All Questions
Tagged with big-list soft-question
317 questions
9
votes
2
answers
473
views
Completing half of Hilbert's program: Foundations that are conservative over Peano Arithmetic
The goal of the Hilbert program was to find a complete and consistent formalization of mathematics. Gödel's first incompleteness theorem establishes that completeness is impossible with first-order ...
- 6,353
9
votes
1
answer
506
views
Current state of the art in geometric complexity theory
I came across this interesting question from almost 7 years ago:
What are the current breakthroughs of Geometric Complexity Theory?
My question is quite simple: Have there been any breakthroughs in ...
14
votes
5
answers
5k
views
Who says understanding physics helps mathematicians? (A reference request) [Take the word "who" literally.]
If I wanted to make a somewhat bold and rather vague claim in print that it is widely acknowledged among mathematicians that knowledge of mechanics (in the sense in which physicists understand that ...
53
votes
1
answer
9k
views
What mathematical problems can be attacked using DeepMind's recent mathematical breakthroughs?
I am a research mathematician at a university in the United States. My training is in pure mathematics (geometry). However, for the past couple of months, I have been supervising some computer science ...
- 765
38
votes
7
answers
18k
views
Daunting papers/books and how to finally read them
Most people throughout their career encounter at least one paper that seems especially daunting to them.
I'm interested in real stories of how you successfully overcame that to extract the knowledge ...
93
votes
9
answers
13k
views
Breakthroughs in mathematics in 2023
At the end of 2021, Johnny Cage asked about breakthroughs in 2021 in different mathematical disciplines. A similar question has been asked at the end of 2022, so it looks like Johnny Cage originated a ...
14
votes
2
answers
1k
views
Famous papers published in annotated form?
I very much enjoyed reading through The Annotated Turing which goes through Turing's "On Computable Numbers, with an Application to the Entscheidungsproblem" in careful detail. I saw this ...
- 323
33
votes
3
answers
2k
views
The probabilistic method outside of discrete mathematics
The probabilitic method is a genius idea in combinatorics, graph theory etc, where instead of constructing something by hand, you construct the thing randomly and show that there is a positive ...
12
votes
7
answers
3k
views
Books containing new results
In Endless controversy about the correctness of significant papers, Denis Serre writes:
The research community is able to point out incorrect statements, at least among those which have some ...
49
votes
2
answers
5k
views
Well known theorems that have not been proved
I believe that there are numerous challenging theorems in mathematics for which only a sketch of a proof exists. To meet the standards of rigor, a complete proof of these theorems has yet to be ...
5
votes
4
answers
499
views
Funding programs for mathematical research [closed]
In the USA, as far as I know, the main grants available to mathematicians are collected on the NSF or the AMS websites [please, correct me if this perception is inaccurate]. On the other hand, for ...
1
vote
1
answer
363
views
Adjunctions in the real world
What concepts in the real world can be described by adjunctions?
For example, parents and children are adjoint to one another. Specifically, work in $ZFC$ plus a finite class of atoms $\mathscr{X}$ (...
- 10.1k
8
votes
1
answer
388
views
Formalisation of intuitive concepts in the language leading to mathematical progress
In his work, Albert Lautman thinks the genesis of some mathematical works as a dialectic that takes place between opposite notions, like between global and local. He argues that while those notions, ...
- 531
1
vote
0
answers
387
views
Mathematical technicalities that few people know [closed]
I am looking for a list of mathematical technicalities that are not so well-known, even in the mathematical community. What I mean is, I am looking for examples of phenomenon where it is important to ...
47
votes
7
answers
8k
views
Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians
I would like to ask a question inspired by the title of a book by Sir Roger Penrose ([1]). The germ of this is to ask about the role, if any, of the fashion in research of pure and applied mathematics....
6
votes
2
answers
447
views
Common/well-known results with natural and/or useful reformulations
$\DeclareMathOperator{\pp}{\mathbb{P}}$My aim here is to have a collection of "natural" not-so-common reformulations/extensions of common/well-known results such that
the reformulation/...
52
votes
14
answers
9k
views
Modern results that are widely known, yet which at the time were ignored, not accepted or criticized
What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It ...
4
votes
0
answers
421
views
What are your common strategies/remedies when your new theory/idea stuck in most cases?
Sorry if this is not a suitable post for MO.
Sometimes after reading the origin of a theory/idea in differential topology I put myself in the shoes of that mathematician and ask myself, Did you do the ...
- 4,195
0
votes
0
answers
214
views
Stories where a different definition lead to an inaccurate conclusion/a misunderstanding/etc
The overall question: What are some good examples where a different understanding of terminology or notation caused you to misinterpret a result in a way that was inaccurate? The intent here is of ...
6
votes
0
answers
287
views
Mathematical questions or areas amenable to AI [duplicate]
This question regards the new paper "Advancing mathematics by guiding human intuition with AI" by Davies et al. (Nature, 2021) (DOI link in open access) in which researchers at Deepmind ...
80
votes
22
answers
15k
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How would you have answered Richard Feynman's challenge?
Reading the autobiography of Richard Feynman, I struck upon the following paragraphs, in which Feynman recall when, as a student of the Princeton physics department, he used to challenge the students ...
40
votes
11
answers
5k
views
Results with short, advanced proofs or long, elementary proofs
Recently I was preparing an undergrad-level proof of (a form of) the Jordan Curve Theorem, and I had forgotten just how much work is involved in it. The proof stored my head was just using Alexander ...
22
votes
1
answer
3k
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What is so special about Chern's way of teaching?
First of all sorry for this non-research post.
I was watching Jeffrey Blitz Lucky documentary movie and it was interesting to me that a winner of Lottery was a math Ph.D. from Berkeley.
In the movie ...
- 4,195
58
votes
5
answers
7k
views
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 ...
30
votes
15
answers
6k
views
Lunch seminars for PhD students
The problem that I would like to ask about is metamathematical, but I hope the question is appropriate.
I would like to know if there exist mathematical departments that run a regular seminar for all ...
17
votes
3
answers
2k
views
Theoretical results on neural networks
With this question I'd like to have a recollection of theoretical rigorous results on neural networks.
I'd like to have results that have been settled, as opposed to hypothesis. As an example, this ...
44
votes
26
answers
9k
views
Theorems with many distinct proofs
I was told that whenever one learns a new technique, it is a good idea to see if one can prove a well-known theorem using the new technique as an exercise. I am hoping to build a list of such theorems ...
52
votes
6
answers
5k
views
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false
What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false?
One interesting example, and the impetus for this question, is work in number ...
53
votes
10
answers
7k
views
Changes forced by the pandemic
The Covid-19 pandemic has changed our work-lives in ways few of us could have anticipated. These exceptional circumstances have forced each one of us and each one of our institutions to adapt, ...
6
votes
3
answers
558
views
Anomalous phenomena [closed]
What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern?
This is most interesting when ...
35
votes
12
answers
3k
views
No canonical isomorphism [duplicate]
I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since ...
92
votes
11
answers
15k
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What are possible applications of deep learning to research mathematics?
With no doubt everyone here has heard of deep learning, even if they don't know what it is or what it is good for. I myself am a former mathematician turned data scientist who is quite interested in ...
10
votes
2
answers
2k
views
How professional mathematicians deal with discouragement? [closed]
All professional mathematicians feel discouraged occasionally due to some issue.
My question is:
How do professional mathematicians deal with discouragement?
In this link , Andrew Wiles say ...
44
votes
10
answers
11k
views
What kid-friendly math riddles are too often spoiled for mathematicians?
Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples:
What is $1+2+\cdots+100$?
Is it possible to tile a mutilated chess board with dominoes?...
14
votes
29
answers
7k
views
Which great mathematicians had great political commitments? [closed]
Some mathematicians claim that their field has nothing to do with political concerns; others are deeply involved in political life.
Are there many great mathematicians with great political commitments?...
195
votes
18
answers
17k
views
Great graduate courses that went online recently
In 09.2020 by pure chance I discovered the YouTube channel of Richard Borcherds where he gives graduate courses in Group Theory, Algebraic Geometry, Schemes, Commutative Algebra, Galois Theory, Lie ...
45
votes
4
answers
4k
views
How to invoke constants badly
In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways ...
132
votes
22
answers
11k
views
Books that teach other subjects, written for a mathematician
Say I am a mathematician who doesn't know any chemistry but would like to learn it. What books should I read?
Or say I want to learn about Einstein's theory of relativity, but I don't even know much ...
2
votes
2
answers
856
views
Easy to explain conjectures that are still unsolved [duplicate]
Mathematics has many open conjectures which are ridiculously hard to even understand. But this is not always the case. An example is:
Collatz conjecture.
I would like to see some more examples. So ...
36
votes
5
answers
4k
views
Advice for researchers outside academia
Perhaps some personal background is relevant to this question. A couple of years ago, I graduated with a master's degree in Applied Mathematics from a good Dutch university. Even though I obtained ...
38
votes
6
answers
3k
views
Online, evolving, collaborative foundational text projects
There are two online, evolving, collaborative "foundational text" projects for research mathematicians that I am aware of:
(1) The Stacks Project for algebraic geometry
(2) Kerodon for ...
58
votes
82
answers
18k
views
Prominent non-mathematical work of mathematicians
First of all, sorry if this post is not appropriate for this forum.
I have a habit that every time I read a beautiful article I look at the author's homepage and often find amazing things.
Recently I ...
147
votes
18
answers
14k
views
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 ...
90
votes
14
answers
9k
views
Time-saving (technology) tricks for writing papers
I have over the years learned some tricks which saves a lot of time,
and I wish I had known them earlier. Some tricks are LaTeX-specific, but other tricks are more general. Let me start with a few ...
6
votes
1
answer
570
views
Explanations simple enough that non-mathematical audiences can understand [closed]
The following (debatable) quote is attributed to Einstein:
"You do not really understand something unless you can explain it to your grandmother."
I feel very enlightened when there is a ...
85
votes
19
answers
15k
views
Each mathematician has only a few tricks
The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
170
votes
47
answers
34k
views
Every mathematician has only a few tricks
In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
53
votes
15
answers
5k
views
Request for examples: verifying vs understanding proofs
My colleague and I are researchers in philosophy of mathematical practice and are working on developing an account of mathematical understanding. We have often seen it remarked that there is an ...
5
votes
5
answers
2k
views
Terminology introduced in recent years with more than one meaning
Suppose a term(inology) is recently (in last 20 years) introduced in research mathematics.
It might happen that some one who wish to use it, in the same area of research, for different purposes or ...
43
votes
9
answers
6k
views
What are some examples of theorem requiring highly subtle hypothesis?
I would like you to expose and explain briefly some examples of theorems having some hypothesis that are (as far as we know) actually necessary in their proofs but whose uses in the arguments are ...