Many processes in the life sciences are inherently multi-scale and dynamic. Spatial structures and patterns vary across levels of organisation, from molecular to multi-cellular to multi-species. With more sophisticated mechanistic models and data available, quantitative tools are needed to study their evolution in space and time. Topological data analysis (TDA) provides a multi-scale summary of data. We review the main tools in topological data analysis and how single and multi-parameter persistent homology provide insights to complex systems.
Recently, there has been growing interest in studying long, slender living worms due to their unique ability to form highly entangled physical structures and exhibit emergent behaviors. These polymer-like organisms can move and assemble into an active three-dimensional soft entity, known as the "blob," which displays both solid-like and liquid-like properties. The blob can respond to external stimuli, such as oxygen concentration, by moving or changing shape. In this talk, I will illustrate how these living worms provide a fascinating experimental platform for exploring the physics of active, polymer-like entities. The combination of activity, long aspect ratios, and entanglement in these systems gives rise to a wide range of emergent behaviors. By investigating both the individual dynamics and the collective behavior of these worm blobs, we can stimulate further research into the physics of entangled active polymers and potentially inform the development of synthetic topological active matter and bioinspired soft robotic collectives.
Control landscape phase transitions (CLPTs) occur as abrupt changes in the cost function landscape upon varying a control parameter, and can be revealed by non-analytic points in statistical order parameters. A prime example are quantum speed limits (QSL) which mark the onset of controllability as the protocol duration is increased. Here we lay the foundations of an analytical theory for CLPTs by developing Dyson, Magnus, and cumulant expansions for the cost function that capture the behavior of CLPTs with a controlled precision. Using linear and quadratic stability analysis, we reveal that CLPTs can be associated with different types of instabilities of the optimal protocol. This allows us to explicitly relate CLPTs to critical structural rearrangements in the extrema of the control landscape: utilizing path integral methods from statistical field theory, we trace back the critical scaling of the order parameter at the QSL to the topological and geometric properties of the set of optimal protocols, such as the number of connected components and its dimensionality. We verify our predictions by introducing a numerical sampling algorithm designed to explore this optimal set via a homotopic stochastic update rule. We apply this new toolbox explicitly to analyze CLPTs in the single- and two-qubit control problems whose landscapes are analytically tractable, and compare the landscapes for bang-bang and continuous protocols. Our work provides the first steps towards a systematic theory of CLPTs and paves the way for utilizing statistical field theory techniques for generic complex control landscapes.