Questions that ask about some aspect of mathematical research or study which doesn't involve the actual mathematics. In general, soft questions can be answered without using mathematical reasoning.

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17
votes
1answer
802 views

Collaboration or acknowledgment?

This post is a sequel of: When should a supervisor be a co-author? A general answer may be a proper adoption of the American patent law rejection (of a coautorship): the alleged research ...
4
votes
0answers
129 views

Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches: Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...
18
votes
4answers
927 views

Companion to theoretical physics for working mathematicians

In the Princeton Companion to Mathematics one reads that even pure mathematicians should know some theoretical physics and applied mathematics. What are some well-organized comprehensive companions to ...
2
votes
0answers
87 views

Comprehensive survey on mathematical modelling of neural networks: from the basic ideas to contemporary research topics

I am looking for a comprehensive survey (paper(s) or book(s)) on mathematical modelling of neural networks (both artificial and biological). It should start from the very basic concepts of modelling ...
12
votes
3answers
506 views

Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
40
votes
13answers
6k views

How has modern algebraic geometry affected other areas of math?

I have a friend who is very biased against algebraic geometry altogether. He says it's because it's about polynomials and he hates polynomials. I try to tell him about modern algebraic geometry, ...
2
votes
2answers
501 views

What's so special about $1$-categories?

I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...
6
votes
2answers
321 views

Generalization of Sylvester-Gallai theorem

The Sylvester-Gallai theorem states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single ...
5
votes
1answer
144 views

Survey paper on isoperimetry

I am searching for a comprehensive survey article (or more different articles) on the subject of isoperimetric problems from ancient Greece to modern mathematical physics. Could you point out some ...
-1
votes
0answers
198 views

Applied Maths author order - PhD collabrating with supervisor [closed]

I see in pure maths, there is a tradition that authors are generally ordered alphabetaly, called Hardy-Littlewood rule. However, to my obervation, in applied mathematics, it may depend: When the paper ...
5
votes
1answer
225 views

Is there a “good” reason that the universal central extension of $SL(2,\mathbb Z)$ is $Br_3$?

Let $$ Br_3 := \langle \tau_1,\tau_2\ :\ \tau_1 \tau_2 \tau_1 = \tau_2 \tau_1 \tau_2 \rangle $$ be the braid group on three strands, and consider the surjection $$\phi : Br_3 \twoheadrightarrow ...
57
votes
13answers
16k 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 ...
17
votes
10answers
3k views

New proofs to major theorems leading to new insights and results?

I am wondering, historically, when has a new proof of an old theorem been particularly fruitful. A few examples I have in mind (all number theoretic) are: First example is classical... which is ...
23
votes
3answers
805 views

Graduate program applications that require questionnaires and other non-letter material

In the December 2014 AMS Notices, a letter to the editor (http://www.ams.org/notices/201411/rnoti-p1311.pdf) by Deconinck and Medlock addresses the problem of (math) graduate programs requiring letter ...
41
votes
46answers
14k 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 ...
100
votes
20answers
22k 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 ...
10
votes
6answers
1k views

Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with intuitionistic logic. It is ...
74
votes
50answers
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 ...
126
votes
36answers
32k 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 ...
16
votes
5answers
1k views

How and how much do the notations and diagrams influence our understanding of mathematical concepts?

How and how much do the notations and diagrams influence our understanding of mathematical concepts? This question was stimulated by the MathOverflow questions Thinking and Explaining and ...
10
votes
1answer
288 views

What are the current views on consistency of Reinhardt cardinals without AC?

It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the ...
3
votes
2answers
457 views

Is a particular type of question about certain infinite sets still being asked?

I apologize in advance if this question is thought to be too soft or otherwise inappropriate for mathoverflow.net. Let M be the infinite set of all homeomorphism types of finite dimensional ...
114
votes
42answers
58k 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 ...
3
votes
1answer
180 views

Decidability of Frankl's union-closed sets conjecture

Is it conceivable that Frankl's union closed sets conjecture is undecidable in $\mathsf{ZFC}$, or is this quite implausible, perhaps due to the "finitistic" nature of the statement, or for some other ...
88
votes
97answers
55k views

Famous mathematical quotes [closed]

Some famous quotes often give interesting insights into the vision of mathematics that certain mathematicians have. Which ones are you particularly fond of? Standard community wiki rules apply: one ...
45
votes
13answers
6k views

Contest problems with connections to deeper mathematics.

Hi, I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay. We all know that problems from, for ...
3
votes
0answers
328 views

Well-known or prolific mathematicians that have never written a sole-author article? [closed]

Dear mathoverflow community, I have a junior colleague who will be coming up for tenure and who has written many articles, but all of them as a co-author. I don't see this as a problem (after all, ...
7
votes
3answers
1k views

The resolution of which conjecture/problem would advance Mathematics the most? [closed]

This is an impossibly broad question, and makes the unwarranted assumption that Mathematics is a uniform field. It might make more sense to ask the same question restricted to, say, Mathematical ...
36
votes
15answers
7k views

Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.] I've read looked at the examples in ...
4
votes
1answer
140 views

What can be said about graphs if there are homomorphisms in both directions?

Let $G,H$ be simple undirected graphs without loops such that there are graph homomorphisms $f_1:G\to H$ and $f_2: H\to G$. An easy argument shows that we have $\chi(G) = \chi(H)$, even for graphs ...
12
votes
2answers
895 views

Group theory required for further study in von Neumann algebra

After over half a year's study on operator algebra (especially on von Neumann algebra) by doing exercises in Fundamentals of the theory of operator algebras 1, 2 --Kadison, I was told that the ...
25
votes
6answers
2k views

What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...
2
votes
1answer
148 views

Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies Normality ...
11
votes
1answer
2k views

ICM 2014 streaming video

Is there a possibility to watch ICM 2014 opening ceremony and the big talks online? I hope there is since it was possible for the previous meeting.
4
votes
0answers
190 views

Etale Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
27
votes
4answers
1k views

Why Cohen-Macaulay rings have become important in commutative algebra?

I want to know the historic reasons behind singling out Cohen-Macaulay rings as interesting algebraic objects. I'm reviewing my previous lecture notes about Cohen-Macaulay rings because now I'm ...
9
votes
2answers
419 views

Model structure for cooperads and for coalgebras

I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like ...
32
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 ...
47
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 ...
16
votes
1answer
725 views

What makes the amenability of Thompsons group $F$ such a tricky problem?

An open problem that seems to get a lot of attention every once in a while is the amenability of Thompsons group $F$. The problem seems to generate both proofs and disproofs at a fairly high rate, ...
12
votes
3answers
773 views

What theorem of Liouville's is Gian-Carlo Rota referring to here?

I am very curious about this remark in Lesson Four of Rota's talk, Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations: "For second order linear differential ...
71
votes
9answers
5k views

Why are flat morphisms “flat?”

Of course "flatness" is a word that evokes a very particular geometric picture, and it seems to me like there should be a reason why this word is used, but nothing I can find gives me a reason! Is ...
26
votes
6answers
2k 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 ...
33
votes
21answers
8k views

Open source mathematical software.

I want some recomendation on which software I should install on my computer, an open source program for general abstract mathematical purposes (as opposed to applied mathematics). I would likely use ...
87
votes
68answers
14k views

Most helpful math resources on the web

What are really helpful math resources out there on the web? Please don't only post a link but a short description of what it does and why it is helpful. Please only one resource per answer and let ...
92
votes
77answers
20k 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 ...
0
votes
0answers
85 views

Degree of Map between Pseudomanifold

There are two different ways to define a degree of map. Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...
16
votes
16answers
2k 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 ...
0
votes
0answers
93 views

Proof of Faà di Bruno's formula with convolution identity

Faa di bruno's formula can be expressed as follows $$ \frac{d^n}{dx^n}[f(g(x))] = \sum_{k=1}^{n}f^{(k)}(g(x)) B_{n,k}(g'(x),....,g^{(n-k+1)}(x)) $$ On this Wikipedia page, there is a convolution ...
11
votes
2answers
566 views

What is a good poster for a math conference?

I'm going to participate to a conference and they ask me to do a poster on my research. I've never made a poster for a conference/seen a poster session in a conference. So what is important? What do ...