Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. In other words, questions that can be answered without making computations or applying theorems and axioms.

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25
votes
3answers
2k views

Is it possible to have a research career while checking the proof of every theorem that you cite?

A colleague raised the above question with me; more precisely he said: Suppose that a mathematician were resolved not to publish any theorems unless she had checked the proof of every theorem ...
-6
votes
0answers
139 views

Great Mathematicians Without a PhD [on hold]

While listing to some music, I was wondering which great mathematicians did not have or do not have a PhD. This is a very subjective question, since "great" is not formally defined. But to describe it ...
29
votes
8answers
4k views

Ubiquity of the push-pull formula

The push-pull formula appears in several different incarnations. There are, at least, the following: 1) If $f \colon X \to Y$ is a continous map, then for sheaves $\mathcal{F}$ on $X$ and ...
18
votes
2answers
2k views

Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...
58
votes
25answers
4k views

What could be some potentially useful mathematical databases?

This is a soft question but it's not meant as a big-list question. I have recently been asked whether I want to provide feedback at the pre-beta stage on a forthcoming website that will provide a ...
22
votes
3answers
2k views

Adapting arguments and plagiarism

I'm currently working on my PhD thesis. I have several suggested problems to work on, some of them are very similar to some problems that my advisor have worked before and published already, either in ...
8
votes
0answers
261 views

Why do we study symplectic geometry? [closed]

What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form? The subject certainly originated from physics, but is there a deeper reason for why it is still an ...
22
votes
24answers
7k views

Is there an image for you that epitomizes mathematics? [closed]

Can you think of an image, whether technical or nontechnical, available for viewing online that says a lot about what you think mathematics or a particular field of mathematics is all about? For ...
3
votes
1answer
166 views

Early examples of problems that are easier in high dimension

In many areas of mathematics, there are problems that admit a natural formulation in any dimension. It often happens that such a problem is easier to solve in dimension $n>k$ as compared to ...
-7
votes
0answers
39 views

Anyone else here believe entangled particles create mini wormholes? [closed]

The properties of their interactions are a little too similar to ignore. Entangled objects ability to align regardless of distance. Their ability to align seemingly even before the other is viewed. ...
3
votes
1answer
179 views

How to learn concepts of Functional Analysis which are common in PDE

I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, ...
64
votes
56answers
14k views

Pseudonyms of famous mathematicians

Many mathematicians know that Lewis Carroll was quite a good mathematician, who wrote about logic (paradoxes) and determinants. He found an expansion formula, which bears his real name (Charles ...
1
vote
0answers
99 views

Referencing your own research paper on a conference board? [closed]

Is it considered poor etiquette to refer a viewer to a research paper while looking at a conference poster? The paper could be placed on the same table so it is readily accessible.
32
votes
13answers
3k views

What math institutes offer research in pairs/research in teams?

Some math institutes offer programs in which a small number of researchers are enabled to meet at the institute for a week or more. A list seemed as if it could be useful.
12
votes
1answer
508 views

Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...
6
votes
3answers
311 views

Connection between solution for Schrödinger equation and solution for heat equation

It's known, that if you write imaginary unit into a heat equation you'll get time-dependent Schrödinger equation. Recently one guy discovered a connection between solutions for these two equations ...
22
votes
10answers
1k views

Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...
42
votes
31answers
7k views

Trichotomies in mathematics

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the ...
2
votes
0answers
286 views

Do Peano curves provide a counterargument to Grothendieck's critique?

This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be ...
7
votes
1answer
264 views

What does “game theory” cover and how should it be called?

There seems to be a huge discrepancy in what people refer to when they speak of "game theory". I tend to think of it as including, among other things: Combinatorial game theory dealing with certain ...
82
votes
7answers
8k views

Mistakes in mathematics, false illusions about conjectures

Since long time ago I have been thinking in two problems that I have not been able to solve. It seems that one of them was recently solved. I have been thinking a lot about the motivation and its ...
18
votes
7answers
3k views

Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
2
votes
0answers
57 views

What does the square root sign tells us in the wave equation? [closed]

I have been reading the paper on wave equations, and I have some confusion in notations. Consider the initial value problem(IVP)(Wave equation): $\frac{\partial ^2 u } {\partial t^2}(x,t) = ...
113
votes
40answers
11k views

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 ...
27
votes
16answers
3k views

What are some examples of narrowly missed discoveries in the history of mathematics?

What are the examples of some mathematicians coming very close to a very promising theory or a correct proof of a big conjecture but not making or missing the last step?
36
votes
5answers
1k views

Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...
43
votes
5answers
2k views

How do you mentor undergraduate research?

Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that. There are two slightly more specific groups of questions I have ...
7
votes
2answers
377 views

Famous results about the value of a given limit assuming it exists

Chebyshev got famous showing that if the limit $l:=\lim_{x\to\infty}\frac{\pi(x)}{x/\log x}$ exists, then necessarily $l=1$, constituting a major breakthrough towards a proof of the famous prime ...
7
votes
0answers
272 views

Errata in EGA, collected

There is an extensive list of EGA's errata on the books themselves, but my question is whether new errata, that is those found by various mathematicians after the publication, are collected somewhere. ...
19
votes
8answers
4k views

MathSciNet vs Google Scholar

What are the pros and cons of the MathSciNet database vs Google Scholar? I don't have access to Mathscinet so this question is out of curiosity, and also this question where MathSciNet is used to ...
8
votes
5answers
429 views

Applications of functional analysis beyond analysis(towards algebra, geometry, number theory…) [closed]

So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional ...
28
votes
36answers
4k views

What are some mathematical sculptures?

Either intentionally or unintentionally. Include location and sculptor, if known.
117
votes
39answers
33k views

Most interesting mathematics mistake?

Some mistakes in mathematics made by extremely smart and famous people can eventually lead to interesting developments and theorems, e.g. Poincaré's 3d sphere characterization or the search to prove ...
127
votes
44answers
75k views

Magic trick based on deep mathematics

I am interested in magic tricks whose explanation requires deep mathematics. The trick should be one that would actually appeal to a layman. An example is the following: the magician asks Alice to ...
2
votes
0answers
305 views

First few research papers [closed]

I was planning on posting this on academia.stackexchange, but I want an answer from mathematicians who've dealt with a similar issue when they were beginning graduate students. If this site doesn't ...
38
votes
9answers
11k views

Is Galois theory necessary (in a basic graduate algebra course)?

By definition, a basic graduate algebra course in a U.S. (or similar) university with a Ph.D. program in mathematics lasts part or all of an academic year and is taken by first (sometimes second) ...
43
votes
10answers
2k views

What advantage humans have over computers in mathematics?

Now that AlphaGo has just beaten Lee Sedol in Go and Deep Blue has beaten Garry Kasparov in chess in 1997, I wonder what advantage humans have over computers in mathematics? More specifically, are ...
79
votes
53answers
26k views

Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in the game's structure, optimal strategies, practical strategies, analysis of the game ...
22
votes
4answers
1k views

Expert, Intuitive, Organizing Analogies

In learning a new area it is very helpful to have high-level intuitive analogies that keep track of the various parts of an important argument or strategy. Experts have a store of such things, and ...
256
votes
72answers
95k views

Video lectures of mathematics courses available online for free

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some ...
7
votes
4answers
1k views

Hecke-algebras in your field of mathematics

(How) do Hecke-algebras arise naturally in your field of mathematics and why are they important? How would you define them and how do you think about them? e.g. generators and relations, functions ...
33
votes
1answer
2k views

Are there any “homotopical spaces”?

This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense. [Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is ...
56
votes
12answers
3k views

Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations. Why was there the necessity of singling ...
11
votes
0answers
184 views

Multiplicative infinitesimals in q-analogs?

Risking to be downvoted, here is a very lightweight question. In various fields - say, algebraic geometry, nonstandard analysis, synthetic differential geometry - infinitely small quantities, i. e. ...
179
votes
36answers
48k views

Widely accepted mathematical results that were later shown wrong?

I wonder if there are any examples in the history of mathematics of a mathematical proof that was initially reviewed and widely accepted as valid, only to be disproved a significant amount of time ...
80
votes
17answers
11k views

Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence: "The unorientable surfaces are never discussed ...
23
votes
4answers
2k views

In what ways did Leibniz's philosophy foresee modern mathematics?

Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...
35
votes
13answers
3k views

Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
45
votes
6answers
16k views

How many mathematicians are there?

Although we are not so numerous as other respected professionals, like for example lawyers, I wonder if we could come up with a reasonable estimate of our population. Needless to say, the question ...
4
votes
1answer
396 views

How many papers are posted a year? [closed]

How many pure math papers are published a year? I vaguely remember seeing a figure of 10,000 but that might be old, and I may be wrong.