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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.

3
votes
3answers
328 views

mixing theorem with definition (definition with proof)

I often find myself writing a definition which requires a proof. You are defining a term and, contextually, need to prove that the definition makes sense. How can you express that? What about a ...
3
votes
1answer
186 views

Generating function for 3 -core partitions: Part II

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length of the cell $u$. We call $\lambda$ a $t$-core partition if none of ...
4
votes
1answer
106 views

Generating function for $3$-core partitions

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length of the cell $u$. We call $\lambda$ a $t$-core partition if none of ...
15
votes
2answers
592 views

How to handle results from an unpublished paper?

I am writing a paper right now, and part of the paper makes use of a (trivial) generalization of a number of really nice theorems and constructions from a paper that was never made public. The author ...
6
votes
1answer
186 views

How to visualize local complete intersection morphisms?

As the question title asks for, how do others visualize local complete intersection morphisms? My experiment in asking people in real life didn't pan out, so I'm consulting the MO algebraic geometry ...
4
votes
1answer
289 views

Question on academic affiliation when submitting paper

I just finished grad school, earning a Phd in mathematics. Currently I do not know if I want to continue my academic career or not, but in the meantime I have written an article containing some of the ...
1
vote
0answers
217 views

Mathematical Physics : a big picture for mathematicians [closed]

I'm a (last year) master's student, so now is the time to choose a subject for pursuing in PhD, and hopefully, in Academia. I do like a lot of mathematics, but in an ideal world, I would like my ...
19
votes
2answers
2k views

Should I inform the editor about a generalized result of a result in a paper under review?

Hoping that my question is appropriate for MO, I would like to ask the following question: I have sent one of the editors of a very good math journal a paper of mine which contains a main result, call ...
4
votes
0answers
89 views

Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory: never perform quotients, add structure instead; never require subobjects, take fibres instead. ...
10
votes
1answer
617 views

Research semester in math

Admittedly, this is a soft question. My own experience in mathematical research has been of long periods of research, mostly characterized in long "blocks" and sporadic breakthrough. How does this ...
8
votes
2answers
641 views

How Much Flesh to the Bones does an Initial Online Publication need?

Background of my question is the following: I have found a solution for my question Smoothness Conditions for Planar “Mock-parametric” Spline Interpolation and while developing the solution, I ...
3
votes
1answer
506 views

CMI at 20 conference [closed]

Just recently (September 24 - 26) there was a conference at Oxford dedicated to 20th anniversary of CMI. (https://www.claymath.org/events/cmi-20) The program looks interesting. Does anyone know if ...
18
votes
2answers
842 views

Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case ...
1
vote
0answers
236 views

Professionalism of using relevant but non-technical images in mathematics presentations to “soften” up some of the slides [closed]

Suppose someone is presenting at a mathematics conference. He or she wants to soften up some of the less-technical slides with relevant images from a site like Pexels. It seems to me that, at least in ...
11
votes
5answers
991 views

Early examples of mathematicians publishing (from home) in a foreign language?

Today this is common, but how exactly did it start? I am looking for examples in various languages, and suggest: Exclude Latin (as more “ancient” or “international” than “foreign”) Exclude French ...
1
vote
1answer
100 views

What is the Essential Difference Between Random Matrices and Random Graphs?

I have the impression, that random graphs and random matrices seem to be perceived and treated as separate areas of interest; I'm not an expert in either of the subjects, so maybe my impression is ...
7
votes
2answers
529 views

Research in applied algebra

I am in my final year of my doctoral study in Mathematics, where my research topic is $p$-groups, specifically classification of $p$-groups by coclass. My work involves a great deal of computation in ...
90
votes
10answers
9k views

Examples of notably long or difficult proofs that only improve upon existing results by a small amount

I was recently reading Bui, Conrey and Young's 2011 paper "More than 41% of the zeros of the zeta function are on the critical line", in which they improve the lower bound on the proportion of zeros ...
1
vote
1answer
481 views

State-of-the-art geometry book? [closed]

For my best friend's birthday, I am looking for a geometry book. He's currently doing his math PhD and is really fond of geometry, especially hyperbolic or higher-dimensional ones, also interested in (...
0
votes
0answers
90 views

Is there a precise relationship between ``Geometric Functional Analysis" and high-dimensional probability/information theory?

The 2009 course on GFA by Roman Vershynin (https://www.math.uci.edu/~rvershyn/papers/GFA-book.pdf) introduced the subject with this line on the course page, "...
51
votes
16answers
4k views

What are examples of books which teach the practice of mathematics?

One may classify the types of mathematics books written for students into two groups: books which merely teach mathematics (i.e., they present theorems and proofs, ready-made, as it were) and those ...
27
votes
2answers
950 views

Should I publish a paper if its results overlap significantly with an earlier paper?

I have a preprint X that is sitting in the ArXiv for which I am not sure if it is still worth publishing. It turns out the paper I wrote has considerable overlap with another preprint Y after one of ...
14
votes
1answer
853 views

What happened to “Research in the Mathematical Sciences”?

The journal "Research in the Mathematical Sciences" was founded in 2014 and originally published by SpringerOpen, a division of Springer supporting Open Access journals. In the first article of the ...
1
vote
1answer
102 views

Intuition for coercive functions

I have been working with $\Gamma$-convergence for some time now; it has lead me to wonder: What is the intuition behind coercive functions?
68
votes
18answers
19k views

What programming language should a professional mathematician know? [closed]

More and more I am becoming convinced that one should know at least one programming language very well as a mathematician of this century. Is my conviction justified, or not applicable? If I am right,...
3
votes
1answer
147 views

Does the space of harmonic forms change continuously with the metric?

Let $(M,g_0)$ be a closed $n$-dimensional Riemannian manifold. Let $1<k<n$ be fixed, and let $\Delta_{g_0}:\Omega^k(M) \to \Omega^k(M)$ be the $g_0$-Laplacian. Let $H^k_{g_0}=\text{ker} \Delta_{...
25
votes
3answers
4k views

Naming in math: from red herrings to very long names

The are some parts of math in which you encounter easily new structures, obtained by modifying or generalizing existing ones. Recent examples can be tropical geometry, or the theory around the field ...
73
votes
5answers
11k 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
416 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 ...
7
votes
0answers
825 views

Work of Caucher Birkar [closed]

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 ...
35
votes
2answers
3k 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....
3
votes
2answers
167 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
587 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
158 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
758 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
890 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 ...
11
votes
2answers
456 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
235 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
219 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
3k 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
192 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
201 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 ...
40
votes
13answers
7k 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
148 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
votes
0answers
178 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
votes
0answers
68 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 \...