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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 …
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 …
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 …
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 …
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 …
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 …
1
vote
Webpages for specialized communities
Galois Theory Web Page
Valuation Theory Home Page
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 …
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 …
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 …
20
votes
Interesting Calculus Questions/Exercises
I have little personal experience with it, but some colleagues and friends hold the following text in high regard:
Robert M. Young, Excursions in calculus.
An interplay of the continuous and the …
33
votes
Teaching undergraduate students to write proofs
This is a great question. In fact, I hope people won't think it over-dramatic if I call it one of the great math education questions of our time.
At the University of Georgia, we have decided as a de …
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 …
41
votes
How to present mathematics to non-mathematicians?
For some reason, many mathematicians have trouble with the idea that when some layman asks them about their work, the appropriate response is not to try to figure out how to describe the latest theore …
11
votes
When do we study maps into an object or from the object to another object?
Let $V$ be a variety over a field $K$. In the category of varieties (or schemes) over $K$. I am very interested in studying all the morphisms from $\operatorname{Spec}(K)$ to $X$. I could hardly be …