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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.
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 …
- 2,821
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
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 …
- 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
1
vote
Theorems that are 'obvious' but hard to prove
The Hodge decomposition theorem
It is obvious that there is a unique point in any given affine plane in a finite-dimensional euclidean vector space which is closest to the origin.
Therefore it would …
- 33.3k
1
vote
Is the dual notion of a presheaf useful?
Edit
I seem to have misunderstood the nature of the duality in the question. This answer is not relevant. I'll keep it in case it has any archaeological interest.
In local quantum field theory, …
- 33.3k
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
2
votes
Are rings really more fundamental objects than semi-rings?
Although not a ring, the renormalisation group of quantum field
theory is really a semigroup. Moreover, there is no compelling
physical reason to add inverses, since in fact physically inverses
need …
- 33.3k
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
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
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
8
votes
Heuristic behind the Fourier-Mukai transform
You may want to look at Tom Bridgeland's PhD thesis.
- 33.3k
2
votes
Justifying a theory by a seemingly unrelated example
Many geometric problems cannot be solved without hard analysis. Perhaps the best known example is the Calabi Conjecture proved by Yau.
- 33.3k
2
votes
Justifying a theory by a seemingly unrelated example
The Poincaré conjecture is an example of a purely topological statement which apparently cannot be proved only by topological means.
- 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 …
- 33.3k