Rethinking geometry in physics: From spatial to spectral geometry.

The role of geometry in physics underwent a significant shift from Newton to Einstein. While Newton considered geometry as an a priori and fixed framework to describe the physical world, Einstein challenged this view and emphasized the importance of linking geometry to physical experiments and operational definitions. This led to the development of Einstein's "practical geometry" which played a crucial role in the development of his theory of general relativity. As we move towards the quantum world, the questions of how to think about geometry and its ontological status become even more crucial. Can we go beyond Einstein's practical geometry? What is the relationship between geometry and quantum theory of matter? These are the questions that I am currently exploring. By examining alternative theories of geometry, such as spectral geometry, we can gain new insights into the ontological status of geometry and its relationship with physical concepts. In this thesis, I aim to provide a critical evaluation of Einstein's practical geometry and explore new avenues for thinking about geometry in the quantum world.