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Results tagged with soft-question
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user 1149
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.
32
votes
Examples of theorems with proofs that have dramatically improved over time
I think that Ax's proof of the Chevalley-Warning Theorem qualifies.
The Chevalley-Warning Theorem is an affirmative solution of a conjecture made by L.E. Dickson in 1909 and taken up more seriously b …
- 4,713
17
votes
Textbook recommendations for undergraduate proof-writing class
If you want a book which is priced under \$30, write it yourself and put it on the internet. Then it's free! (This is not a quip or a dismissive comment: please do actually do this. I have done this s …
- 1,161
47
votes
Why should one still teach Riemann integration?
Here are some unpolemical facts concerning the Riemann integral:
The Riemann integral has a geometric interpretation which is different than that of the Lebesgue integral and is certainly useful in s …
CommunityBot
- 1
11
votes
Complex Analysis applications toward Number Theory
I think basic is on the right track. The two big classical theorems in analytic number theory whose classical proofs use some complex analysis are Dirichlet's Theorem on primes in arithmetic progress …
CommunityBot
- 1
85
votes
What are the most misleading alternate definitions in taught mathematics?
I increasingly abhor the introduction of the finite ring $Z_n$ not as $\mathbb{Z}/n\mathbb{Z}$ but as the set $\{0,\ldots,n-1\}$ with "clock arithmetic". (I understand that if you want to introduce m …
CommunityBot
- 1
55
votes
Fundamental Examples
The Fermat Equation xn + yn - zn = 0.
This has truly been much more than an example in both algebra and number theory: it was one of the main motivations to develop the theory of unique factorizati …
- 65.4k
18
votes
An example of a proof that is explanatory but not beautiful? (or vice versa)
1) "There is no simple group of order $n$" (for various composite values of $n$ in the interval $[50,200] \setminus \{60,168\}$ or so). These arguments are explanatory but not beautiful. They seem v …
- 65.4k
30
votes
Consolidation: Aftermathematics of fads
Since Quinn's article is a long opinion piece which he says is 90% complete and welcomes comments, it seems entirely appropriate to contact him for clarification on this point. He would probably be h …
- 65.4k
23
votes
Why are finiteness conditions important (and how to recognize them)?
The fact that various finiteness conditions lead to good theorems which are manifestly false in their absence seems like a good explanation of why they are important. (In fact, I am having trouble th …
- 65.4k
20
votes
Awfully sophisticated proof for simple facts
I claim that the rational canonical model of the modular curve $X(1) = \operatorname{SL}_2(\mathbb{Z}) \backslash \overline{\mathcal{H}}$ is isomorphic over $\mathbb{Q}$ to the projective line $\mathb …
- 65.4k
15
votes
Possibility of an Elementary Differential Geometry Course
I see that someone has mentioned this in the comments already, but I think it deserves to be left as an answer.
Here at UGA we do have a regular undergraduate course fitting your approximate descript …
- 65.4k
12
votes
Should there be a specified standard knowledge of mathematicians?
Like M. Emerton, Pierre's question makes me reflect on what most departments actually do to enforce common knowledge among (future) working mathematicians: their qualifying exams. I hope that most de …
- 65.4k
1
vote
8
votes
Graduate School
The advice to apply separately for a master's program is very good. If you can take the GREs (general and math subject) and do well, then many institutions will be willing to take a chance on you as …
- 65.4k
12
votes
What makes a theorem *a* "nullstellensatz."
For a field $k$, by a "Nullstellensatz" over $k$, I mean an explicit description of the Galois connection between subsets of $k^n$ and ideals in the polynomial ring $k[x_1,\ldots,x_n]$. See this MO q …
CommunityBot
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