The mathematical analysis of self-organized patterning in the limb bud is a well-tread problem in mathematical biology. Since the 80s, it has been understood that motile dermal cells undergo spontaneous condensation, leading to the formation of segregated regions localizing hard and soft tissues within the limb. In this talk, we report updates to this basic picture, motivated by new experimental information made possible by a recently developed multi-tissue in vitro assay. Based on this evidence, we propose a continuum model of chemomechanical pattern formation in which the robust establishment of tissue domains depends on the complementary actions of direct cell-cell adhesion and integrin-mediated cell-ECM interactions. Primarily through numerical analysis, we examine the feedback mechanisms required to regulate these processes and, ultimately, to establish cell-fate transitions as observed in experiment.  Our results highlight the growing understanding of mechanotransductive transitions in development and, in particular, establish an alternative role for classic morphogen pathways as modulators of local mechanical interactions rather than as direct determinants of cell fate. 

We prove that if an orientable 3-manifold $M$ admits a complete Riemannian metric whose scalar curvature is positive and has at most $C$-quadratic decay at infinity for some $C > \frac{2}{3}$, then it decomposes as a (possibly infinite) connected sum of spherical manifolds and $\mathbb{S}^2\times \mathbb{S}^1$ summands. Consequently, $M$ carries a complete Riemannian metric of uniformly positive scalar curvature. The decay constant $\frac{2}{3}$ is sharp, as demonstrated by metrics on $\mathbb{R}^2 \times \mathbb{S}^1$. This improves a result of Balacheff, Gil Moreno de Mora Sard{\`a}, and Sabourau, and partially answers a conjecture of Gromov. The main tool is a new exhaustion result using $\mu$-bubbles.