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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1
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0answers
132 views

“For sufficiently large” vs. “For all sufficiently large” [on hold]

A purely grammatical question: Do people generally prefer: "For sufficiently large x,..." or "For all sufficiently large x,..." or not care? Or might you use either according to context? The meaning ...
9
votes
3answers
2k views

Two questions about combinatorics journals

Hello, I have two questions regarding combinatorics journals. I hope that this is the right place for such questions. Which combinatorics/DM journals would you consider as the "top tier"? I tried ...
1
vote
0answers
49 views

Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional. It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is: ...
83
votes
25answers
32k 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 ...
25
votes
2answers
617 views

Different styles of writing/reading articles

Recently, I discovered a rather unexpected thing. We are writing an article in collaboration and we permanently have some discussions about how to write, in which order, how to organize material etc. ...
6
votes
2answers
664 views

Publication in proceedings

Why and how publishing a paper in proceedings? What are the difference with a "classical" journal? What's the list of the main proceedings in which one can publish? Do proceedings papers (never, ...
29
votes
16answers
23k views

Undergraduate math research

I believe this is the right place to ask this, so I was wondering if anyone could give me advice on research at the undergraduate level. I was recently accepted into the McNair Scholars program. It ...
70
votes
25answers
9k views

What is the most useful non-existing object of your field?

When many proofs by contradiction end with "we have built an object with such, such and such properties, which does not exist", it seems relevant to give this object a name, even though (in fact ...
45
votes
22answers
9k views

How to respond to “I was never much good at maths at school.” [closed]

We've all heard it. I even got it in Norwegian recently. It's number 1 on the list of responses to the statement "I'm a mathematician.". Does anyone have any good comebacks? What other responses ...
118
votes
43answers
61k 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 ...
51
votes
30answers
5k views

books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is ...
8
votes
3answers
214 views

How do small central extensions drop the dimension of a faithful representation?

Apologies in advance that this is a very soft question. I might be talking complete nonsense. But I hope I am talking about something that has even been studied... I am interested in the phenomenon ...
38
votes
18answers
5k views

What are some deep theorems, and why are they considered deep?

All mathematicians are used to thinking that certain theorems are deep, and we would probably all point to examples such as Dirichlet's theorem on primes in arithmetic progressions, the prime number ...
58
votes
3answers
2k views

How do you select an interesting and reasonable problem for a student?

I am interested in how to select interesting yet reasonable problems for students to work on, either at Honours (that is, a research-based single year immediately after a degree) or PhD. By this I ...
1
vote
0answers
167 views

Intuitive Approach to Sheaf and Cech Cohomology [closed]

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
5
votes
2answers
312 views

Applications of cohomology to probability and statistics

Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory? By "interesting/useful", I mean "not merely ...
61
votes
61answers
24k 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 ...
33
votes
7answers
4k views

What is DAG and what has it to do with the ideas of Voevodsky?

In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
4
votes
3answers
299 views

Need examples of homotopy orbit and fixed points

I am no expert in equivariant homotopy theory. Let's say, I am planing to give a talk on homotopy fixed points and orbits. My audience will be graduate students who are doing algebraic topology or ...
18
votes
10answers
3k views

Research Experience for Undergraduates: Summer Programs

Some time ago, I found this list of REU programs held in 2009. The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more ...
79
votes
30answers
12k views

Most harmful heuristic?

What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?
1
vote
1answer
240 views

How to decide whether the journal is pure or applied?

I am a beginner in research and have draft ready for my first article. I have a little confusion about the pure and applied journal in Mathematics. My work belongs to pure mathematics(I think). And ...
61
votes
14answers
17k views

Top specialized journals

In geometry/topology, there are (at least) three specialized journals that end up publishing a large fraction of the best papers in the subject -- Geometry and Topology, JDG, and GAFA. What journals ...
20
votes
8answers
2k views

Does your dissertation matter for industry research jobs?

Suppose Mr A. is a graduate student who, for some reason, does not want to go into teaching or academic research. Mr A. did a Doctoral dissertation, but it is in a narrow area. This area is not ...
3
votes
1answer
396 views

Mathematics of Computer science and AI [closed]

Computer science and Artificial Intelligence have been fertile grounds for research for decades, not only for Engineers but particularly for Mathematicians. What kinds of Mathematics have emerged ...
8
votes
0answers
526 views

How to approach the stigma of not having a math degree? [closed]

I am a faculty member in a department that is not mathematics, but is highly-ranked in my field. I greatly enjoy working with mathematicians, and have had a number of successful collaborations. ...
3
votes
3answers
311 views

Reference request: a guide through quantum probability

Could you point out a comprehensive reference book (or more than one, if it is the case) on Quantum Probability that introduces the subject and then gradually builds up to the edges of contemporary ...
18
votes
18answers
3k views

Math books for advanced high school students

I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...
7
votes
2answers
803 views

Surveys of the items of Erdős' “toolbox”

Could you point out some survey papers and monographs that highlight the kernel of tricks, techniques, and tools that Paul Erdős employed the most in his research work (in particular in graph theory, ...
2
votes
2answers
232 views

Heuristics for 2-morphisms of (algebraic) stacks

For topological spaces and simplicial sets one can consider each pair of parallel morphisms $f,g:X\rightrightarrows Y$ as equipped with a set of 2-morphisms given by homotopies $H:f\simeq g$ (let's ...
5
votes
2answers
854 views

Physicist trying to understand modern mathematics

I'm a physicist trying to gain a deep understanding of mathematics that is required for my work.I intend to specialize in string theory which is a very math intensive branch of theoretical physics ...
3
votes
1answer
92 views

Reference for exercises with solutions for affine Lie algebras

I am doing self-study of affine Lie algebras, and while I have all recommended reference material, I find it hard to prepare for the exam. It was quite easy to study finite-dimensional simple Lie ...
75
votes
18answers
7k views

How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...
8
votes
9answers
1k views

Examples where adding complexity made a problem simpler

I can think of a few situations in math where a problem becomes easier or an object becomes simpler when some complexity is added. Examples: $S^n$ is never contractible, but $S^{\infty}$ is. The ...
2
votes
2answers
399 views

Popular books written by great mathematicians [closed]

I read: H. Poincare. Value of science F. Klein. Development of Mathematics in the 19th Century J.E. Littlewood. A Mathematicians Miscellany G.H. Hardy. A Mathematician’s Apology R. Courant, ...
4
votes
1answer
1k views

What would an ideal mathematical note-taking/organizer/PIM software look like?

I'm struggling with this problem for a long time, and I'm sure a lot of you out there are having similar problems too: when studying from a e-textbook I read and annotate in one app and make ...
7
votes
2answers
205 views

Recent trends in effective analysis

The references listed at http://en.wikipedia.org/wiki/Computable_analysis have all been published 30-15 years ago. Are the approaches which these references expose still up-to-date and relevant to the ...
104
votes
68answers
20k views

Which math paper maximizes the ratio (importance)/(length)?

My vote would be Milnor's 7-page paper "On manifolds homeomorphic to the 7-sphere", in Vol. 64 of Annals of Math. For those who have not read it, he explicitly constructs smooth 7-manifolds which are ...
13
votes
7answers
765 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, ...
3
votes
3answers
262 views

Survey papers of results (relevant to mathematics) produced during research on the relationship between mathematics and music theory

It is well-known that there is a relationship between music and mathematics, and there are many references that explore this topic (for example, Benson's book). However, I would like to ask if there ...
1
vote
0answers
82 views

Why are they called 'pernicious' numbers?

The definition of a pernicious number: In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime. The meaning of ...
34
votes
10answers
3k views

Presenting a paper: Do's and Don'ts?

I was wondering what would be the best way to present your paper at a conference, if your paper is selected for "short communication", lasting for about 15 minutes? Should you concentrate on the main ...
51
votes
6answers
3k views

What does it take to run a good learning seminar?

I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might ...
94
votes
81answers
71k views

Do good math jokes exist? [closed]

Have a good joke? Share. I know this is subjective, but the principle "should be of interest to mathematicians" trumps. (I hope.)
8
votes
1answer
409 views

Open access journals in number theory

I'm a phd student in number theory, and I'm required (by the funding council) to publish any article I write in open access journals. The problem is that all journals I can find are either out of my ...
40
votes
12answers
6k views

japanese/chinese for mathematicians?

I'd like to learn to read math articles in Japanese or Chinese, but I am not interested in learning these languages from usual textbooks. Exist suitable texts, specialized for the needs for reading ...
3
votes
1answer
377 views

Do hom-sets really live in the category Set?

This isn't really a research-level question (sorry!), but I asked on math.se (link), and though the question was upvoted a few times, I didn't get any answers. So since there may well be more ...
77
votes
52answers
14k 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 ...
103
votes
83answers
22k views

Examples of great mathematical writing

This question is basically from Ravi Vakil's web page, but modified for Math Overflow. How do I write mathematics well? Learning by example is more helpful than being told what to do, so let's try to ...
14
votes
4answers
1k views

How Many 4-Manifolds are Symplectic?

As an honest question (probably with some subjectivity), how many smooth oriented 4-manifolds are actually symplectic? Can I say half (perhaps under some mild assumptions)? I ask this question because ...