In this talk, we discuss the notion of localization in ring theory through examples. We begin with the familiar construction of the rational numbers from the integers and then consider what happens when only selected elements of (\mathbb{Z}) are to be inverted. This leads to the general concept of localization. Through an example, we illustrate why this process is known as localization.

TBA

Gauge theories provide the theoretical framework for explaining exotic physical phenomena at different length scales, ranging from the femtometer scale in a relativistic heavy-ion collision experiment to mesoscopic scales in strongly correlated systems such as cuprates. In my pre-synopsis seminar, I will discuss two problems, one in each of these areas and how we address them based on non-perturbative ab-initio lattice gauge theory techniques. I begin by discussing the formalism we have developed to understand the dynamics of a heavy quark inside a strongly interacting non-Abelian plasma, both in and out of thermal equilibrium conditions. I will highlight my new results on the momentum and flavor dependence of the heavy quark diffusion coefficient. I will also discuss how we can understand thermalization in such a strongly interacting gauge theory and provide an estimate of the thermalization time. In the second half of my seminar, I will discuss a quantum field theory based on non-Abelian gauge interactions, which has been developed to understand the phase diagram of cuprates at low hole doping. I will elaborate on the Monte-Carlo algorithm that we have developed to unambiguously demonstrate a Kosterlitz-Thouless transition from the low-temperature d-wave superconducting phase to the high-temperature pseudogap phase in the presence of different competing orders. I will discuss some of the properties of the pseudogap phase that we observe. We could also show, for the first time, nucleation of period-4 charge order at the core of the vortices in the superconducting phase, resembling the observations made through scanning tunneling microscope studies of cuprate samples.