Mathematics Colloquium

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Topology, geometry and adaptivity in soft and living matter

Speaker: Vishal Patil, Stanford

Location: Warren Weaver Hall 1302

Date: Monday, February 26, 2024, 3:20 p.m.

Synopsis:

Topology and adaptivity play fundamental roles in controlling the dynamics of biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. How topological rules give rise to adaptive, self-optimizing dynamics in soft and living matter remains poorly understood. Here we investigate the interplay between topology, geometry and reconfigurability in knotted and tangled matter. We first identify topological counting rules which predict the relative mechanical stability of human-designed knots, by developing a mapping between elastic knots and long-range ferromagnetic spin systems. Building upon this framework, we then examine the adaptive topological dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments controls their emergent topological state. To further understand how tangled systems, along with more general biological networks, adapt to their surroundings, we introduce a theory of adaptive elastic networks which can self-learn mechanical information. By identifying how topology and adaptivity produce stable yet responsive structures, these results have applications in understanding broad classes of adaptive, self-optimizing systems.