Explore the concept of simplex-preserving maps and their applications in studying Čech and Vietoris–Rips complexes in this 45-minute conference talk. Delve into the definition of simplex-preserving maps and their significance in analyzing infinite-dimensional complexes. Examine the moment curve map as a classic example and investigate the challenges of finding simplex-preserving maps for various domains. Learn how the existence of these maps may be influenced by the dimension of the codomain. Cover topics such as simplicial complexes, geometric realization, metric thickening, Euclidean realization, trigonometric moment curves, and centrally-symmetric trigonometric moment curves.