Rigidity is the ability of a system to resist imposed stresses before ultimately undergoing failure. However, disordered materials often contain both rigid and floppy subregions that complicate the utility of taking system-wide averages. I will talk about 3 frameworks capable of connecting the internal structure of disordered materials to their rigidity and/or failure under loading, and describe how my collaborators and I have applied these frameworks to laboratory data on laser-cut lattices and idealized granular materials. These are, in order of increasing physics content: (1) centrality within an adjacency matrix describing its connectivity, (2) Maxwell constraint counting on the full network of frictional contact forces, and (3) the vibrational modes of a synthetic dynamical matrix (Hessian). The first two rely primarily on topology, and the second two contrast the utility of considering interparticle forces (Coulomb failure) vs. the energy landscape. All three methods, while successfully elucidating the origins of rigidity and brittle vs. ductile failure, also provide interesting counterpoints regarding how much information is enough to make predictions.

Diversity in the natural world emerges from the collective behavior of large numbers of interacting objects. Statistical physics provides the framework relating microscopic to macroscopic properties. A fundamental assumption underlying this approach is that we have complete knowledge of the interactions between the microscopic entities. But what if that, even though possible in principle becomes impossible in practice ? Can we still construct a framework for describing their collective behavior ? Dense suspensions and granular materials are two often quoted examples where we face this challenge. These are systems where because of the complicated surface properties of particles there is extreme sensitivity of the interactions to particle positions. In this talk, I will present a perspective based on notions of constraint satisfaction that provides a way forward. I will focus on our recent work on the emergence of elasticity in the absence of any broken symmetry, and sketch out other problems that can be addressed using this perspective.