QCD phase structure & equation of state: A functional perspective

The phase structure of QCD remains an open fundamental problem of standard model physics. In particular at finite density, our knowledge is limited. Yet, numerous model studies point towards a rich and complex phase diagram at large density. Functional methods like the functional renormalization group and Dyson-Schwinger equations offer a way to study hot and dense QCD matter directly from first principles. I will discuss the phase structure of QCD and its experimental signatures through the lens of these methods.