We can see the simplest setting of persistence from a functional point of view: given a fixed finite simplicial complex, we have the barcode function which, given a filter function over this complex, returns the corresponding persistent diagram. The bottleneck distance induces a topology on the space of persistence diagrams, and makes the barcode function a continuous map: this is a consequence of the stability Theorem. In this presentation, I will present ongoing work that seeks to deepen our understanding of the analytic properties of the barcode function, in particular whether it can be said to be smooth. Namely, if we smoothly vary the filter function, do we get smooth changes in the resulting persistent diagram? I will introduce a notion of differentiability/smoothness for barcode valued maps, and then explain why the barcode function is smooth (but not everywhere) with respect to the choice of filter function. I will finally explain why these notions are of interest in practical optimisation/learning situations. 

The amount and complexity of biological data has increased rapidly in recent years with the availability of improved biological tools. When applying persistent homology to large data sets, many of the currently available algorithms however fail due to computational complexity preventing many interesting biological applications. De Silva and Carlsson (2004) introduced the so called Witness Complex that reduces computational complexity by building simplicial complexes on a small subset of landmark points selected from the original data set. The landmark points are chosen from the data either at random or using the so called maxmin algorithm. These approaches are not ideal as the random selection tends to favour dense areas of the point cloud while the maxmin algorithm often selects outliers as landmarks. Both of these problems need to be addressed in order to make the method more applicable to biological data. We study new ways of selecting landmarks from a large data set that are robust to outliers. We further examine the effects of the different subselection methods on the persistent homology of the data.

In the first part of the talk, I will describe surface colonization strategies of the motile bacteria Pseudomonas aeruginosa. During early stages of biofilm formation, the majority of cells that land on a surface eventually detach. After a prolonged lag time, cells begin to cover the surface rapidly. Reversible attachments provide cells and their descendants with multigenerational memory of the surface that primes the planktonic population for colonization. Two different strains use different surface sensing machinery and show different colonization strategies. We use theoretical modelling to investigate how the hydrodynamics of type IV pili and flagella activity lead to increased detachment rates and show that the contribution from this hydrodynamic effect plays a role in the different colonization strategies observed in the two strains.

In the second part of the talk, I will show that when cells migrate through constricting pores, there is an increase in DNA damage and mutations. Experimental observations show that this breakage is not due to mechanical stress. I present an elastic-fluid model of the cell nucleus, coupled to kinetics of DNA breakage and repair proposing a mechanism by which nuclear deformation can lead to DNA damage. I show that segregation of soluble repair factors from the chromatin during migration leads to a decrease in the repair rate and an accumulation of damage that is sufficient to account for the extent of DNA damage observed experimentally.