Filaments and fibers represent a fundamental unit of biological matter, from chromosomal DNA and microtubule networks to cilia carpets and worm collectives. How topology and elasticity mediate coordination in biological filaments remains poorly understood. To uncover the topological principles at play in living matter, we first examine the adaptive topological dynamics exhibited by California blackworms, which form tangled aggregates in minutes but can rapidly untangle in milliseconds. By constructing stochastic trajectory equations, we build a predictive model which explains how the dynamics of individual active filaments controls their emergent topological state. We then examine the elastodynamics and coordination of individual filaments more closely, and exhibit mathematical principles underlying knot-formation in filamentous organisms across length scales. By identifying how topology and elasticity produce stable yet responsive structures, these results have applications in understanding organization and self-optimization in broad classes of biological systems.

In his paper on free subgroups of linear groups, Tits proved his famous alternative: a linear group is either virtually solvable or contains a free subgroup. Since then, Tits’ work has been generalized and applied in many different ways.

One remaining open question in this field was the one asked by de la Harpe: let $G$ be a semisimple Lie group without compact factors and with trivial center and let $\Gamma$ be a Zariski-dense subgroup of $G$. Given a prescribed finite subset $F$ of $G$, is it always possible to find an element $\gamma \in \Gamma$ such that for any $h \in F$, the subgroup generated by $h$ and $\gamma$ is freely generated (in that case, we say $h$ and $\gamma$ are ping-pong partners).

In the talk, we will discuss a variant of the question of de la Harpe, where $F$ is a finite set of finite subgroups $H_i$ of $G$. Using careful refinements of the main steps of Tits’ proof of the alternative (which we will recall), we give sufficient conditions for the existence of ping-pong partners for the $H_i$ in any Zariski-dense subgroup $\Gamma$.

We will also show that these conditions are satisfied for products of copies of $\mathrm{SL}_n$ over division $\mathbb{R}$-algebras.

The existence of such free products has applications in the theory of integral group rings of finite groups, which will be briefly mentioned.

Joint work with G. Janssens and D. Temmerman.