Topology, a cornerstone of modern mathematics, provides a universal framework for classifying defects in physical systems, e.g., nematic disclinations, spin/orbital angular momentum, and spatial vortices etc.  Its integration with band theory has revolutionized our understanding of momentum space as a parameter space, where band degeneracies (Dirac/Weyl points) are characterized by topological invariants constructed from Berry phase, exemplified by winding numbers and Chern classes.  The recent incorporation of algebraic topology tools, particularly homotopy and homology theories, has further enriched the classification of topological phases.  In this talk, we introduce the application of catastrophe theory–a distinct mathematical branch–in band theories.  Focusing on PT-symmetric non-Hermitian systems, we demonstrate that catastrophe singularities in the ADE classification (cuspoids, umbilics, etc.) universally manifest as symmetry protected band degeneracies, defining unprecedented topological phase classes.  Crucially, these catastrophe-engineered degeneracies host novel topological edge states that persist in gapless regimes, challenging the conventional bulk–boundary correspondence.  Our findings establish connections between catastrophe theory and topological matter.  This cross-disciplinary synthesis opens new avenues for topological physics and enriches the fundamental understanding of singularity-associated topological phenomena.