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Results tagged with big-picture
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Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
77
votes
Accepted
What is an integrable system?
This is, of course, a very good question. I should preface with the disclaimer that despite having worked on some aspects of integrability, I do not consider myself an expert. However I have thought …
- 33.3k
68
votes
Accepted
Fundamental Examples
The harmonic oscillator is a fundamental example in both classical and quantum mechanics.
53
votes
How to find ICM talks?
Years ago I received for free a CD-ROM from World Scientific containing a PDF
file with some Fields medalists' talks at the ICM. It contains talks
up to and including the 1994 ICM. I have just manag …
- 33.3k
35
votes
The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Alge...
Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware o …
- 33.3k
35
votes
Fundamental Examples
The cotangent bundle is the fundamental example of symplectic manifold/phase space.
23
votes
Why is it useful to study vector bundles?
Although not a complete answer to the question, let me just point out that vector bundles are sometimes forced upon you.
For instance, you may start with an honest function $f$ defined on a manifold …
- 33.3k
22
votes
Fundamental Examples
It has often been said that if you understand $su_3$ you understand all simple Lie algebras, so that should make it the fundamental example. (Personally I think that it suffices to understand $su_2$! …
19
votes
Accepted
What are important examples of filtered/graded rings in physics?
It is debatable that a physicist would use those very words, and if they did one would hope their meaning would be the same as for a mathematician, since it means that they are trying to speak the sam …
- 33.3k
11
votes
Justifying a theory by a seemingly unrelated example
There are a number of algebraic theorems which are easier to prove using topology. The best known is probably the fundamental theorem of algebra, but there are others. For example, an $n\times n$ ma …
10
votes
Linear/Non-linear sigma model
The sigma model started life as a model for pions and derives its name from one of the fields in the theory (denoted $\sigma$). This, however, is another story and not the reason why I would get "exc …
- 33.3k
9
votes
Surprising and Useful Physical Intuition for Mathematical Objects
Mr. McGuire: Benjamin.
Benjamin: Mr. McGuire.
Mr. McGuire: Benjamin.
Benjamin: Mr. McGuire.
Mr. McGuire: I want to say two words to you.
Benjamin: Yes, sir.
Mr. McGuir …
8
votes
Heuristic behind the Fourier-Mukai transform
You may want to look at Tom Bridgeland's PhD thesis.
- 33.3k
6
votes
Fundamental Examples
The complex projective space is the fundamental example in toric geometry, symplectic and GIT quotients,...
5
votes
Doing geometry using Feynman Path Integral?
You might find Witten's lectures on the The Dirac index on manifolds and loop spaces from the IAS course on quantum field theory useful.
- 33.3k
5
votes
What do whitehead towers have to do with physics?
I finally got a hold of the paper in question: Edward Witten's Global anomalies in string theory, where the cases of the particle and the string are discussed.
The model in question is a supersymmetr …
- 33.3k