quasar987 said:I was under the impression that only differential geometry was involved in calculating the shape of space time. Can anyone confirm what Danger said?
This intervention is not a professional one and many other people could give a better answer to your question than me. But if you think to the notion of parallel transport and to its consequence, the Berry's phase for example, you get a first concrete application of the indirect effect of the topology on physical phenomenon. In my non-specialist mind, geometry and topology are notions very closed together even if any specialist will immediately contradicts my point of view. Hope I could help you!quasar987 said:What is the motivation for a physicist to learn topology?
Are there fields of physics that make explicit use of the concept of topology? (which ones)
Do the ideas of topology give any insights into any topic of physics?
etc.
Topology is a branch of mathematics that studies the properties of geometric objects that do not change under continuous transformations. In physics, topology is used to understand the underlying structure and behavior of physical systems, such as the shape and connectivity of materials or the topology of spacetime.
The use of topology in physics was motivated by the need to explain certain phenomena that could not be understood using traditional methods. For example, topological invariants were found to be useful in describing the behavior of electrons in a magnetic field, which led to the discovery of the quantum Hall effect. This showed that topology can play a crucial role in understanding physical systems.
Topology has a wide range of applications in physics, including condensed matter physics, particle physics, and cosmology. It has been used to explain the behavior of superconductors, the properties of topological insulators, and the structure of the early universe. It is also being explored as a potential tool for quantum computing.
Topology allows us to classify and understand the properties of materials based on their underlying structure. For example, topological insulators have a unique electronic structure that makes them insulating in the bulk but conductive on their surface. This property is protected by the topology of the material and can be used for various applications, such as in quantum computing.
Some current research areas include topological phases of matter, topological quantum field theory, and topological defects in materials. There is also ongoing research on the impact of topology on cosmology and the possible connections between topology and the fundamental forces of nature.