Abstract
Symmetry plays a central role in our understanding of quantum many-body systems. In the first part of this talk, I will provide an overview of my past work, illustrating how spontaneous symmetry breaking offers a predictive framework for organizing phases and determining their low-energy excitations, and how symmetry serves as a guiding principle for classifying and diagnosing topological phases of matter.

I will then turn to my recent research on frustration-free quantum systems --- tractable yet surprisingly rich models that uncover new theoretical insights. I will describe a universal lower bound on their spectral gap, an unexpected correspondence between quantum and classical dynamics, and the resulting implications for the dynamical critical exponent governing classical Ising dynamics.

I will conclude with a forward-looking perspective, using insights from frustration-free systems as a stepping stone toward broader studies of open quantum systems and quantum information theory.