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1500 questions
150
votes
31 answers
Extremely messy proofs
Currently in my undergraduate courses I am being taught how to set up various machinery using slick, short proofs and then how to apply that machinery. What I am not being taught, largely, is what came before these slick, short proofs. What did…
Qiaochu Yuan
- 114,941
150
votes
12 answers
"Philosophical" meaning of the Yoneda Lemma
The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward.
Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning how it has deeper implications into how to think…
Sam Derbyshire
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150
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45 answers
Nontrivial theorems with trivial proofs
A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because its proof is trivial.
I don't remember who said…
Michael Hardy
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150
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26 answers
Has philosophy ever clarified mathematics?
I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any mathematical insights from them.
My naïve expectation would…
Daniel Moskovich
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149
votes
38 answers
Computer algebra errors
In the course of doing mathematics, I make extensive use of computer-based calculations. There's one CAS that I use mostly, even though I occasionally come across out-and-out wrong answers.
After googling around a bit, I am unable to find a list of…
Kevin O'Bryant
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148
votes
7 answers
Homotopy groups of Lie groups
Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, perhaps like a more clever version of the proof that…
Matt Noonan
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148
votes
31 answers
What are the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in practice. Are there examples of equivalent…
QPeng
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148
votes
26 answers
Good "casual" advanced math books
I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks rigorously, and I find them too dense to parse…
user3002473
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148
votes
71 answers
Nonequivalent definitions in Mathematics
I would like to ask if anyone could share any specific experiences of
discovering nonequivalent definitions in their field of mathematical research.
By that I mean discovering that in different places in the literature, the same name is used for…
Angeliki Koutsoukou Argyraki
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148
votes
2 answers
What is a Frobenioid?
Since there will be a long digression in a moment, let me start by reassuring you that my intention really is to ask the question in the title.
Recently, there has been a flurry of new discussion surrounding Shinichi Mochizuki's interuniversal…
Minhyong Kim
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147
votes
11 answers
Is Fourier analysis a special case of representation theory or an analogue?
I'm asking this question because I've been told by some people that Fourier analysis is "just representation theory of $S^1$."
I've been introduced to the idea that Fourier analysis is related to representation theory. Specifically, when considering…
David Corwin
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147
votes
43 answers
Where does a math person go to learn quantum mechanics?
My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn quantum for some time now, and after seeing it…
Qiaochu Yuan
- 114,941
146
votes
18 answers
Suggestions for special lectures at next ICM
(I am posting this in my capacity as chair of the ICM programme committee.)
ICM 2022 will feature a number of "special lectures", both at the sectional and plenary level, see last year's report of the ICM structure committee. The idea is that these…
Martin Hairer
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146
votes
66 answers
Important formulas in combinatorics
Motivation:
The poster for the conference celebrating Noga Alon's 60th birthday, fifteen formulas describing some of Alon's work are presented. (See this post, for the poster, and cash prizes offered for identifying the formulas.) This demonstrates…
Gil Kalai
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146
votes
21 answers
Mathematical software wish list
Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course $\LaTeX$ extensively. When I chat with colleagues about such software tools, very often someone has an idea of how to extend an existing tool, what…
eins6180
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