This work briefly reviews the main achievements of theoretical particle physics obtained using approaches based on symmetry and topology, and then highlights the weighty errors present in these theories. Particle theories are very abstract and developed within the framework of physical mathematics, which has no solid physical basis. However, real particles and real physical processes require finding a real basis on which fundamental physics should stand, and this is very important. The primary mathematical structure must be such that it is possible to apply mathematical physics to describe the physical picture of the world in real parameters that are actually present in particles and their interactions. It is precisely such a mathematical theory that is described in this work, the suitable geometry and topology are revealed, and references are provided to the corresponding physical theories developed by the author within the framework of mathematical physics, which already have a number of reliable experimental confirmations.