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Results tagged with big-list
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user 766
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
19
votes
Trichotomies in mathematics
After passing to a subsequence if necessary, a sequence of real numbers either (a) converges to a real number; (b) diverges to $+\infty$; or (c) diverges to $-\infty$. In a similar vein, a sequence …
- 4,713
53
votes
Colloquial catchy statements encoding serious mathematics
Complete disorder is impossible.
This is the standard way of summing up Ramsey theory in a succinct sentence (according to that Wikipedia article, the above quote is due to Motzkin).
- 4,713
91
votes
Colloquial catchy statements encoding serious mathematics
"Can you hear the shape of a drum?"
This was Kac's famous way of asking whether the shape of a two-dimensional domain could be reconstructed from the spectrum of the Laplacian on that domain. (The …
- 4,713
44
votes
Mathematicians who were late learners?-list
According to this Notices article, Raoul Bott was undistinguished in high school, but displayed impressive talent once he reached graduate school (though his thesis was actually in electrical engineer …
- 4,713
57
votes
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums …
- 114k
23
votes
When forgetting structure doesn't matter
By the positive solution to Hilbert's fifth problem (as well as the theorem of Cartan that continuous homomorphisms between Lie groups are automatically smooth), the forgetful functor from the categor …
- 114k
34
votes
Proofs that require fundamentally new ways of thinking
It seems that certain problems seem to induce this sort of new thinking (cf. my article "What is good mathematics?"). You mentioned the Fourier-analytic proof of Roth's theorem; but in fact many of t …
- 12.9k
49
votes
Suggestions for good notation
I am fond of subscripting asymptotic notation with the parameters that the implied constant is allowed to depend on (and on the asymptotic parameter, if needed). e.g.
$X = O_k(Y)$ (or $X \ll_k Y$, o …
- 114k
29
votes
Cardinalities larger than the continuum in areas besides set theory
One of the quickest ways to demonstrate that there exist Lebesgue measurable subsets of the real line that are not Borel measurable is to compute the cardinality of the Lebesgue $\sigma$-algebra and t …
- 4,713
40
votes
Taking a theorem as a definition and proving the original definition as a theorem
Many of the standard abstract mathematical structures were first defined and studied "externally" (in terms of some sort of concrete representation) and only later defined "internally" (as abstract sp …
- 114k
64
votes
Real-world applications of mathematics, by arxiv subject area?
math.AC Commutative algebra
Reed-Solomon codes (a type of error correction codes based on polynomials over finite fields - this is why CDs and DVDs still work even after being scratched!)
- 4,713
46
votes
Real-world applications of mathematics, by arxiv subject area?
math.DG Differential geometry
Lie groups are used in robotics (to find the most efficient way to maneuver a robotic arm, for instance).
Spherical trigonometry is essential for navigation (a few centu …
- 4,713
30
votes
Real-world applications of mathematics, by arxiv subject area?
math.AG Algebraic geometry
Elliptic curve cryptography
Motion planning: Configuration spaces of robot arms are semi-algebraic sets, and algebraic geometry (especially the Cylindrical Algebraic Decomp …
- 4,713
31
votes
Real-world applications of mathematics, by arxiv subject area?
math.ST Statistics
almost everywhere
Accurate polling
Fraud detection: whether financial (e.g. via Benford's law) or voter fraud.
Used in the world of finance, economics and gambling on a daily basis …
- 4,713
21
votes
Real-world applications of mathematics, by arxiv subject area?
math.NA Numerical analysis
Linear programming algorithms are used in compressed sensing, which is now being used in MRI and imaging to increase resolution and/or decrease the number of measurements r …
- 4,713