Five connected topics.
Move from rigorous topology to geometry, band topology, strongly correlated quantum matter, and technological applications.
Topic 01 · 8 conceptsFoundations of TopologyGeometry measures distances and angles; topology tracks properties unchanged by continuous deformation.Topic 02 · 6 conceptsGeometric Quantum MechanicsTwo state vectors differing only by a nonzero complex factor represent the same pure quantum state.Topic 03 · 12 conceptsTopological Band TheoryPeriodic lattices let us label electronic states by crystal momentum.Topic 04 · 8 conceptsInteracting Topological PhasesStrongly interacting electrons can form incompressible liquids with fractionally charged excitations.Topic 05 · 8 conceptsQuantum Phenomena and ApplicationsQuantum superposition, phase, and interference make eigenstates vary continuously across parameter spaces; global patterns can then become topological.