models of subglacial drainage systems are unable to cope with such a large and sudden volume of water. This motivates the idea of a 'subglacial blister' which propagates and slowly dissipates underneath the ice sheet. We present a basic hydrofracture model for understanding this process, before carrying out a number of extensions to observe the effects of turbulence, topography, leak-off and finite ice thickness.

Estimating the correlation $\sum_{n \le x} d_k(n)d(n+h)$ is a central problem in analytic number theory. In this talk, I will present a method to obtain an asymptotic formula for a smoothed version of this sum. A key feature of the result is a power-saving error term whose exponent does not depend on $k$, improving earlier bounds where the quality of the saving deteriorates with $k$. The argument relies on balancing three distinct bounds for the remainder term according to the sizes of the factors of $n$.

The flow of viscous fluid through a soft porous medium exerts drag on the matrix and induces non-uniform deformation. This behaviour can become increasingly complicated when the medium has a complex rheology, such that deformations exhibit elastic (reversible) and plastic (irreversible) behaviour, or when the rheology has a viscous component, making the response of the medium rate dependent. This is perhaps particularly the case when compaction is repeated over many cycles, or when additional forces (e.g. gravity or an external load) act simultaneously with flow to compact the medium, as in many industrial and geophysical applications. Here, we explore the interaction of viscous effects with elastic and plastic media from a theoretical standpoint, focussing on unidirectional compaction. We initially consider how the medium responds to the reversal of flow forcing when some of its initial deformation is non-recoverable. More generally, we explore how spatial variations in stress arising from fluid flow interact with the stress history of the sample when some element of its rheology is plastic and rate-dependent, and characterise the response of the medium depending on the nature of its constitutive laws for effective stress and permeability.

Migrant communities shape cross-border investment to their country of origin by reducing

information frictions and attitudes bias. Whether these benefits spill over to locals depends

not only on the size of the diaspora but also on the intensity of interaction between migrants

and locals in the host country. I present a theoretical model with agent-based simulation to

study how homophily between migrants and locals affects information and attitude diffusion

in the host society. I implement varying homophily preferences in a Schelling-style

segregation model and compare two diffusion processes: (i) a simple susceptible–infected

(SI) model for information diffusion; (ii) an adoption-threshold model for attitude diffusion.

For information diffusion, preliminary results indicate that higher homophily slows the

spread and confines diffusion within the migrant group, especially under high segregation. In

the attitude model, adoption varies non-monotonically with homophily. I also provide an

initial analysis of how these patterns interact with different migrant population shares and

seeding rules.

TBA

Major depressive disorder (MDD) is a heterogeneous mental disorder. International guidelines present overall symptom severity as the key dimension for clinical characterisation. However, additional layers of heterogeneity may reside within severity levels related to how symptoms interact with one-another in a patient, called symptom dynamics. We investigate these individual differences by estimating the proportion of patients that display differences in their symptom dynamics while sharing the same diagnosis and overall symptom severity. We show that examining symptom dynamics provides information about the person-specific psychopathological expression of patients beyond severity levels by revealing how symptoms aggravate each other over time. These results suggest that symptom dynamics may serve as a promising new dimension for clinical characterisation. Areas of opportunity are outlined for the field of precision psychiatry in uncovering disorder evolution patterns (e.g., spontaneous recovery; critical worsening) and the identification of granular treatment effects by moving toward investigations that leverage symptom dynamics as their foundation. Future work aimed at investigating the cascading dynamics underlying depression onset and maintenance using the large-scale (N > 5.5 million) CIPA Study are outlined.