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.

learn more… | top users | synonyms

4
votes
1answer
155 views

soft: Reference/ Suggested Read: Homological Algebraic techniques in PDEs

I was reading this article on wikipiedia and was interested by the apparent link between Homological Algebra and PDEs. What is an accessible reference which showcases the link between these topics? ...
48
votes
14answers
5k views

How to write popular mathematics well?

Recently, some classmates and I were lamenting the fact that our classmates in other disciplines had almost no conception of what we did, despite the large mathematics population at Waterloo. Instead ...
81
votes
53answers
15k views

Counterexamples in Algebra?

This is certainly related to "What are your favorite instructional counterexamples?", but I thought I would ask a more focused question. We've all seen Counterexamples in Analysis and Counterexamples ...
36
votes
23answers
9k views

Open source mathematical software

I want some recommendation on which software I should install on my computer. I'm looking for an open source program for general abstract mathematical purposes (as opposed to applied mathematics). I ...
2
votes
0answers
127 views

Why only Normed Linear Spaces? [on hold]

It is well known that "Norm on a vector space can be used to obtain a metric on that space." I think easily we can generalize the notion of norms to groups and rings. My questions are, Why ...
18
votes
5answers
4k views

number of zeroes in 100 factorial.

I was on math.stackexchange the other day and i found a question that said How many zeroes are there in 100!. I quickly factored it out and said that there where 24 zeroes. However thats only the ...
60
votes
32answers
10k 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 ...
10
votes
0answers
365 views

Which journals publish applied mathematics with mostly pure mathematics content?

In the spirit of Which journals publish expository work? please advise: What consistently high quality journals$^1$ today publish results that would otherwise go to a pure mathematics journal if ...
3
votes
0answers
21 views

Characterization of complete lattices with join-incomplete lattice endomorphisms

Let $L$ be an complete lattice. A lattice homomorphism $f: L\to L$ is said to be join-incomplete if there is an infinite set $S \subseteq L$ such that $f(\bigvee_L S) > \bigvee_L f(S).$ How can ...
32
votes
2answers
875 views

When to postpone a proof?

One possible practice in writing mathematics is to prove every theorem and lemma right after stating it. A long, technical proof — and sometimes even a short one — can interrupt the flow ...
11
votes
1answer
322 views

What's the role of $H^{p}(\mathbb{R}^{n})$ in modern (harmonic) analysis?

The classical theory of $H^p$,due to it's heavy dependence on the complex function theory(such as Blaschke products), seemed to have an insurmountable obstacle barrying its extension to several ...
4
votes
5answers
5k views

Beginners text on Calculus of Variations

I want to begin learning Calculus of Variations. What texts would MathOverflow recommend? Amazon shows up quite a few options: http://tinyurl.com/36koaq4 I work on Machine Learning, and that where ...
29
votes
15answers
1k views

Discovering and selecting conferences

Last summer, there were several excellent summer schools in my field that I learned of only after the application date. The events I did attend were chosen without too much care. I'm planning for the ...
21
votes
3answers
755 views

What technical and/or theoretical challenges are involved in automatically extracting proofs from books and papers into Coq code?

Over the years, advances in machine learning has allowed us to communicate and interact, using the same natural language, more and more semantically with computers, e.g. Google, Siri, Watson, etc. On ...
22
votes
7answers
6k views

Changing Careers: Becoming a Professional Mathematician

First, I apologize if this question is too soft or doesn't comport precisely with what is considered a good question, but I know of no other place to ask it. A little background: I obtained an ...
0
votes
0answers
99 views

Is there a rather natural space an automorphism of which is the Mellin transform?

Disclaimer: this question might be a little too vague and thus not suitable for this site despite the soft-question tag. If so, feel free to migrate it to MSE. I just read this and, trying to find ...
0
votes
0answers
112 views

Newer list of open problems in model theory

In the book Model Theory by C. C. Chang and H. J. Keisler, there is a list of open problems in model theory. More exactly, this list is called "Open problems in classical model theory" (on page 597, ...
12
votes
1answer
426 views

Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...
29
votes
8answers
3k views

Should one attack hard problems? [closed]

When I applied for a PhD student position I had an interview with two professors. Somehow we touched the problem if $P$ is $NP$ and, once we got there, for some reason both professors made it clear ...
33
votes
30answers
11k views

Most intricate and most beautiful structures in mathematics

In the December 2010 issue of Scientific American, an article "A Geometric Theory of Everything" by A. G. Lisi and J. O. Weatherall states "... what is arguably the most intricate structure known to ...
27
votes
22answers
7k views

Titles composed entirely of math symbols

I apologize for burdening MO with such a vapid, nonresearch question, but I have been curious ever since Suvrit's popular October 2010 Most memorable titles MO question if there were any ...
27
votes
35answers
4k views

What are some mathematical sculptures?

Either intentionally or unintentionally. Include location and sculptor, if known.
0
votes
0answers
122 views

How to prepare a radical change of research field after the PhD [duplicate]

I am in the middle of my PhD in functional analysis. My undergraduate studies were focused on pure theory and so it was logical to continue in this direction. However, recently I got into contact with ...
19
votes
2answers
1k views

History of Geometric Analogies in Number Theory

My question, put simply, is: When did mathematicians/number theorists begin viewing questions in number theory through a geometric lens? For example, was it before Grothendieck introduced schemes to ...
63
votes
61answers
25k views

Favorite popular math book [closed]

Christmas is almost here, so imagine you want to buy a good popular math book for your aunt (or whoever you want). Which book would you buy or recommend? It would be nice if you could answer in the ...
12
votes
0answers
497 views

“To operate the machine, it is not necessary to raise the bonnet.”

The quotation in the title is attributed to Frank Adams and appears in several places: In the preface of [2002, Operads in algebra, topology and physics]: "to operate the machine, it is not ...
12
votes
3answers
632 views

How to write an abstract for a math paper? [closed]

How would you go about writing an abstract for a Math paper? I know that an abstract is supposed to "advertise" the paper. However, I do not really know how to get started. Could someone tell me how ...
2
votes
1answer
435 views

Mathematics equivalent of Feynman's Lectures in Physics? [closed]

I'm looking for an equivalent of "Feynman's Lectures in Physics" in mathematics. I'm specifically looking for book/books that delve into, using Feynman's words, "the meaning of things".
5
votes
2answers
195 views

Do computational geometers use Lagrange multipliers?

Can anyone point me to an example of a problem that (more or less) originated in computational geometry whose solution requires the use of Lagrange multipliers (or Kuhn-Tucker conditions, or dual ...
111
votes
23answers
26k views

Examples of unexpected mathematical images

I try to generate a lot of examples in my research to get a better feel for what I am doing. Sometimes, I generate a plot, or a figure, that really surprises me, and makes my research take an ...
19
votes
7answers
6k views

Advice on doing mathematical research

Please share any general tips or advice you have on doing mathematical research. How do you identify good problems to work on or to think about? What do you do when you get stuck on a problem? Etc.
51
votes
8answers
6k views

Least collaborative mathematician

The recent question about the most prolific collaboration interested me. How about this question in the opposite direction, then: can anyone beat, amongst contemporary mathematicians, the example of ...
7
votes
0answers
267 views

What would you do if you improve your own result that is submitted but not publishied?

Here is a hypothetical situation: You have proved a result and written up a paper about it. You have submitted your article to some journal and it is being reviewed. While you are waiting, you have ...
17
votes
3answers
2k views

Style of mathematical writing vs. too many lemmas

I work in PDEs. I have now written 3 papers. I find my style is of the form: introduction, statement of results, paragraphs to introduce something, lemma, more text, lemma, more text, lemma, more ...
32
votes
17answers
5k views

Great mathematical figures and/or diagrams?

Most math papers have few figures, if any, although sometimes a well-chosen figure can be a tremendous help in understanding mathematical concepts. Does anyone have any examples of notable uses of ...
12
votes
2answers
778 views

Describe the desired features of a “Mathematics Colloquium”?

I'm now a member of my department's colloquium committee. Our task is to make a great colloquium series. I thought that the first step would be to come up with an appropriate definition of ...
25
votes
13answers
9k views

How to draw knots with Latex?

I am writing an exam for my students, and the topic is intro knots theory. I have no idea how to put knots into the file, but I know many MO users who can draw amazing diagrams in their papers. Can ...
1
vote
0answers
54 views

Precise statement of Gersho's conjecture

Here is the Gersho's conjecture from his paper "Asymptotically optiaml block qunatization" "For $N$ sufficiently large the optimal(distortion-minimizing) quantizer for a random vector uniformly ...
5
votes
2answers
153 views

Separable coordinate systems for the Laplace and Helmholtz equations?

According to Mathworld, in three dimensions there are 13 coordinate systems in which Laplace's equation is separable, and 11 for the Helmholtz equation. I've read the relevant chapters of the book by ...
28
votes
5answers
1k views

The unpublished papers in reference to the published papers

Sometimes it happens that a published paper refers to an unpublished paper for a result used. In this case, if we want to check this result by ourselves, we need to access to this unpublished paper. ...
45
votes
19answers
6k views

Mathematicians whose works were criticized by contemporaries but became widely accepted later

Gauss famously discarded Abel's proof that an algebraic equation of degree five or more cannot have a general solution (Abel himself had rejected divergent series as the work of the devil). Cantor's ...
5
votes
3answers
646 views

“Family Tree” of Theorems

Is anyone aware of any attempt to describe the dependencies of theorems (perhaps in mathematics generally, perhaps in some limited areas) in the form of a "family tree"? That is, each node on the ...
10
votes
2answers
1k views

Should we post on arXiv only papers in publishable shape (or very close)?

Question: Should we post on arXiv only papers in publishable shape (or very close)? This question should be distinguished from the following: Should one post a paper on the arXiv if it is not ...
165
votes
109answers
44k views

What are some examples of colorful language in serious mathematics papers? [closed]

The popular MO question "Famous mathematical quotes" has turned up many examples of witty, insightful, and humorous writing by mathematicians. Yet, with a few exceptions such as Weyl's "angel of ...
147
votes
36answers
40k 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 ...
46
votes
46answers
16k views

An example of a beautiful proof that would be accessible at the high school level?

The background of my question comes from an observation that what we teach in schools does not always reflect what we practice. Beauty is part of what drives mathematicians, but we rarely talk about ...
110
votes
15answers
12k views

When and how is it appropriate for an undergraduate to email a professor out of the blue?

This may not be appropriate for MathOverflow, as I haven't seen precedent for this type of question. But the answer is certainly of interest to me, and (I think) would be of interest to many other ...
17
votes
11answers
5k views

Open Questions in Riemannian Geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.
24
votes
1answer
1k views

Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. ...
28
votes
6answers
5k views

What is the etymology of the term “perverse sheaf”?

Grothendieck famously objected to the term "perverse sheaf" in Récoltes et Semailles, writing "What an idea to give such a name to a mathematical thing! Or to any other thing or living being, except ...