5 ms·

In the past, topology was little more than an amusing diversion for mathematicians doodling about the difference between donuts and dumplings. ...what?

• shardling 14y ago
  Obviously it is super-important in all sorts of areas of mathematics, but I can't think of any branches of physics that explicitly invoke any interesting topology. (Except probably string theory.) (And perhaps some areas of solid state do?)
  • xyzzyz 14y ago
    There are lots of examples. For instance, you can interpret Maxwell equations as certain statements about de Rham cohomology classes of certain 2-forms on 4-manifolds. Fiber bundles are used to describe local symmetries in gauge theories, and since fiber bundles are connected with homotopy groups, they also find their use. Even K-theory is used, in string theory. The list could easily go on. EDIT: fixed mistake.
    • aroberge 14y ago
      The list could go on ... if you are a mathematician. If you are a physicist (not working on string theory), you rarely use topological concepts - if at all. While working towards my Ph.D. on some topic in finite temperature effects in gauge theories, I did not know anything about fiber bundles and still know nothing about de Rham cohomology classes. My non-theorist physicists friends knew even less about those mathematical topics.
      • xyzzyz 14y ago
        Sure. That's why math is great: after you have spent enough time familiarizing with concepts, you can easily understand so many things in terms of concepts you already feel very comfortable with. You see electromagnetism, realize that electromagnetic field is just 2-form on 4-manifold, and see that Maxwell equations just state that both this form and its Hodge dual are closed, so that they represent de Rham cohomology classes. Of course, you can also represent this result in a classical way, but creating a bridge translating physical concepts into well studied mathematical frameworks has the advantage enabling you to also pass the bridge in the other direction: sometimes you can find physical interpretation concepts that arised in the abstract setting. So, we know that forms representing electromagnetic fields are closed, but what does it mean in physics when they are exact? How to interpret Mayer-Vietoris sequence in electromagnetism terms? What the induced maps do with forms representing electromagnetic fields, and how to interpret homotopy? What about Poincare duality? It's such a great feeling to realize that your favourite toy, after playing with it for years, is actually able to do a whole lot more stuff than you were previously aware of.
    • shardling 14y ago
      Sure, but in no point in the pursuit of a PhD in theoretical physics did I actually use any of that machinery. Nor did it ever come up in any papers I've read, nor any talk I've seen. The point is this -- if you want to extend our model of the universe, knowing the more abstract machinery is useful; by generalising your understanding (and a hell of a lot of math is basically generalising to more abstract systems) you gain insight into how alternative models might be constructed. If you want to understand our current model, even at a quite deep level, you can get by without knowing any topology at all.