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1500 questions
110
votes
4 answers
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a nutshell, is there a natural condition to impose on a…
Daniel Moskovich
- 21,887
110
votes
34 answers
Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher might do is ask students to calculate the…
Steven Sam
- 10,197
109
votes
10 answers
Set theories without "junk" theorems?
Clearly I first need to formally define what I mean by "junk" theorem. In the usual construction of natural numbers in set theory, a side-effect of that construction is that we get such theorems as $2\in 3$, $4\subset 33$, $5 \cap 17 = 5$ and $1\in…
Jacques Carette
- 11,669
109
votes
29 answers
Open problems with monetary rewards
Since the old days, many mathematicians have been attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express their belief in the magnitude of the difficulty of the…
Unknown
- 2,815
109
votes
6 answers
When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion-free connection $\nabla_g$, the Levi-Civita connection, that is compatible with the metric.
I was wondering if one can reverse this situation: Given a manifold with $M$…
Jean Delinez
- 3,339
109
votes
19 answers
Why were matrix determinants once such a big deal?
I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. I am curious about why determinants were such a…
Jiahao Chen
- 1,870
109
votes
89 answers
Tweetable Mathematics
Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty.
I am looking for examples that the essence of a notable mathematical…
Gil Kalai
- 24,218
109
votes
11 answers
Why do Groups and Abelian Groups feel so different?
Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly.
However, this heuristic breaks down in the case of many abelian groups. …
Greg Muller
- 12,679
108
votes
20 answers
Mathematical habits of thought and action which would be of use to non-mathematicians
Once again I come to MO for help with something I'm writing for the public.
Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a community, the way we decide on our goals, whatever --…
JSE
- 19,081
108
votes
27 answers
Why should one still teach Riemann integration?
In the introduction to chapter VIII of Dieudonné's Foundations of Modern Analysis (Volume 1 of his 13-volume Treatise on Analysis), he makes the following argument:
Finally, the reader will probably observe the conspicuous absence of
a…
Harry Gindi
- 19,374
108
votes
15 answers
Are there any good websites for hosting discussions of mathematical papers?
I was wondering if there are any websites out there which
systematically provide space for the discussion of mathematics articles (particularly those on the arXiv, though not necessarily just those), and
have a large enough user base to have some…
maxdev
- 136
108
votes
11 answers
Examples of notably long or difficult proofs that only improve upon existing results by a small amount
I was recently reading Bui, Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros on the critical line of the Riemann…
Klangen
- 1,943
- 2
- 23
- 34
108
votes
9 answers
How do you not forget old math?
I am trying to not forget my old math. I finished my PhD in real algebraic geometry a few years ago and then switched to the industry for financial reasons. Now I get the feeling that I want to do a postdoc and I face the dilemma that I actually…
Jose Capco
- 2,175
107
votes
32 answers
The half-life of a theorem, or Arnold's principle at work
Suppose you prove a theorem, and then sleep well at night knowing that future generations will remember your name in conjunction with the great advance in human wisdom. In fact, sadly, it seems that someone will publish the same (or almost the same)…
Igor Rivin
- 95,560
107
votes
6 answers
How small can a sum of a few roots of unity be?
Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How small can $|S|$ be?
$S$ is an algebraic integer…
Terry Tao
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- 31
- 432
- 517