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1500 questions
63
votes
7 answers
What well known results with countability assumptions can be naturally extended to uncountable settings?
In many of the common categories of spaces (or algebras) in mathematics, one often restricts attention to those spaces or algebras which are "countable" or "countably generated" in some sense. For instance:
When studying topological spaces, it is…
Terry Tao
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63
votes
27 answers
Golden ratio in contemporary mathematics
A (non-mathematical) friend recently asked me the following question:
Does the golden ratio play any role in contemporary mathematics?
I immediately replied that I never come across any mention of the golden ratio in my daily work, and would guess…
johhnyelgerton
- 113
63
votes
52 answers
Colloquial catchy statements encoding serious mathematics
As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as answers so they may be voted up and down with the…
Armin Straub
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63
votes
2 answers
Geometric interpretation of characteristic polynomial
The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its intermediate coefficients?
For a linear operator…
Per Vognsen
- 2,041
63
votes
2 answers
Guessing each other's coins
I recently thought about the following game (has it been considered before?).
Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an integer $a$. Similarly, Bob observes a sequence of…
Guillaume Aubrun
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63
votes
3 answers
Class field theory - a "dead end"?
I found the claim in the title a bit astonishing when I first read it recently in an interview with Michael Rapoport in the German magazine Spiegel (8 February 2019). And I was wondering how he comes to that conclusion.
Here is the article, but the…
wood
- 2,714
63
votes
6 answers
Why isn't integral defined as the area under the graph of function?
In order to define Lebesgue integral, we have to develop some measure theory. This takes some effort in the classroom, after which we need additional effort of defining Lebesgue integral (which also adds a layer of complexity). Why do we do it this…
user57888
- 1,229
63
votes
20 answers
What should we teach to liberal arts students who will take only one math course?
Even professors in academic departments other than mathematics---never mind other educated people---do not know that such a field as mathematics exists. Once a professor of medicine asked me whether it is necessary to write a thesis to get a Ph.D.…
Michael Hardy
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63
votes
1 answer
Feit-Thompson conjecture
The Feit-Thompson conjecture states:
If $p
Mare
- 25,794
63
votes
10 answers
Fascinating moments: equivalent mathematical discoveries
One of the delights in mathematical research is that some (mostly deep) results in one area remain unknown to mathematicians in other areas, but later, these discoveries turn out to be equivalent!
Therefore, I would appreciate any recollections…
T. Amdeberhan
- 41,802
63
votes
3 answers
A roadmap to Hairer's theory for taming infinities
Background
Martin Hairer gave recently some beautiful lectures in Israel on "taming infinities," namely on finding a mathematical theory that supports the highly successful computations from quantum field theory in physics.
(Here are slides of a…
Gil Kalai
- 24,218
63
votes
5 answers
Intuition about the cotangent complex?
Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as geometric ones (like "homology detects holes"), as are…
Peter Arndt
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63
votes
7 answers
Theorems demoted back to conjectures
Many mathematicians know the Four Color Theorem and its history: there were two alleged proofs in 1879 and 1880 both of which stood unchallenged for 11 years before flaws were discovered.
I am interested in more such examples, especially, in former…
Moritz
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63
votes
10 answers
Mathematical applications of quantum field theory
I understand that quantum field theories are interesting as physics; however, there is also a large community of mathematicians who are interested in them. For someone who is not at all interested in physics, what are some compelling mathematical…
Sarah
- 279
63
votes
5 answers
What is modern algebraic topology(homotopy theory) about?
At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other areas of mathematics, especially, in geometry(I…
BASIL478
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