Explore practical applications of computational topology and mathematics in solving real-world problems across various scientific fields. Delve into a series of case studies spanning material science, dynamical systems, and brain research. Discover the symbiotic relationship between theoretical concepts and algorithmic implementations, understanding how they collaborate to tackle complex challenges. Examine the role of persistence diagrams, numerical invariance, and high-dimensional synthesis in advancing scientific understanding. Learn about innovative approaches to material optimization, neural network architecture analysis, and the emerging mathematics of shape. Gain insights into the computational difficulties encountered and strategies for overcoming them in applied topology research.