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Results tagged with soft-question
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user 3272
Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.
114
votes
Demystifying complex numbers
The nicest elementary illustration I know of the relevance of complex numbers to calculus
is its link to radius of convergence, which student learn how to compute by various tests, but more mechanical …
- 50.6k
81
votes
Accepted
Elementary results with p-adic numbers
Introduce ${\mathbf Z}_p$ as "formal" infinite base $p$ expansions where you add and multiply by carrying (any other description will probably take too long and not be concrete). Show them the series …
- 50.6k
7
votes
Do empirical studies have a place in contemporary mathematics research?
The Birch and Swinnerton-Dyer conjecture was formulated on the basis of substantial computer calculations. They looked at the growth of products $\prod_{p \leq x} N_p(E)/p$ for large $x$ and various …
- 50.6k
27
votes
Swimming against the tide in the past century: remarkable achievements that arose in contras...
In the first decades of the 20th century, $p$-adic analysis (or valuation theory more generally) was regarded by many as rather exotic. After Hensel's work there was a steady development by Strassmann …
- 12.8k
23
votes
Modern results that are widely known, yet which at the time were ignored, not accepted or cr...
Does acceptance of conjectures before they became theorems count?
Example 1. The Artin reciprocity law. When Artin went around to other people describing what he was trying to show, nobody else belie …
- 50.6k
24
votes
How would you have answered Richard Feynman's challenge?
Here are two questions, and both are about math that was known long before Feynman passed away.
Explain to him what unique factorization into irreducibles means (including the ambiguity from multipli …
- 50.6k
16
votes
Are there any "related rates" calculus problems that don't feel contrived?
Here are two examples I think are interesting. (Update: I have added a third example at the end, more intricate than the other two.)
A ladder that is leaning against a wall starts slipping down. If t …
- 50.6k
9
votes
Golden ratio in contemporary mathematics
In every real quadratic field $K$, the unit group of its ring of integers $\mathcal O_K$ is known to have the form $\pm u^\mathbf Z$ for a unique number $u > 1$, which is called the fundamental unit o …
- 50.6k
31
votes
Is pure mathematics useful outside of mathematics itself?
Why do you want current work in pure math to "immediately benefit the population at large in a direct and obvious way"? Applications of pure math might take decades or centuries. As much as you may wi …
- 50.6k
21
votes
Widely accepted mathematical results that were later shown to be wrong?
Any rational function field over a finite field has genus $0$ and class number $1$, where the class number of a function field over a finite field is the number of degree-zero elements of the divisor …
- 4,713
227
votes
Widely accepted mathematical results that were later shown to be wrong?
Mathematicians used to hold plenty of false, but intuitively reasonable, ideas in analysis that were backed up with proofs of one kind or another (understood in the context of those times). Coming to …
- 50.6k
10
votes
What is the point of reading classics over modern treatments?
A question very similar to yours was asked quite a few years ago on MO here. Take a look at the answers there.
- 50.6k
28
votes
Examples of improved notation that impacted research?
There is a notation that had an immediate and profound impact on research in algebraic topology, later algebraic geometry, and was eventually adopted by all areas of mathematics: the introduction of a …
- 50.6k
40
votes
Why certain diophantine equations are interesting (and others are not) ?
The question that was asked compares Diophantine equations to differential equations, with the famous differential equations first arising due to physical arguments before taking on a life of their ow …
- 50.6k
16
votes
Noteworthy, but not so famous conjectures resolved recent years
In number theory, the Sato-Tate conjecture about elliptic curves over $
\mathbf Q$ was a problem from the 1960s and Serre's conjecture on modularity of odd 2-dimensional Galois representation was a co …
- 50.6k