Explore the cutting-edge developments in defining the space of gapped lattice systems and their implications for classifying gapped phases of matter in this comprehensive 1 hour 50 minute lecture by Anton Kapustin from Caltech. Delve into the recent challenges to the belief that Topological Quantum Field Theory (TQFT) could classify these phases, and examine the plausible connection between Short-Range Entangled (SRE) phases and invertible TQFT. Investigate the cobordism conjecture and its suggestion that SRE phases are classified by homotopy groups of certain Omega-spectra. Learn about the construction of "higher Berry classes" and their equivariant versions, including Hall conductance and nonabelian analogs, which arise from infinite-dimensional spaces of SRE systems. Discover the key role of differential graded Frechet-Lie algebra in these constructions. Cover topics such as local and almost local observables, accuracy approximation, locality, Hamiltonians, uniformly almost local, ambiguity, analogs, analogy, local conservation, lambda, chain complex, Lee algebra, Goldstone theorem, and invariance. Read more