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1500 questions
53
votes
16 answers
Undergraduate math research
I believe this is the right place to ask this, so I was wondering if anyone could give me advice on research at the undergraduate level.
I was recently accepted into the McNair Scholars program. It is a preparatory program for students who want to…
user7504
53
votes
7 answers
Are there any undecidability results that are not known to have a diagonal argument proof?
Is there a problem which is known to be undecidable (in the algorithmic sense), but for which the only known proofs of undecidability do not use some form of the Cantor diagonal argument in any essential way?
I will freely admit that this is a…
Terry Tao
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53
votes
6 answers
Does homeomorphic and isomorphic always imply homeomorphically isomorphic?
Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that
$(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$.
Does it follow that $(G,\cdot,T)$ and $(H,\star,S)$ are isomorphic as topological groups?
If no,…
user5810
53
votes
21 answers
Interesting applications of the pigeonhole principle
I'm a little late in realizing it, but today is Pigeonhole Day. Festivities include thinking about awesome applications of the Pigeonhole Principle. So let's come up with some. As always with these kinds of questions, please only post one answer per…
Anton Geraschenko
- 23,718
53
votes
14 answers
Modern results that are widely known, yet which at the time were ignored, not accepted or criticized
What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It can be a theorem, a proof method, an algorithm or a…
alhal
- 419
53
votes
6 answers
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false?
One interesting example, and the impetus for this question, is work in number theory based on the assumption that Siegel zeros…
53
votes
10 answers
Changes forced by the pandemic
The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, sometimes in creative ways. I would like to compile a…
André Henriques
- 42,480
53
votes
1 answer
Why are there 1024 Hamiltonian cycles on an icosahedron?
Fix one edge $e$ of the graph (1-skeleton) of an icosahedron.
By a computer search, I found that there are 1024 Hamiltonian cycles that include $e$.
[But see edit below re directed vs. undirected!]
With the two endpoints of $e$ fixed, there are 10…
Joseph O'Rourke
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53
votes
0 answers
What is the current status of derived differential geometry?
I hope you will excuse this naive and general question. I've read from many places (e.g. Dominic Joyce's website, John Pardon's thesis, etc.) that the/a "right" foundations for many constructions in sympletic geometry, such as Fukaya category, and…
davik
- 2,035
53
votes
15 answers
Request for examples: verifying vs understanding proofs
My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an important difference between merely verifying that a…
Becky
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53
votes
3 answers
On which regions can Green's theorem not be applied?
In elementary calculus texts, Green's theorem is proved for regions enclosed by piecewise smooth, simple closed curves (and by extension, finite unions of such regions), including regions that are not simply connected.
Can Green's theorem be…
GermanJablo
- 619
53
votes
11 answers
What definitions were crucial to further understanding?
Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a problem and then ask, ‘What if I generalize this a…
Marcel
- 2,510
53
votes
8 answers
Modern algebraic geometry vs. classical algebraic geometry
Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar with something like the contents of Eisenbud's…
Amitesh Datta
- 1,983
53
votes
2 answers
Prime numbers as knots: Alexander polynomial
A naive and idle number theory question from a topologist (but not a knot theorist):
I have heard it said (and this came up recently at MO) that there is a fruitful analogy between Spec $\mathbb Z$ and the $3$-sphere. I gather that from an etale…
Tom Goodwillie
- 54,421
53
votes
3 answers
Does every real function have this weak continuity property?
In my research I came across the following question :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, converging to some real number $c$, such that the…
Dattier
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