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

108
votes
39answers
32k 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 ...
25
votes
5answers
1k views

Can one make high-level proofs about chess positions?

I realize this question is risky (as the title and the tags indicate), but hopefully I can make it acceptable. If not, and the question cannot be salvaged, I'm sorry and ready to delete it or accept ...
67
votes
11answers
8k views

How has “what every mathematician should know” changed?

So I was wondering: are there any general differences in the nature of "what every mathematician should know" over the last 50-60 years? I'm not just talking of small changes where new results are ...
24
votes
1answer
2k views

Community experiences writing Lamport's structured proofs

About two years ago, I came across this paper by Lamport http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf on writing proofs hierarchically. It changed how I wrote ...
0
votes
1answer
586 views

Hard maths on viXra? [on hold]

A few years ago a nice paper surveyed the differences in quality between papers submitted to arXiv and those submitted to arXiv's rough cousin, viXra. However, that paper was about generic ...
144
votes
77answers
24k 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 ...
1
vote
1answer
828 views

What is the modern consensus on the difficulty of infinitesimals?

At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...
0
votes
0answers
68 views

chromatic number of a graph and C++ programming [on hold]

Is there a graph library which already computes the chromatic number of a graph?...Maybe a home-made extension using Boost?... I just want to know if there exists exacts algorithms already ...
60
votes
9answers
6k views

Have you solved problems in your sleep?

I have hit upon major (for me—relative to my trivial accomplishments) insights in my research in various sleep-deprived altered states of consciousness, e.g., long solo car-drives extending ...
35
votes
7answers
6k views

There must be a good introductory numerical analysis course out there!

Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant ...
4
votes
2answers
532 views

What is the mathematical significance of the IHES logo?

The logo of the IHES http://www.ihes.fr/jsp/site/Portal.jsp (upper left) is lovely, but what exactly does represent mathematically? (There's a slightly larger version at ...
5
votes
2answers
893 views

motive of a modular form

What is the idea behind a "motive of a modular form"? (I know what a motive is and what a Weil cohomology is. I want to know how to get the motive [what is the idea of Scholl?], and why this is ...
-9
votes
0answers
159 views

Korean Mathematics [closed]

The Korean language is very geometric in its written form. I am no expert on the topic, but I have Korean nieces and a Korean sister-in-law (from South Korea) so I studied Korean somewhat. I know at ...
4
votes
0answers
55 views

The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens: Some elements are considered equivalent, in some topological or algebraic sense We take the quotient The metric is usually not ...
0
votes
0answers
15 views

“Can't” and “don't” in a mathematical paper [migrated]

I am refereeing a paper (a good one) for a mathematical journal. The author repeatedly writes like this: "we can't do", "we don't know" etc. He is not a native English speaker, and this paper is his ...
15
votes
2answers
947 views

Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
25
votes
0answers
581 views

Greatly expanded new edition of a Bourbaki chapter on algebra?

Recently I discovered by accident that Bourbaki issued in 2012 a radically expanded version of their 1958 Chapter 8 Modules et anneaux semi-simples (like other chapters, initially in French) within ...
-1
votes
1answer
183 views

Everyday, real-life applications of mathematical concepts, and human intuition vs mathematical analysis [closed]

I'm working on an educational project about the applications of reasonably 'lofty', high-ish-level mathematical concepts in the real world. I've already scoured these links (1) (2) (3) after ...
33
votes
8answers
3k views

What makes a theorem *a* “nullstellensatz.”

I know what the (Hilbert) Nullstellensatz says. A MathSciNet search on "nullstellensatz" turns up nearly 200 papers, with only a minority offering either new proofs or new applications of the classic ...
2
votes
2answers
421 views

Is there a countable pseudocharacter Hausdorff space,such that…?

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two ...
-1
votes
0answers
89 views

Soft question: take complex analysis or cryptology? [migrated]

I am math major junior considering math grad school. I need to decide whether to take complex analysis or cryptology this semester. Complex analysis seems to be a recommended course for people ...
28
votes
1answer
4k views

Math and Wormholes

Hopefully, MathOverflow is the correct place for this. I had a student approach me and ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any ...
125
votes
27answers
30k 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 ...
3
votes
0answers
113 views

Pronunciation of ¡ (inverted exclamation mark, historically used for subfactorial)

For anyone who uses ¡ (inverted exclamation mark) in a mathematical context, how do you pronounce it? Background: I have privately been using ¡ in a couple of notations for a while, and am ...
57
votes
41answers
9k views

Examples of theorems with proofs that have dramatically improved over time

I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider ...
8
votes
3answers
1k views

Why are they called L-functions?

I was hoping to see this pop up on the recent big list question about etymology or terms and symbols. Since it has not, and I can't find an answer, I will ask: What is the reason for the $L$ in ...
21
votes
4answers
1k views

What are the advantages of the more abstract approaches to nonstandard analysis?

This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ...
12
votes
5answers
1k views

How to cite authors from any country correctly?

It has always seemed to me that the Mathematical Community gives a high importance to the act of properly citing an author (Do not write Erdos! It's Erdős. Cauchy must be read as in French, not as in ...
36
votes
6answers
4k views

Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
4
votes
0answers
103 views

Geometric Characterization of Martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as Geodesics in a very large dimensional manifold. My question is, is there any research studying this idea? ...
9
votes
1answer
226 views

Literature that helps explain what the theory of numerosities contributes with

Since 2003 a group of Italian mathematicians (Benci, Di Nasso and Forti) has developed a new measure for infinite sets that satisfies the Euclidian principle: The whole is greater than the part. The ...
17
votes
1answer
1k 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 ...
104
votes
59answers
18k views

Jokes in the sense of Littlewood: examples? [closed]

First, let me make it clear that I do not mean jokes of the "abelian grape" variety. I take my cue from the following passage in A Mathematician's Miscellany by J.E. Littlewood (Methuen 1953, p. 79): ...
22
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 ...
15
votes
5answers
2k views

Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?

Many topologists express a clear preference for working with CW complexes instead of simplicial sets. One of the reasons is that the cellular chain complex of a CW complex is often easier to work ...
112
votes
27answers
46k 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 ...
31
votes
7answers
3k views

Is it worthwhile to give off-topic talks?

I am a graduate student. Occasionally for some reason I am asked to give a talk on my research at a conference whose stated purpose is almost completely unrelated to my research. To preserve my ...
20
votes
2answers
2k views

Does this approach for the Poincaré conjecture work?

Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. ...
29
votes
8answers
5k views

What are some important but still unsolved problems in mathematical logic?

In the past, First order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ...
1
vote
1answer
161 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
-4
votes
1answer
155 views

When do Theorems (or Algorithms or Methods) Become Celebrated? [closed]

I recently noticed that certain theorems (e.g. Tutte's 1-factor theorem or, Edmond's Blossom algorithm) are attributed celebrated. A quick search on the internet yields further examples: ...
11
votes
3answers
675 views

Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham. Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in finding ...
31
votes
1answer
2k views

Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
168
votes
36answers
46k 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 ...
9
votes
0answers
168 views

Algebraic K-theory of a ring.

I started to learn some algebraic $K$-theory and its relation to geometric topology problems. My question is : What is the list of rings such that all their algebraic $K$-theory groups are known ? I ...
80
votes
17answers
21k 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 ...
6
votes
7answers
3k views

Notable math from those without math PhDs [duplicate]

Possible Duplicate: What recent discoveries have amateur mathematicians made? Can anyone give historical examples of people outside of mathematics that have contributed to the literature? ...
11
votes
2answers
4k views

“You can't push a rope”

"You can't push a rope" is a wisdom saying that some engineering teachers pass along to their students. Since I'm not an engineer, I can only guess at what they mean, but it sounds to me like code ...
5
votes
1answer
115 views

Historical refererences for Castelnuovo-Mumford regularity

Does anyone know a good reference to understand the historical background of Castelnuovo-Mumford regularity? I know the backgound for the modern commutative-algebra approach (using free graded ...
12
votes
5answers
3k views

How to be updated with current advances in mathematics

Hello, I wanted to know that how researchers in mathematics keep updates in their field of interest. (original question by Rahul Gupta)