talks

This talk showcases a methodology for the rapid computation of persistent homology in pandemic-scale time series data of SARS-CoV-2 genomes. By focusing only on small genomic distance scales, which capture the selective advantages and deleterious effects of individual mutations relative to its progenitor, and encoding temporal information through a deformation of the metric, we are able to efficiently calculate homology classes for millions of unique viral sequences. We discuss the implementation challenges and solutions that have enabled us to process data at this scale and explain the biological insights we obtain about the evolutionary fitness landscape during the pandemic. Our method exemplifies the application of topology to large datasets, streamlining data analysis at unprecedented scales, while also enhancing our understanding of viral evolution by providing insights into mutation dynamics at a daily resolution.

The Haydys-Witten equations are gauge theoretic partial differential equations on five-dimensional Riemannian manifolds that are equipped with a non-vanishing vector field. Conjecturally, their solutions on M^5=IR×W^4 determine the Floer differential in a gauge-theoretic approach to homological knot invariants. In this talk, I will provide a brief overview of this ‘instanton Floer theory of four-manifolds’ and then focus on a ‘decoupled’ version of the Haydys-Witten equations that emerges in the geometric setup for knot invariants. Since the latter exhibit a Hermitian Yang-Mills structure, these results may offer novel insights into the conjectured relationship to knot invariants.

RNA velocity data provides a snapshot of cell states and their current rate of change. It promises insights into the behaviour of individual cells and the dynamics governing cell division and differentiation processes. To explore single cell RNA-sequence data one often relies on low-dimensional visualizations, e.g. tSNE or UMAP. A priori it's not obvious how RNA velocities carry over to such representations and current methods have several drawbacks. It was recently suggested that one should fix a biologically motivated, low-dimensional manifold and infer RNA velocities strictly in terms of an embedding of the data in that manifold. I expand on that idea and argue that low-dimensional representations of position-velocity pairs should utilize the Sasakian geometry on the tangent bundle of curved target spaces. Moreover, I propose that non-linear neighbour embeddings into low- or middle-dimensional symmetric spaces provide a geometric representation of the principal dynamical components in the data. This geometrization provides interesting future directions regarding the analysis of the dynamical processes captured in single cell data.

The Haydys-Witten equations are partial differential equations on five-dimensional Riemannian manifolds that are equipped with a non-vanishing vector field v. Conjecturally, their solutions on M5=R×W4 determine the Floer differential in a gauge-theoretic approach to homological knot invariants. In this talk, I will provide a brief overview of this ‘instanton Floer theory of four-manifolds’ and then focus on a ‘decoupled’ version of the Haydys-Witten equations that emerges in the geometric setup for knot invariants. Since the latter exhibit a Hermitian Yang-Mills structure, these results may offer novel insights into the conjectured relationship to knot invariants

The Haydys-Witten equations are partial differential equations in gauge theory, defined on five-dimensional Riemannian manifolds that are equipped with a non-vanishing vector field $v$. Conjecturally, their solutions on $M^5 = \mathbb{R} \times W^4$ determine the Floer differential in a gauge-theoretic approach to homological knot invariants. In this talk, I will provide a brief overview of this 'instanton Floer theory of four-manifolds' and then focus on a 'decoupled' version of the Haydys-Witten equations that appears in the context of knot invariants. I will discuss conditions under which the full equations simplify to the decoupled form. Since the latter exhibit a Hermitian Yang-Mills structure, these results may offer novel insights into the conjectured relationship to knot invariants.

The application of persistent homology to viral evolution was suggested some years ago at the example of genomic data of flu viruses. It was observed that long-lived topological features detect reassortment and recombination events, which provide much of the variability of flu viruses. Here, I instead present a careful investigation of short-lived features, typically interpreted as noise. Our results show that persistent homology efficiently identifies recurrent mutations in the evolution of SARS-CoV-2. This provides insights into how distinct viral strains exhibit convergent genomic structures, which is a hallmark of adaptation processes. Moreover, I will explain how our analysis can easily incorporate temporal information despite the general pitfalls of multi-persistent homology. In that way it is possible to monitor changes in convergent behaviour over the course of the pandemic.