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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.

0
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0answers
207 views

How does one publish a Springer GTM? [on hold]

I am planning to write a textbook that I would like to publish as a Springer GTM. I am wondering how to get started. The guidelines on Springer website are very general and apply to all subjects. Is ...
68
votes
5answers
10k views

Has incorrect notation ever led to a mistaken proof?

In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, ...
10
votes
2answers
365 views

What kind of computer tools topologists/geometrists use to visualize the objects they deal with?

I have recently started to read a bit about geometry and topology. Hopf fibration, Lense spaces, CW complexes, stuff that are discussed in Hatcher's Algebraic Topology and other things that require ...
-3
votes
0answers
59 views

Transform a function with a homothecy [closed]

What expression describes how a function $y(x)$ is transformed under a general homothetic transformation? I'd like to prove that if I apply a homothetic transformation to a particular solution I ...
3
votes
0answers
587 views

Work of Caucher Birkar

I am asking this since the work of this Fields medallist was not covered in the related question on work of 2018 ICM plenary speakers below. Work of plenary speakers at ICM 2018 Terry Tao has some ...
31
votes
2answers
2k views

ICM 2018 lecture videos

Is there a place to watch ICM 2018 plenary lectures (and other lectures if possible)? Here is the official Youtube channel of the ICM but they don't seem to be posting the lectures. https://www....
0
votes
0answers
56 views

Local and global analysis of PDEs [closed]

The general theory of linear partial differential equations is largely addressed to local problems, i.e.,to the study of solutions in a suitably small neighbourhood of $x_0\in \mathbb{R}^n$. It is ...
3
votes
2answers
152 views

Best notation for fibrant/cofibrant replacement

In Quillen's original text on model categories (homotopical algebra) he uses $Q$ and $R$ to denote cofibrant and fibrant replacement respectively. This notation has been used by several other ...
12
votes
3answers
555 views

General principles which lead to good questions in many concrete situations [closed]

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...
5
votes
1answer
145 views

about morphisms of affine formal schemes $\mathrm{Spf}(B)\to \mathrm{Spf}(A)$

It is well known that there is a correspondence between homomorphism of rings $A\to B$ and morphism of affine schemes $\mathrm{Spec}(B) \to \mathrm{Spec}(A)$. Question: (1) In analogy, is there ...
41
votes
13answers
4k views

Examples of “miraculous” proofs [closed]

Concerning the proof that $\zeta(3)$ is irrational, Van der Poorten famously noted that "Apéry's incredible proof appears to be a mixture of miracles and mysteries". Indeed, many ideas introduced in ...
14
votes
1answer
737 views

Math journal publishing work related to combinatorics, probability, counting problems etc.?

I'm a high school student. My peer and I have done some work on the Ballot Theorem counting problem and Catalan Numbers. We have come up with a new proof to the Ballot Theorem and we demonstrate the ...
10
votes
3answers
842 views

In a publication, should an `\ldots` always be followed by a period? [closed]

The Latex command \ldots is often used to denote "and so forth". For instance, $$ \pi \approx 3.1415\ldots $$ When a sentence ends with an ...
10
votes
2answers
431 views

Reference Request: Theoretical Mixing Times Research in Machine Learning / Artificial Intelligence (AI)

I'm doing a PhD in probability theory, focusing mostly on mixing times. It's a pure maths PhD, considering precise models and showing rigorous mixing results. I'm also interested in stuff like machine ...
15
votes
3answers
1k views

Where can I read reviews of mathematical theories? [closed]

I'm really enjoying the AMS column "What is ..." (http://arminstraub.com/math/what-is-column) and The Princeton Companion to Mathematics. I am looking for something similar. I'd like to acquire some ...
1
vote
0answers
224 views

Mathematical expressions involving weird constants [closed]

I hope this is a question that fits here and is not duplicated. Also that is clear since it can be a little ambiguous. I was wondering if you know deep expressions, theorems, isomorphisms or ...
7
votes
2answers
216 views

complicated combinatorial algorithms with good descriptions

For educational purposes, I am looking for an example of a complicated, elementary, but very well-explained combinatorial algorithm. Such an example might be a bijection between two easily described ...
43
votes
5answers
2k views

Explaining the Main Ideas of Proof before Giving Details

I'll be the first to admit that this is a risky question to try to get away with on math overflow, but I'm going to give it a shot anyway. Roughly speaking, the question is this: Is it good to try to ...
1
vote
0answers
178 views

Do there exist similar programs which connect different field of Mathematics like Langlands program? [closed]

$2018$ Abel prize is awarded to Robert P. Langlands for his visionary program connecting representation theory to number theory. In particular, his program predicts the existence of a tight web of ...
31
votes
2answers
2k views

Diplomacy when reporting errors

I am a young researcher and sometimes I face an uncomfortable situation : I find an error in a research paper! Of course, most of the time, it is all just my misunderstanding but it happens that after ...
8
votes
0answers
182 views

étale vs syntomic vs flat cohomology

Let $\mathscr{A}/X$ be an abelian scheme over $X$ of characterisitic $p$. The étale topology is not fine enough for the Kummer sequence for $\mathscr{A}$ to be (right) exact, but the syntomic and flat ...
32
votes
8answers
6k views

PhD dissertations that solve an established open problem

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (or with collaboration of her/his supervisor). In my question I search for every possible ...
3
votes
0answers
146 views

curve blow ups of toric Fano $3$-folds

Suppose $X$ is a smooth toric Fano $3$-fold, and $D$ is a torus invariant divisor corresponding to a face of the polytope associated to $X$. I would like to search for (smooth) curves $C \subset D$, ...
29
votes
2answers
1k views

Is it possible to cite a page in n-Lab in a research paper?

One of the concepts I need in my paper is best summarized on the n-Lab page. I tried my best to find a similar description of the same concept in a more "classical" reference, such as a paper or a ...
0
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0answers
175 views

Important papers that were rejected several times before 1950 [duplicate]

There is no reasson to assume that an important paper has to be well written. For example, when Galois submited his paper on his theory of solvability by radicals it was rejected several times, and ...
0
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0answers
66 views

A system similar to the Keller-Segel one in $\mathbb{R}^N$

Can you point out references on existence and uniqueness of solutions to the following system? $$u_t = \nabla(h(u)\nabla v) $$ $$v = \Delta u$$ $$u(0,\cdot) = u_0(\cdot)$$ in $(0,\infty) \times \...
17
votes
3answers
6k views

What is so special about set theory anyway? [closed]

(Later edit - tried to clarify a couple of vague places concerning interpretations of theories that became evident in comments (thanks to Andrej Bauer, Mauro ALLEGRANZA and Emil Jeřábek). (To closers ...
12
votes
1answer
856 views

Historically, which came first: the Lie algebras or their classification?

The classification of the complex simple Lie algebras by their Dynkin diagrams gives rise to five exceptional complex simple Lie algebras: $F_4, G_2, E_6, E_7$ and $E_8$. I am trying to find out ...
2
votes
1answer
100 views

Exponential Decay on nolinear Schrodinger type equation with negative potential

Given a unbounded domain $\Omega$(a region or an open manifold) with $\dim \geq3$, consider the equation $$A u+Vu+|u|^ku=0,$$ here $A=\sum_{ij}\partial_ia^{ij}(x)\partial_j$ is an elliptic operator ...
39
votes
6answers
2k views

What are good articles/books on the psychology of mathematical research?

I am thinking about advanced texts similar to Polya's 'How to solve it?'. Quite a few good articles of such a kind are published under Philosophy of Mathematics, but that dwells on a very different ...
6
votes
0answers
144 views

Can a topological cat count the number of yarn windings around a ball?

This is a follow-up question to my previous question on "winding numbers" of curves in higher-dimensional space. I thought it best to post a new question rather than changing the setup of the previous ...
6
votes
2answers
352 views

Is there a sensible notion of a winding number of a closed curve in $\mathbb{R}^n$, $n\geq 3$, with respect to a point not lying on it?

I have been browsing "Topological Degree Theory and Applications" by Cho, Chen and O'Regan as well as "Mapping Degree Theory" by Outerelo and Ruiz, but I have not been able to quite answer myself the ...
7
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3answers
2k views

Which affiliation to use when publishing, when invited professor at second university

I am a PhD student at one university and an invited professor at a second, i.e. I do not have a permanent position in the second one. Now I need to indicate an affiliation in a journal paper but I do ...
2
votes
0answers
34 views

What are Interesting Questions about Families of Graphs?

In my recreational occupation with the TSP I encountered a family of graphs, which I would like to call "Möbius Sponges" because they generalize the Möbius Ladder graphs in a way, that may provide an ...
32
votes
2answers
1k views

When to publish minor results?

One of the main things I enjoy about MO and MSE is the chance to solve somewhat difficult problems. On a couple of occasions, I've found proofs of results (posed as questions mostly on MSE) which ...
8
votes
2answers
1k views

Collection of Mathematical Constants

I recently stumbled over the large collection of mathematical constants of Mauro Fiorentini; it is in Italian and appears to be something in the vein of the famous OEIS, however maintained by a single ...
6
votes
3answers
1k views

Is there a generalization (surely there is) of this simple combinatorial identity?

I was just doing some algebra on a paper and obtained: $$\sum_{l=0}^{n-1} {{n+l} \choose l}={2n \choose {n+1}}$$ Are there some generalizations of this identity? One possible generalization would be ...
76
votes
6answers
6k views

What’s the etiquette on using diagrams that need color to be understood?

I’m working on a paper that makes heavy use of colorful diagrams to supplement the text. For most of these it would probably not be possible to create grayscale versions that convey the same ...
102
votes
10answers
14k views

Do you know important theorems that remain unknown?

Do you know of any very important theorems that remain unknown? I mean results that could easily make into textbooks or research monographs, but almost nobody knows about them. If you provide an ...
19
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8answers
3k views

Mathematical theory of aesthetics

The notion of beauty has historically led many mathematicians to fruitful work. Yet, I have yet to find a mathematical text which has attempted to elucidate what exactly makes certain geometric ...
7
votes
4answers
2k views

Would mathematics be different if not written one-dimensionally? [closed]

Mathematics is written one-dimensionally, using symbols that make sense when put together on a line. The 2d sheets of paper that we use don't have enough room to write mathematics two-dimensionally. ...
23
votes
3answers
1k views

Advanced software for OEIS?

Is there (if not, why?) a software where I can input a sequence of integers, like into the OEIS, and then it makes some simple transformations on it to check whether the sequence can be obtained from ...
3
votes
2answers
209 views

References on thin film equation: derivation and properties

Where can I find a derivation of the thin film equation $$u_t = - \mathrm{div} (u^m\nabla\Delta u)$$ from a physical model? a good introduction to its properties (e.g. conserved quantities and ...
4
votes
1answer
156 views

Why is the definition of entropy solution necessary to prove uniqueness for hyperbolic conservation laws?

I'm aware that there are a lot of counterexamples to show that distributional solutions for hyperbolic (scalar) conservation laws are not unique. However, I'd like to ask: Conceptually, at ...
2
votes
1answer
188 views

Is this approach for establishing regularity of harmonic maps between manifolds valid?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ While trying to understand some regularity results, I thought about the following "naive" approach for establishing regularity of weakly ...
147
votes
21answers
24k views

How can a mathematician handle the pressure to discover something new?

Suppose I'm an aspiring mathematician-to-be, who started doing research. Although this is really what I love doing, I found that one disturbing point is that there's always the pressure of discovering ...
8
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3answers
2k views

Where to publish new mathematical identities?

Similar questions have been asked before regarding journals that publish: expository work, recreational mathematics, computational results, new proofs of old theorems, and even math textbooxs. ...
19
votes
4answers
700 views

Illustrating mathematics with wysiwyg tools

What tools are out there for creating mathematical illustrations in a what-you-see-is-what-you-get mode? Having struggled with tikz for several years, I've found creating figures in Omnigraffle (...
5
votes
0answers
527 views

Why should one subscribe to print Journals [closed]

It seems obvious to me that having print journals in a library is beneficial. Yes, Arxiv, MathSciNet, Blogs and lecture notes by Mathematicians, Math Overflow, Wikipedia and Scholarpedia all of these ...
2
votes
1answer
71 views

Meaningful generalization of viscosity solutions to higher order equations

Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?