All Questions
Tagged with big-list big-picture
71 questions
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 ...
- 8,435
201
votes
67
answers
47k
views
Examples of eventual counterexamples
Define an "eventual counterexample" to be
$P(a) = T $ for $a < n$
$P(n) = F$
$n$ is sufficiently large for $P(a) = T\ \ \forall a \in \mathbb{N}$ to be a 'reasonable' conjecture to ...
- 12.9k
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 ...
- 6,349
131
votes
14
answers
30k
views
Why are modular forms interesting?
Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about ...
- 35.5k
41
votes
26
answers
12k
views
What are some slogans that express mathematical tricks?
Many "tricks" that we use to solve mathematical problems don't correspond nicely to theorems or lemmas, or anything close to that rigorous. Instead they take the form of analogies, or general methods ...
- 850
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 ...
- 1,019
-8
votes
2
answers
860
views
Homotopy theory and algebraic topology last 10 years. Is it a dying field? [closed]
I'm under the impression that algebraic topology is a dying field in mathematics. That was my impression but I think I'm wrong. As every person I do need some evidence that my impression is not ...
- 63.9k
7
votes
1
answer
2k
views
Which revolutions in topology and geometry can we expect in the next 20 years? [closed]
In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
- 30.3k
399
votes
23
answers
69k
views
Thinking and Explaining
How big a gap is there between how you think about mathematics and what you say to others? Do you say what you're thinking? Please give either personal examples of how your thoughts and words differ, ...
123
votes
9
answers
14k
views
Breakthroughs in mathematics in 2021
This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research.
So, generally speaking, which have been important ...
- 19.6k
0
votes
1
answer
454
views
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
- 188k
115
votes
32
answers
21k
views
What notions are used but not clearly defined in modern mathematics?
"Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions."
Felix Klein
What notions are used but not ...
- 6,406
230
votes
89
answers
45k
views
Your favorite surprising connections in mathematics
There are certain things in mathematics that have caused me a pleasant surprise -- when some part of mathematics is brought to bear in a fundamental way on another, where the connection between the ...
- 21.3k
15
votes
4
answers
6k
views
What is the interface between functional analysis and algebraic geometry?
This is a very open ended curiosity of mine and I would be grateful to hear any comments in this direction. In particular I am interested in functional analysis/algebraic geometry books/papers ...
- 6,367
54
votes
30
answers
7k
views
What are examples of good toy models in mathematics?
This post is community wiki.
A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to ...
- 1,362
40
votes
17
answers
10k
views
Interesting mathematical topics arising from biology
I've heard that there's a relatively new field of science called mathematical biology.
It will certainly apply well known and less known mathematical techniques to the understanding of some biological ...
- 10.5k
154
votes
26
answers
44k
views
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 ...
- 23
96
votes
36
answers
17k
views
The concept of duality
I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come....
172
votes
36
answers
35k
views
Proposals for polymath projects
Background
Polymath projects are a form of open Internet collaboration aimed towards a major mathematical goal, usually to settle a major mathematical problem. This is a concept introduced in 2009 by ...
- 103
18
votes
4
answers
4k
views
What are the "hot" topics in mathematical QFT at the time?
I am currently finishing my Master's studies in mathematical physics. One topic which always interested me a lot were modern mathematical approaches to Quantum Field Theory (QFT) as well as the ...
- 1,429
127
votes
23
answers
37k
views
Collection of equivalent forms of Riemann Hypothesis
This forum brings together a broad enough base of mathematicians to collect a "big list" of equivalent forms of the Riemann Hypothesis...just for fun. Also, perhaps, this collection could include ...
- 35.5k
72
votes
13
answers
11k
views
The use of computers leading to major mathematical advances II
I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances.
This is a continuation of a question ...
- 3,499
91
votes
24
answers
22k
views
Examples of major theorems with very hard proofs that have not dramatically improved over time
This question complement a previous MO question: Examples of theorems with proofs that have dramatically improved over time.
I am looking for a list of
Major theorems in mathematics whose proofs are ...
- 123
156
votes
52
answers
24k
views
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 ...
- 2,821
13
votes
3
answers
6k
views
Linear/Non-linear sigma model
This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model.
I gauge from some other interactions that it has considerable ...
- 2,821
4
votes
1
answer
369
views
Examples of rich theories that started out as an infinite-dimensional inquiry
It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it ...
- 485
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 ...
- 1,068
25
votes
3
answers
2k
views
Interpretations and models of permanent
The standard interpretation of permanent of a $0/1$ matrix if considered as a biadjacency matrix of a bipartite graph is number of perfect matchings of the graph or if considered as a adjacency matrix ...
- 13.9k
41
votes
4
answers
6k
views
Linear algebra in terms of abstract nonsense?
The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think.
I was wondering what portions of basic linear algebra (first couple of courses) fall ...
- 258
5
votes
2
answers
499
views
Critical points in $ZF$ without Choice
Recall the definition of critical point for set theory:
A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...
- 236k
75
votes
13
answers
13k
views
What precisely Is "Categorification"?
(And what's it good for.)
Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
- 39.9k
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 ...
- 12.9k
37
votes
14
answers
5k
views
What are interesting families of subsets of a given set?
Motivation
The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$.
Indeed, one defines a topology on $S$ to be a family of subsets ...
- 5,407
42
votes
12
answers
7k
views
Why is the definition of the higher homotopy groups the "right one"?
If someone asked me the question for the fundamental group, I would answer as follows:
The connection to classification of covering spaces.
The fundamental group of many spaces is an object of ...
- 14.4k
101
votes
10
answers
16k
views
Why do Bernoulli numbers arise everywhere?
I have seen Bernoulli numbers many times, and sometimes very surprisingly. They appear in my textbook on complex analysis, in algebraic topology, and of course, number theory. Things like the criteria ...
- 1,020
5
votes
3
answers
810
views
Update on "Hopf algebras: their status and pervasiveness" by Hazewinkel
Hazewinkel wrote this article in 2005. Perhaps it's time for an update.
For example, updating item
34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of ...
- 149
5
votes
1
answer
291
views
The Idea of Kroneckerian geometry
Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$.
Let's ...
- 7,746
14
votes
3
answers
1k
views
Can there be a polymath project for mathematical physics?
My hunch is that it might be possible to create something like https://polymathprojects.org/ for mathematical physics and I'd like to know whether MathOverflow users can recommend some appropriate ...
- 219
42
votes
26
answers
8k
views
Where can square roots come from when they are not distances?
In a recent survey "Supergeometry in Mathematics and Physics", Kapranov points out cases in which observable quantities of immediate interest are represented as bilinear combinations of more ...
- 8,992
142
votes
17
answers
23k
views
What makes four dimensions special?
Do you know properties which distinguish four-dimensional spaces among the others?
What makes four-dimensional topological manifolds special?
What makes four-dimensional differentiable manifolds ...
- 83
220
votes
140
answers
49k
views
Fundamental Examples
It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)
I'd love to learn about ...
50
votes
48
answers
13k
views
Describe a topic in one sentence. [closed]
When you study a topic for the first time, it can be difficult to pick up the motivations and to understand where everything is going. Once you have some experience, however, you get that good high-...
CommunityBot
- 1
40
votes
6
answers
8k
views
Doing geometry using Feynman Path Integral?
I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space.
Coming from a background of studying Quantum Field Theory from the books like ...
- 10.5k
251
votes
29
answers
168k
views
Intuitive crutches for higher dimensional thinking
I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...
CommunityBot
- 1
50
votes
37
answers
6k
views
Structures that turn out to exhibit a symmetry even though their definition doesn't
Sometimes (often?) a structure depending on several parameters turns out to be symmetric w.r.t. interchanging two of the parameters, even though the definition gives a priori no clue of that symmetry. ...
- 12.9k
96
votes
50
answers
43k
views
Theorems that are 'obvious' but hard to prove
There are several well-known mathematical statements that are 'obvious' but false (such as the negation of the Banach--Tarski theorem). There are plenty more that are 'obvious' and true. One would ...
CommunityBot
- 1
4
votes
2
answers
214
views
Viewing parts of $\mathbb{V}$ 'from the top down' or 'from the bottom up'
I am curious about instances where we can glean nontrivial information about a certain piece of the universe by viewing it as being 'built over' a smaller part of the universe, or alternatively ...
- 236k
45
votes
8
answers
10k
views
What is Realistic Mathematics?
This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...
- 505
18
votes
8
answers
2k
views
Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found.
A good example is Lovász's proof of ...
- 3,652
4
votes
1
answer
325
views
Maximality without Zorn
When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma.
I am interested in instances of proving the existence of ...
- 4,713