A Landau–Siegel zero, or exceptional zero, is a potential counterexample to the generalized Riemann hypothesis for Dirichlet $L$-functions associated with quadratic number fields. Assuming the existence of a Landau-Siegel zero, we establish an explicit Deuring--Heilbronn zero repulsion phenomenon for Dirichlet $L$-functions modulo $q$. Our estimate is uniform in the entire critical strip. It is a joint work with Kubra Benli, Henry Twiss, and Asif Zaman.

Amorphous solids, unlike crystals, lack long-range order, and their mechanical stability arises from disordered networks of force-balanced constituents and quenched internal stresses. In this talk, I will discuss how these internal stresses influence the mechanical response of athermal jammed solids. I will introduce a stress-based theoretical framework for describing elasticity in systems without a well-defined stress-free reference state, in which mechanical equilibrium takes the form of Gauss's law in a rank-2 tensor theory with vector charges, with internal stresses playing a role analogous to polarization in a dielectric medium. I will then discuss how incorporating a characteristic microscopic length scale leads to a scale-dependent response, capturing the crossover from continuum behavior at long wavelengths to the suppression of stress fluctuations at short distances. Finally, I will discuss the response to external forces and the behavior of the emergent elasticity as the system approaches the unjamming transition.