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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. In other words, questions that can be answered without making computations or applying theorems and axioms.

1
vote
1answer
135 views

On the 2002 paper “Dynamics of polynomial automorphisms of $\mathbb{C}^k$” by Guedj and Sibony

I desperately need to read the paper [1] before meeting a would-be supervisor, but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, not even one ...
16
votes
1answer
408 views

Commutative rings : Topoi = Fields :?

The following is probably a bad question, but hopefully, it might have a very good answer. In category theory there is a quite famous analogy between topoi and commutative rings, I was never ...
1
vote
2answers
297 views

When was the generalization easier to prove than the specific case? [duplicate]

I distinctly remember from my long-ago undergraduate math that there were some interesting cases where a solution (proof) was sought for some specific thing but it wasn't easy to find - and in a few ...
90
votes
18answers
10k views

Good “casual” advanced math books

I'm curious if there are any good math books out there that take a "casual approach" to higher level topics. I'm very interested in advanced math, but have lost the time as of late to study textbooks ...
11
votes
2answers
2k views

Why not a Stacks project for Homotopy Theory?

The lack of resources bridging the gap between what one finds in Hatcher's algebraic topology text and modern research on homotopy theory has been brought several times before on MathOverflow [1, 2, 3]...
-2
votes
0answers
237 views

Why is the notion of compactness so powerful? [closed]

According to you, what are the deep reasons, at a very fundamental level, that makes the notion of compactness so useful and ubiquitous throughout modern mathematics: theory of compact groups, compact ...
6
votes
0answers
347 views

Is there any theorem achieving Conway's “Mathematician's Liberation Movement”

John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
14
votes
0answers
412 views

Does inner model theory seek canonical models for large cardinals?

Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in ...
80
votes
23answers
12k views

Noteworthy, but not so famous conjectures resolved recent years

Conjectures play important role in development of mathematics. Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in ...
6
votes
1answer
125 views

Name for topological spaces where “every point has a local base wellordered by reverse inclusion”?

There are many properties regarding local bases of a topological space, like first countable if every point has a countable local base. Is there a similar name for a space where "every point has a ...
9
votes
0answers
361 views

Why is Planar algebras I (by Vaughan Jones) not published?

On Saturday 4 September 1999, Vaughan Jones put on arXiv a paper entitled Planar algebras, I. Until now, this preprint was cited 343 times (according to Google Scholar). It is often cited with the ...
5
votes
0answers
289 views

What solutions to useful computational problems could be rewarded through cryptocurrency smart contracts?

What kinds of cryptocurrency smart contracts could be used to reward people for solving specific kinds of useful computational problems? Background In this question, I asked for proposals for useful ...
12
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2answers
364 views

Category theory & geometric measure theory?

My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research ...
4
votes
0answers
287 views

Better names for Lie groups

After reading this question I was wondering whether mathematicians tried to invent better names for exceptional simple Lie groups $F_4, E_6, E_7, E_8$ ? These names seems a bit obscure and does not ...
3
votes
1answer
133 views

Metric Measure Space: A Basic Question

I understand the basic definition of a metric measure space to be the following: A metric measure space is a triple of a space $X$, metric $d$, and measure $m$: $(X,d,m)$ in the sense that the ...
4
votes
0answers
122 views
+100

Local “boundary comparison principle” for harmonic functions

Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = - ...
0
votes
1answer
78 views

Name for Directed Edges in Digraphs

Graph theory originated in German speaking countries and there directed edges are called "Pfeil" which translates to "arrow", which makes sense, because arrows have distinguishable front end and rear ...
4
votes
0answers
106 views

Variational formulation for elliptic interface problem

Where can I find a paper that deals with the following interface problem with variational methods? In particular, what is the correct variational formulation of the problem (that is, a functional ...
8
votes
0answers
124 views

Mirror site for the NIST Digital Library of Mathematical Functions (DLMF)

For research I use the NIST Digital Library of Mathematical Functions (https://dlmf.nist.gov/) quite often for looking up basic facts about special functions. At the moment, however, this gives ...
4
votes
1answer
169 views

On the roots of Bernoulli polynomials

Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $\mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,\dots,N$ for some enough large $N$. It ...
2
votes
0answers
133 views

Rough paths theory - why is it natural?

First a disclaimer, I know very little rigorously about the subject and I may have many misconceptions in what follows so please correct me if I’m wrong about anything. Also none of this is meant to ...
16
votes
1answer
628 views

What is known about the common knowledge of mathematicians outside their field?

When giving a talk or writing a paper intended for non-specialist (i.e., mathematicians not specializing in the topic being discussed), the question inevitably occurs of what one can assume to be "...
29
votes
2answers
886 views

Littlewood’s three precepts of refereeing in mathematics: is it (1) new, (2) correct, (3) interesting?

I have a question regarding Littlewood’s three precepts of refereeing a mathematical paper, namely whether it is (1) new, (2) correct, and (3) interesting. I have found these mentioned in the ...
4
votes
0answers
568 views

How should one approach reading Spectral Algebraic Geometry by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory. This MathOverflow question asked for a roadmap to Lurie's Higher Algebra. Still another question asked for a ...
11
votes
1answer
837 views

How should one approach reading Higher Algebra by Lurie?

A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT ...
10
votes
2answers
211 views

Sum of squared nearest-neighbor distances between points in a square

Let $\square_2=\{(x,y): 0\leq x, y\leq1\}$ be the unit square in $\mathbb{R}^2$. Take $n>1$ points $P_1, \dots, P_n\in\square_2$. Denote the distances $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...
6
votes
0answers
187 views

Does anyone use non-sober topological spaces?

Recall that a sober space is a topological space such that every irreducible closed subset is the closure of exactly one point. Is there any area of mathematics outside of general topology where non-...
2
votes
1answer
217 views

Thematic programs for collaborative research (similar to the one in Bonn) [closed]

The Hausdorff Institute in Bonn periodically organizes thematic junior trimester programs, which "give young mathematicians (postdocs, junior faculty) the opportunity to carry out collaborative ...
0
votes
0answers
91 views

(Semi-)Riemannian geometry for working PDE analysts

What is a good reference on (semi-)Riemannian geometry written for PDE analysts (that is, with main focus on analytical problems and approaches)? The closest thing I know to this, are two books by ...
0
votes
0answers
60 views

Critical growth and geodesic connectedness in Lorentz manifold

What is the deep ("heuristic") reason why the quadratic growth of $\beta$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $\mathcal M = \mathcal M_0 \times \...
24
votes
1answer
448 views

“Matchmaking website” for project-specific mathematical collaborations [closed]

It seems that even in the age of Internet, most mathematical collaborations are born (and pursued) off-line. However, I wonder if there exists a "matchmaking website" for mathematical ...
9
votes
2answers
647 views

How to organize collaborations? Managing shared library and LaTeX document

What is an effective way to organize collaborations with several people on the same paper? How do you arrange the $\LaTeX$ document, the shared (digital) papers library, and other aspects? More in ...
0
votes
0answers
147 views

MathJobs: Should I update submitted material after deadline?

I am using MathJobs to apply for research postdocs in mathematics and I have a doubt. Suppose I have applied for a postdoc at University X by submitting all requested material (CV, research statement,...
46
votes
8answers
4k views

Your professional $\LaTeX$ experiences that saves your time in typesetting

In $\LaTeX$ typesetting, when we repeat a long and complex formula in long documents, it is appropriate to create a new command that just by calling this new command we get the desired output. For ...
4
votes
1answer
131 views

Compilation of representations of holomorphic functions

Holomorphic functions are my muse. As my muse, I love drawing them different ways. Allow me to frame this as though an artist talking about his muse. A holomorphic function $f$ on the unit disk $\...
3
votes
1answer
112 views

Interpretation of free energy for Ising models

In thermodynamics, the physical meaning of free energy is the maximum amount of work that can be extracted from a system. Now if we take an Ising model on a graph, with interaction weights on each ...
0
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0answers
42 views

Elliptic Dirichlet problems with measure boundary data

Can you point out any references on the Dirichlet problem for divergence-type elliptic operators with a Radon measure as boundary data?
3
votes
1answer
90 views

Boundary condition for elliptic problems and domain decomposition

This question is motivated by one that has been previously asked on this website: Elliptic problem on a domain split in two subdomains Consider an open domain $U$ split in two non-overlapping ...
3
votes
1answer
190 views

Elliptic problem on a domain split in two subdomains

Consider the following elliptic problem in a split domain: $$ (\ast) \quad\begin{cases} -\Delta u=f_1 \quad &\text{ in } U_1\\ -\Delta u =f_2 & \text{ in } U_2\\ u=g & \text{ on } \...
7
votes
4answers
676 views

Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$

Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...
3
votes
1answer
201 views

Heuristics for boundary Harnack inequality

What is the heuristic idea of the proof of the boundary Harnack inequality presented in the appendix of Caffarelli's 1998 lectures on the obstacle problem (page 38 here)?
14
votes
1answer
588 views

“Gauss trick” vs Karatsuba multiplication

This question is inspired by article Alexander Shen "Gauss multiplication trick?" (submitted to "Mathematical Enlightenment"). Dasgupta, Papadimitriou, Vazirani, Algorithms (2008) Ch. 2: The ...
15
votes
2answers
1k views

Integrating over a hypercube, not a hypersphere

Denote $\square_m=\{\pmb{x}=(x_1,\dots,x_m)\in\mathbb{R}^m: 0\leq x_i\leq1,\,\,\forall i\}$ be an $m$-dimensional cube. It is all too familiar that $\int_{\square_1}\frac{dx}{1+x^2}=\frac{\pi}4$. ...
112
votes
44answers
14k views

Old books you would like to have reprinted with high-quality typesetting

There are some questions on mathoverflow such as What out-of-print books would you like to see re-printed? Old books still used with answers that tell us things such as: Mathematicians prefer to ...
54
votes
5answers
6k views

What to do if you notice a substantial improvement to a result in a paper whilst refereeing it?

What would you do/have you done in such a situation? Hand out the improvement for free in your report Wait until the result is published and then submit elsewhere Inform the editor about the ...
4
votes
0answers
208 views

Is there a theory behind these puzzles? (communicating by modifying data)

Consider the following puzzles: Problem 1: Alice is given two data by Zora: a binary string $w$ of length $2^r$, and a position $p$ in the string (which we can view as an integer $0\leq p<2^r$). ...
46
votes
11answers
5k views

What definitions were crucial to further understanding?

Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
6
votes
0answers
211 views

Best software to do big number calculations quickly [closed]

I am trying to do some work on some math conjecture. I am testing the conjecture numbers using very large math numbers (100+ digits ). I am currently using python to test these numbers. In the ...
30
votes
5answers
3k views

Should computer code be included within publications that present numerical results?

Many research papers include numerical results obtained through computation. Most of the time such computations are performed using software that is used by many mathematicians, i.e., Maple, ...
10
votes
1answer
294 views

Polymath type websites for specialized areas

This question is inspired by the success and more importantly, the democratizing affect of the polymath projects and Mathoverflow. I put the idea in my NSF proposals a few times, but the panels don't ...