Questions tagged [soft-question]

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. Do not use this tag for easy or supposedly easy mathematical questions.

Filter by
Sorted by
Tagged with
51
votes
7answers
8k views

When is one 'ready' to make original contributions to mathematics?

This is quite a philosophical, soft question which can be moved if necessary. So, basically I started my PhD 9 months ago and have thrown myself into learning more mathematics and found this an ...
9
votes
1answer
1k views

How reliable is arXiv to use as a reference in a paper?

I'm writing a small and simple paper to finish my graduation, I never wrote or published anything before. Is about some irrational/transcendental numbers. I was mentioning some numbers that it's not ...
8
votes
0answers
467 views

When is a paper finished? [closed]

How much truth is there to Chomsky's remark that "mathematicians stop working when things get too difficult"? https://youtu.be/atupfHizJxM?t=453 Is this true in your own work? How do you know you ...
2
votes
1answer
413 views

Cute/striking application(s) of snake lemma outside homological algebra

I already asked this question on MSE here https://math.stackexchange.com/questions/3254184/cute-striking-applications-of-snake-lemma-outside-homological-algebra, but still received no answer. I hope I ...
7
votes
0answers
308 views

The meaning of this mysterious remark in Littlewood's Miscellany

In the well known book by Littlewood (Mathematician's Miscellany, or the later edition called Littlewood's Miscellany) there is a remark made in the chapter 'A Mathematical education', the meaning of ...
37
votes
3answers
3k views

A map of non-pathological topology?

I think of topological spaces as coming in several "islands of interestingness" (the CW island, the Zariski archipelago,...) dotting a vast "pathological sea" (the long line ocean, the gulf of the ...
19
votes
0answers
816 views

What examples of existence forcing proofs are there?

Forcing proofs tend to be fairly constructive, in the sense that if I claim that there is a forcing that does something, I usually prove this by constructing that forcing. There are only a handful of ...
5
votes
2answers
409 views

Tricks for getting a creative idea [closed]

Caveat: I fear that people will criticize me for asking this potentially inappropriate question here, but I guess that the community here is quite unique in the ability of potentially answering my ...
16
votes
2answers
2k views

Is differential topology a dying field? [closed]

I recently had a post doc in differential topology advise against me going into the field, since it seems to be dying in his words. Is this true? I do see very little activity on differential topology ...
2
votes
0answers
368 views

Reading list in dynamical systems

So I’ve managed to gather from various sources, a plethora of books in dynamical systems. Now I’m wondering which of them to read, and in what order. So far these are the books I’ve found/been ...
0
votes
0answers
74 views

Fractional and Exterior Calculus

I am interested in fractional calculus and was wondering is there an equivalent/analagous fractional exterior calculus or fractional theory of differential forms? If so can someone point me out to ...
25
votes
5answers
4k views

Is the field of q-series 'dead'? [closed]

I had a discussion with my advisor about what am I interested as my future research direction and I said it is special functions and q-series. He laughed and said that the topic is essentially dead ...
0
votes
0answers
81 views

Alternate Ways of Writing Complex Summations

In my writing, I often have to consider summation formulas where describing the terms of the summation make the equation rather weirdly spaced out. For instance, I am currently considering the formula ...
12
votes
4answers
3k views

Grade-school elementary algebra presented in an abstract-algebra style?

I remember once hearing a (probably apocryphal) story about a university math professor that tried to teach a gradeschool class about algebra by telling them a few simple axioms and definitions and ...
0
votes
1answer
71 views

Focusing and nonfocusing nonlinear terms

What is the mathematical and physical meaning of the terms focusing and nonfocusing when they refer to nonlinear terms in a dispersive equations?
1
vote
1answer
385 views

Rejection of article without mentioning specific reason [closed]

After 3 months of submission I got this mail from EJC without any specific reason of rejection. "We regret to inform you that we have decided not to proceed with publication of your submission "......
2
votes
0answers
175 views

Lecture notes in complex algebraic geometry (which might follow Voisin's book)

I have two questions here, Whats the difference in the target audience/required-background between Voisin's two volumes and Griffiths-Harris' book on algebraic geometry which also seems highly tied ...
22
votes
8answers
3k views

How do we explain the use of a software on a math paper?

Suppose one has written a math/computer science paper that is more focused in the math part of it. I had a very complicated function and needed to find its maximum, so I used Mathematica (Wolfram) to ...
3
votes
0answers
68 views

Is projection method really applicable for numerical solution of linear integral equations in $ L^p \ (p \neq 2)$ setting?

Projection method is a traditional method to numerically handle problem of linear integral equation. The routine way is to do it in $ L^2 $ setting. For example: Let $ A:L^2(a,b) \to L^2(a,b) $ be a ...
8
votes
1answer
663 views

How many graduate students in mathematics are there worldwide?

There is already a question about the number of research mathematicians here, but I was wondering : is it possible to estimate the number of graduate students in mathematics worldwide in 2019 ? (...
2
votes
0answers
434 views

Does a result remain known after everyone who knew it has died? [closed]

When working on a research project, one tries to spend their time answering questions that have not yet been answered. There enters the terminology of "known" versus "unknown" results, which we ...
8
votes
1answer
365 views

Reciprocal sum of binomials and divisibility by $3$

We all know that $\sum_{k=0}^n\binom{n}k$ is not divisible by $3$. QUESTION. Is it true that the numerator of $a_n$ (in reduced form) is never divisible by $3$? $$a_n=\sum_{k=0}^n\frac1{\binom{n}...
23
votes
3answers
4k views

How come mathematicians published in Annals of Eugenics?

I was surprised to see that On the construction of balanced incomplete block designs by Raj Chandra Bose was published (in 1939) in a journal named Annals of Eugenics (see here) (published between ...
3
votes
2answers
192 views

One generating function, two-fold sums

This comes out of a series of transformations, so I'll just get to the main focus here. Define the functions $$F_n(x)=\frac12\left(x+2+2\sqrt{x+1}\right)^n+\frac12\left(x+2-2\sqrt{x+1}\right)^n. \...
4
votes
3answers
524 views

Problem based representation theory book

I am trying to find books similar in the spirit of Ram Murty's Problems in Analytic Number theory or Murty Esmonde's Problems in Algebraic number theory in the field of Representation Theory (of ...
115
votes
17answers
14k views

Pressure to defend the relevance of one's area of mathematics

I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “...
2
votes
0answers
51 views

Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an $s$-dimensional Hausdorff measure restricted to the Koch curve?

Motivated by my previous question Alberti rank-one theorem and irregular jump discontinuities, I'd like to ask the following: Is there a function $u \in BV_{loc}(\mathbb{R}^2)$ such that $Du$ is an ...
2
votes
0answers
52 views

Alberti rank-one theorem and irregular jump discontinuities

Is it fair to say that Alberti rank one theorem means that a BV functions $u \in BV(\mathbb{R}^2)$ has $Du = D^{cantor}u$ if and only if it has a jump discontinuity across a curve that is not smooth (...
2
votes
2answers
246 views

Concrete example of BV function $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}$ with singular derivative? More precisely, I'd like to see an example (and a plot using Mathematica or Matlab) of a function $$...
2
votes
0answers
56 views

“Almost” absolute continuity of derivative of BV function if ${\rm Tr}\,D_Sf = 0$

Let $f: \mathbb R^N \to \mathbb R^N$ be a $BV$ function. Suppose that $\mathrm{div} f$ is absolutely continuous with respect to the Lebesgue measure: $\operatorname{div} f \ll \mathcal L^N$. This ...
7
votes
1answer
135 views

BV functions and wave equation

What is the role played by BV functions in the study of (possibly nonlinear) wave equations? I think that one would need assume small initial data in $L^1$ or $H^1$ to get a well-posedness result (...
2
votes
0answers
168 views

What are the points about representation of groups? [closed]

For a fixed (let say finite-, or Lie-, to respect the historical motivations) group, why does the study of all its linear representations over a fixed field, leads to some knowledge about its ...
3
votes
0answers
218 views

Research on formalization of what constitutes a good theorem

This question is about automating theorem proving. Now automated theorem proving, as currently understood in CS literature, has little to do with pure mathematics as practiced on this site (I believe ...
16
votes
2answers
919 views

Examples of problems where considering “discrete analogues” has provided insight or led to a solution of the original problem

The Kakeya conjecture posits that any Kakeya set in $\mathbb{R}^n$ has dimension $n$. A discrete (finitized?) version of this problem is the Finite Field Kakeya conjecture, which was proved by Dvir ...
5
votes
0answers
151 views

Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
2
votes
1answer
335 views

Mathematical background required to learn about sheaves

Due to my interest in type theory (and higher type theory), I have found that learning about sheaves might be useful (for, e.g., sheaf models of type theories). There is Kashiwara and Schapira's ...
3
votes
0answers
136 views

Partially BV vector fields and renormalization

Why does the approach used to prove Theorem 4.1 in the paper by Le Bris and Lions on Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications not work ...
12
votes
2answers
1k views

Where are Serre’s lectures at Collège de France to be found?

Having run into several references, at various places and occasions, to "Serre’s Course at Collège de France, 19xy-19xy+1" for various values of xy, I would genuinely like to know where these lectures ...
3
votes
0answers
174 views

Has the external knit product been used to construct a previously unknown group?

In the Wikipedia article Zappa–Szép product , the knit product (a.k.a. Zappa–Szép product, Zappa–Rédei-Szép product, general product, exact factorization) is defined, and its basic properties are laid ...
3
votes
1answer
141 views

Lusin Lipschitz approximation in BV and Sobolev space

Theorem 5.34 in Functions of bounded variation by L. Ambrosio, N. Fusco and D. Pallara states that Let $u \in [BV(\mathbb{R}^N)]^m$. Then there exists a constant $\kappa>0$ such that for every $...
4
votes
1answer
148 views

Alberti rank one theorem and a blow-up argument

In this paper, it is written that Alberti’s rank says that the singular part $D^s u$ with respect to $\mathcal L^d$ of the distributional derivative $Du$ of a function $u \in BV_{loc}(\mathbb R^d; \...
6
votes
0answers
157 views

The distributional gradient of the closest isometry to the differential of a smooth map

The setting-a "linear algebra" fact: Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
11
votes
6answers
997 views

Interesting examples of non-locally compact topological groups

Harmonic analysis is concentrated mostly on studying locally compact groups. I would like to ask people about examples of non-locally compact topological groups that are interesting in connection with ...
16
votes
0answers
405 views

Reference request for Grothendieck's work on “Integration with values in a topological group”

Disclaimer. This question was already asked in Mathematics Stack Exchange (see the link here). I wanted the question to be migrated here but I was told by a moderator that a question that old is ...
7
votes
1answer
390 views

Divisibility of sum of multinomials

Let $n, m$ and $t$ be positive integers. Define the multi-family of sequences $$S(n,m,t)=\sum_{k_1+\cdots+k_n=m}\binom{m}{k_1,\dots,k_n}^t$$ where the sum runs over non-negative integers $k_1,\dots,...
8
votes
2answers
424 views

Hyperbolic PDE in mathematics

Hyperbolic PDE (like the wave equation) are roughly speaking, PDE that satisfy the “finite propagation speed of information” property. They are ubiquitous in mathematical physics (essentially, most ...
10
votes
0answers
2k views

Story of “Grothendieck's prime number” 57

I asked this question earlier, at hsm.stackexchange.com without much luck. Maybe somebody can answer it here. There is a story about Alexander Grothendieck and the "Grothendieck Prime" 57, which ...
0
votes
0answers
335 views

The collected works of John von Neumann

Might there be an online collection of John von Neumann's collected works in pdf format? I'm particularly interested in his approach to applied mathematics(ex. shockwaves, hydrodynamics). Note: I ...
22
votes
3answers
2k views

Why is the standard definition of a $(p, q)$-tensor so bizarre?

At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows. Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{...
30
votes
8answers
6k views

Example of a Mathematician/Physicist whose Other Publications during their PhD eclipsed their PhD Thesis

I am wondering if there is some example of a mathematician or physicist who published other papers at the same time as their PhD work and independently of it which actually eclipsed the content of the ...

1
3 4
5
6 7
38