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Results tagged with big-picture
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user 4721
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.
131
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14
answers
29k
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Why are modular forms interesting?
Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about m …
75
votes
14
answers
6k
views
Why is Set, and not Rel, so ubiquitous in mathematics?
The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling …
73
votes
5
answers
18k
views
Mathematics of path integral: state of the art
I was told that one of the most efficient tools (e.g. in terms of computations relevant to physics, but also in terms of guessing heuristically mathematical facts) that physicists use is the so called …
63
votes
11
answers
8k
views
Why certain diophantine equations are interesting (and others are not) ?
It is quite clear why certain differential equations, among the jungle of possible diff equations that is possible to conceive, are studied: some come from physical problems, or from "spontaneous" mat …
- 23.3k
41
votes
1
answer
3k
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Are there any "homotopical spaces"?
This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense.
[Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is th …
- 23.3k
40
votes
17
answers
10k
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Interesting mathematical topics arising from biology
I've heard that there's a relatively new field of science called mathematical biology.
It will certainly apply well known and less known mathematical techniques to the understanding of some biological …
37
votes
1
answer
3k
views
Various flavours of infinitesimals
I'm not sure if this is a soft question, and whether it may be too broad or, on the contrary, too localized. Well, in Mathematics the concept of "infinitesimal" has been of extreme importance for cent …
- 23.3k
37
votes
3
answers
3k
views
"Softness" vs "rigidity" in Geometry
According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ t …
- 23.3k
31
votes
2
answers
3k
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Current status of Grothendieck's homotopy hypothesis and Whitehead's algebraic homotopy prog...
(Disclaimer: I'm no expert in homotopy theory nor in higher categories!) If I understand it correctly, Grothendieck's homotopy hypothesis states that there should be an equivalence (of $(n+1)$-categor …
- 23.3k
27
votes
14
answers
2k
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Characterizing specific "concrete" mathematical objects by abstract general properties
In this note by Tom Leinster the Banach space $\mathrm{L}^1[0,1]$ is recovered by "abstract nonsense" as the initial object of a certain category of (decorated) Banach spaces. So a function space, tha …
23
votes
6
answers
2k
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Are rings really more fundamental objects than semi-rings?
The discovery (or invention) of negatives, which happened several centuries ago by the Chinese, Indians and Arabs, has of course be of fundamental importance to mathematics.
From then on, it seems th …
5
votes
2
answers
561
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Can one make a category concrete by "enlarging the universe"?
This is more or less a followup of this question. There, it was established that (it is well known that) the homotopy category of topological spaces is not concrete, in other words, there is no faithf …
- 23.3k
4
votes
2
answers
382
views
About the cone being unique up to non-unique isomorphism
In an answer to this MO question [link] Fernando Muro sais:
the mapping cone of a morphism in a triangulated category is unique up
to non-unique isomorphism. This fact has originated a lot of re …
- 23.3k
4
votes
0
answers
250
views
Finite subgroups of the unimodular group
This is related to this MO question (and others as well).
Hoping that this will not turn out to be too broad, I would like to know about the 'state of the art' of:
1) The problem of classifying f …