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user 6153
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.
4
votes
The resolution of which conjecture/problem would advance Mathematics the most?
I conjecture the question to be premature ...
I would nominate the Riemann Hypothesis, since it is clear that something occurs that we fundamentally don't understand. But folding other things in with …
7
votes
How does "modern" number theory contribute to further understanding of $\mathbb{N}$?
A short answer: the kind of "structure" recognised in the analytic number theory of the period 1900 to 1930, successful as that theory was, doesn't go far enough. You need at least functions of severa …
- 12.6k
3
votes
Why are finiteness conditions important (and how to recognize them)?
Two types of points, I think. (1) Counting dimensions is usually a lot more interesting for finite-dimensional vector spaces than for the rest. (2) Where finiteness conditions can be removed, as often …
4
votes
Narratives in modular curves
"Moduli" are parameters that algebraic varieties depend on - continuous invariants if you like, as opposed to discrete invariants. Or the same for complex structures on a given topological manifold, i …
5
votes
Why certain diophantine equations are interesting (and others are not) ?
Picking up on the theme of the Hilbert problem on diophantine sets: we do know that they comprise all recursively enumerable sets. A diophantine set being only slightly more sophisticated than a given …
- 12.6k
9
votes
How do you decide whether a question in abstract algebra is worth studying?
Not sure I agree with the whole post in detail. Distinguish "pure algebra" from "applied algebra"; and within "pure algebra" distinguish "structural" issues from "combinatorial" ones such as the Burns …
1
vote
Defining variable, symbol, indeterminate and parameter
Of the various types of "placeholder", certainly a couple have definite mathematical meanings. In logic, the meaning of free and bound variables is set out in detail. And I take "indeterminate" to be …
- 12.6k
16
votes
To what extent is it true that "number theory = mathematics"?
I just don't think it's true, despite my own tastes in topics. Such formulations are substantially a matter of fashion.
There is one basic axis, running from very detailed information at one end (wh …
10
votes
Why are modular forms interesting?
Bryan Birch's view is that they form a bottomless area for research problems. All answers to the question fall into two types: showing examples of why this is true, and asking why it should be true. G …