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1500 questions
58
votes
3 answers
What is the geometry of an undecidable diophantine equation?
As an arithmetic algebraic geometer of the highest moral fiber, I am trained to look at Diophantine equations in terms of the geometry of the corresponding scheme. For instance, if the Diophantine equation comes from a curve, I know that I should…
Will Sawin
- 135,926
58
votes
4 answers
Why do Todd classes appear in Grothendieck-Riemann-Roch formula?
Suppose for some reason one would be expecting a formula of the kind
$$\mathop{\text{ch}}(f_!\mathcal F)\ =\ f_*(\mathop{\text{ch}}(\mathcal F)\cdot t_f)$$
valid in $H^*(Y)$ where
$f:X\to Y$ is a proper morphism with $X$ and $Y$ smooth and…
Ilya Nikokoshev
- 14,934
57
votes
30 answers
Examples of conjectures that were widely believed to be true but later proved false
It seems to me that almost all conjectures (hypotheses) that were widely believed by mathematicians to be true were proved true later, if they ever got proved. Are there any notable exceptions?
Eric
- 2,601
57
votes
3 answers
Every prime number > 19 divides one plus the product of two smaller primes?
This is a part of my answer to this question I think it deserves to be treated separately.
Conjecture Let $A$ be the set of all primes from $2$ to $p>19$. Let $q$ be the next prime after $p$. Then $q$ divides $rs+1$ for some $r,s\in A$.
I wonder…
user6976
57
votes
5 answers
Why are spectral sequences so ubiquitous?
I sort of understand the definition of a spectral sequence and am aware that it is an indispensable tool in modern algebraic geometry and topology. But why is this the case, and what can one do with it? In other words, if one were to try to do…
Akhil Mathew
- 25,291
57
votes
1 answer
Every real function has a dense set on which its restriction is continuous
The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|_D$ is continuous.
Or so I'm told, but this leaves me stumped. Apart from the rather trivial fact that…
Gro-Tsen
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57
votes
23 answers
A good book of functional analysis
I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics). What I am looking for is good books that I…
dan232
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57
votes
8 answers
There must be a good introductory numerical analysis course out there!
Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant engineering majors and some majors of mathematics.…
Nilima Nigam
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57
votes
25 answers
Cocktail party math
Ok, hotshots. You're at a party, and you're chatting with some non-mathematicians. You tell them that you're a mathematician, and then they ask you to elaborate a bit on what you study, or they ask you to explain why you like math so much.
What are…
Kevin H. Lin
- 20,738
57
votes
14 answers
Open problems in Euclidean geometry?
What are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit what I mean by "Euclidean geometry". By this term I…
Qfwfq
- 22,715
57
votes
5 answers
How to make Ext and Tor constructive?
EDIT: This post was substantially modified with the help of the comments and answers. Thank you!
Judging by their definitions, the $\mathrm{Ext}$ and $\mathrm{Tor}$ functors are among the most non-constructive things considered in algebra:
(1)…
darij grinberg
- 33,095
57
votes
6 answers
Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?
If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The existence of a non-zero linear functional can be…
Konrad Swanepoel
- 3,481
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57
votes
10 answers
Books/websites which have motivating stories of mathematicians overcoming hardships in life
Edit 1: I have received a lot of great answers. I am not accepting any answer because I think there might be in future that some user want to contribute any new answer, as in my opinion some users might find the answers useful to them to keep their…
Arnold
- 668
57
votes
5 answers
What about a mathematics journal for 'negative' results?
In the empirical sciences, there are a number of journals that publish 'negative' results. Negative or null results occur when researchers are unable to confirm the findings obtained from earlier published reports. In the applied sciences, they may…
Max Muller
- 4,505
57
votes
5 answers
Examples where "thin + thin = nice and thick"
I'm interested in examples where the sum of a set with itself is a substantially bigger set with nice structure. Here are two examples:
Cantor set: Let $C$ denote the ternary Cantor set on the interval $[0,1]$. Then $C+C = [0,2]$. There are several…
Marc Chamberland
- 644