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1500 questions
158
votes
48 answers
Generalizing a problem to make it easier
One of the many articles on the Tricki that was planned but has never been written was about making it easier to solve a problem by generalizing it (which initially seems paradoxical because if you generalize something then you are trying to prove a…
gowers
- 28,729
157
votes
28 answers
How To Present Mathematics To Non-Mathematicians?
(Added an epilogue)
I started a job as a TA, and it requires me to take a five sessions workshop about better teaching in which we have to present a 10 minutes lecture (micro-teaching).
In the last session the two people in charge of the workshop…
Asaf Karagila
- 38,140
155
votes
11 answers
Why are flat morphisms "flat?"
Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason!
Is there some geometric property corresponding to…
Harrison Brown
- 12,543
154
votes
26 answers
What recent discoveries have amateur mathematicians made?
E.T. Bell called Fermat the Prince of Amateurs. One hundred years ago Ramanujan amazed the mathematical world. In between were many important amateurs and mathematicians off the beaten path, but what about the last one hundred years? Is it still…
user37691
154
votes
54 answers
Old books you would like to have reprinted with high-quality typesetting
There are some questions on mathoverflow such as
What out-of-print books would you like to see re-printed?
Old books still used
with answers that tell us things such as:
Mathematicians prefer to use older books because of some old books are full…
C.F.G
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154
votes
7 answers
Where to buy premium white chalk in the U.S., like they have at RIMS?
While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white chalk I've seen anywhere. It's slightly larger than…
Allen Knutson
- 27,645
154
votes
4 answers
Does there exist a bijection of $\mathbb{R}^n$ to itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ is connected.
Question: Assume…
Willie Wong
- 37,551
154
votes
52 answers
Experimental mathematics leading to major advances
I would like to ask about examples where experimentation by computers has led to major mathematical advances.
A new look
Now as the question is five years old and there are certainly more examples of mathematical advances via computer…
Gil Kalai
- 24,218
153
votes
4 answers
Analytic tools in algebraic geometry
This is not a very precise question, but I hope it will get some good answers.
As someone with a background in smooth manifold theory, I have experienced algebraic geometry as a beautiful but foreign territory. The strangeness has a lot to do with…
Tom Goodwillie
- 54,421
153
votes
7 answers
Consequences of the Riemann hypothesis
I assume a number of results have been proven conditionally on the Riemann hypothesis, of course in number theory and maybe in other fields. What are the most relevant you know?
It would also be nice to include consequences of the generalized…
Andrea Ferretti
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152
votes
5 answers
What makes dependent type theory more suitable than set theory for proof assistants?
In his talk, The Future of Mathematics, Dr. Kevin Buzzard states that Lean is the only existing proof assistant suitable for formalizing all of math. In the Q&A part of the talk (at 1:00:00) he justifies this as follows:
Automation is very…
MWB
- 1,617
152
votes
6 answers
Proofs shown to be wrong after formalization with proof assistant
Are there examples of originally widely accepted proofs that were later discovered to be wrong by attempting to formalize them using a proof assistant (e.g. Coq, Agda, Lean, Isabelle, HOL, Metamath, Mizar)?
Mei Zhang
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151
votes
13 answers
Why is the fundamental group of a compact Riemann surface not free ?
Consider a compact Riemann surface $X$ of genus $g$.
It is well-known that its fundamental group $\pi_1(X)$ is the free group on the generators $a_1,b_1,...,a_g,b_g$ divided out by the normal subgroup generated by the single relator…
Georges Elencwajg
- 46,833
151
votes
18 answers
Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ones of $p=1$, $p=2$, and $p=\infty$. I don't…
Timothy Chow
- 78,129
150
votes
26 answers
A soft introduction to physics for mathematicians who don't know the first thing about physics
There have been similar questions on mathoverflow, but the answers always gave some advanced introduction to the mathematics of quantum field theory, or string theory and so forth. While those may be good introduction to the mathematics of those…
James D. Taylor
- 6,178