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.

Elliptic division fields, obtained by adjoining p-torsion points of an elliptic curve are natural objects of study since they are analogous to cyclotomic fields. We establish a conductor discriminant bound for such family of number fields. We also formulate and conditionally prove Ihara's conjectures for Euler-Kronecker constants in this setting. This is ongoing work with Anup Dixit. https://www.imsc.res.in/~anupdixit/IMSc-CMI-NT-seminar.html