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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.

3
votes
0answers
36 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 ...
7
votes
0answers
40 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 ...
5
votes
0answers
104 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) \...
189
votes
141answers
39k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...
4
votes
0answers
62 views
+50

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; \...
1
vote
0answers
82 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 ...
5
votes
0answers
37 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 ...
5
votes
1answer
89 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 $...
3
votes
0answers
96 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 ...
10
votes
6answers
603 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 ...
26
votes
17answers
5k views

Research-only permanent positions worldwide

Most academic jobs involve some amount of teaching. Post-docs generally do not, but they are only short-term positions. Question: in which countries can one obtain a research-only permanent ...
6
votes
6answers
522 views

Reference Request: Perspective Painting

What is a good book/article explaining the mathematics behind perspective painting? I have already looked at the Wikipedia article on the topic, so I am looking for something more advanced than this. ...
-4
votes
0answers
98 views

Icons for mathematical concepts [closed]

Question migrated to User Experience. With the advent of mobile devices and small screens, symbols are very commonly used for abstract concepts. As I type this, the MathOverflow editor has symbols ...
3
votes
0answers
90 views

Pohozaev identity and related non-existence result for a nonlinear problem

Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional $$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \...
18
votes
0answers
276 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 ...
163
votes
45answers
88k 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 ...
6
votes
2answers
472 views

Where can I find resources for creating a mathematics “bridge course”?

My department is in the very early stages of developing a "bridge course" or "introduction to proofs" course, motivated by our lower-level courses not currently doing a good job of preparing our ...
49
votes
48answers
12k views

Describe a topic in one sentence. [closed]

When you study a topic for the first time, it can be difficult to pick up the motivations and to understand where everything is going. Once you have some experience, however, you get that good high-...
66
votes
22answers
9k views

Essays and thoughts on mathematics

Many distinguished mathematicians, at some point of their career, collected their thoughts on mathematics (its aesthetic, purposes, methods, etc.) and on the work of a mathematician in written ...
28
votes
9answers
5k views

How to explain to an engineer what algebraic geometry is?

This question is similar to this one in that I'm asking about how to introduce a mathematical research topic or activity to a non-mathematician: in this case algebraic geometry, intended as the most ...
7
votes
1answer
365 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,...
17
votes
1answer
905 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}_{...
21
votes
3answers
2k views

Why is free probability a generalization of probability theory?

Note: This question was already asked on Math.SE nearly a week and a half ago but did not receive any responses. To the best of my knowledge, free probability is an active topic of research, so I hope ...
27
votes
4answers
3k views

Is there an RSS reader for mathematicians?

For a while, I have used Google Reader to stay on top of several math blogs. Unfortunately, Google will pull the plug on Reader one month from today, so I need to find an alternative fast. I was ...
18
votes
4answers
1k views

Betting markets for unsolved problems?

I'm not sure if this is appropriate for Math Overflow. I was curious about whether any attempts have been made to set up betting markets for getting mathematicians to put wagers on what they think ...
11
votes
14answers
13k views

Movies about mathematics/mathematicians [closed]

I would like to watch a movie about mathematics/mathematicians (english/french language is OK, italian would be the best! Both real and invented stories are OK, maybe I would prefer something based on ...
8
votes
2answers
321 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 ...
147
votes
58answers
74k views

Interesting mathematical documentaries

I am looking for mathematical documentaries, both technical and non-technical. They should be "interesting" in that they present either actual mathematics, mathematicians or history of mathematics. I ...
-1
votes
0answers
62 views

How to “cite” MathOverflow answer? [migrated]

I am writing a paper at the moment and for one proof I use an idea which I got from a MathOverflow answer (at the time I was asking idle questions but later of course found a way to use it). I of ...
5
votes
0answers
426 views

Story of Grothendieck's Prime Number

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 ...
30
votes
8answers
5k 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 ...
11
votes
1answer
376 views

Is there a “formula” for the point in $\text{SO}(n)$ which is closest to a given matrix?

$\newcommand{\Sig}{\Sigma}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\distSO}[1]{\dist(#1,\SO)}$ $\newcommand{\distO}[1]{\text{dist}(#1,\On)}$ $\newcommand{\tildistSO}[1]{\operatorname{...
0
votes
0answers
142 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 ...
14
votes
8answers
1k views

Applications of the idea of deformation in algebraic geometry and other areas?

The idea of proving something by deforming the general case to some special cases is very powerful. For example, one can prove certain equalities by regarding both sides as functions/sheaves, and show ...
3
votes
1answer
126 views

Why control a continuous approximation of stochastic gradient descent instead of just the SGD?

In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in $$x_{k+1} = x_k - \eta u_k \nabla f_{\...
2
votes
0answers
164 views

Characteristic classes in a categorical framework [closed]

How does the notion of Characteristic Classes behave in a Categorical framework? That is if instead of Manifolds and Smooth maps if we use Categories and Functors(Smooth in some way) and instead of ...
24
votes
19answers
14k 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 ...
6
votes
2answers
571 views

Books on the History of math research at European universities

Are there good books that cover the history of math and mathematical science (ex. physics, chemistry, computer science) PhD programs in the Occident? My primary motivation is to figure out how the PhD ...
8
votes
0answers
402 views

Visualization and new geometry in higher stacks (soft question)

I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond manifolds ...
13
votes
1answer
774 views

Journal losing indexing services

I recently had a paper accepted by a journal. When I looked it up on the AMS’ Mathematical Reviews, I noticed that it was previously indexed by the service but, at present, is it not. The journal is ...
7
votes
5answers
1k views

Is there any physical or computational justification for non-constructive axioms such as AC or excluded middle?

I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, ...
1
vote
1answer
189 views

Reference request: Gauge theory [closed]

What are some good introductory texts to gauge theory? I have some basic differential geometry knowledge, but I don’t know any algebraic geometry. Also, as a side question, what intuitively is a ...
3
votes
0answers
192 views

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 = ...
101
votes
13answers
28k views

What are some noteworthy “mic-drop” moments in math?

Oftentimes in math the manner in which a solution to a problem is announced becomes a significant chapter/part of the lore associated with the problem, almost being remembered more than the manner in ...
20
votes
1answer
424 views

Videos of Gian-Carlo Rota Lectures

I apologize if this is off topic. I think most of his listeners would agree with me that Gian-Carlo Rota had a wonderful style of lecture delivery. I have heard him lecture, both as an undergraduate ...
116
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 ...
99
votes
20answers
15k views

Mathematical habits of thought and action which would be of use to non-mathematicians

Once again I come to MO for help with something I'm writing for the public. Which habits of mathematicians -- aspects of the way we approach problems, the way we argue, the way we function as a ...
70
votes
36answers
15k views

Demystifying complex numbers

At the end of this month I start teaching complex analysis to 2nd year undergraduates, mostly from engineering but some from science and maths. The main applications for them in future studies are ...
98
votes
38answers
18k views

What are some very important papers published in non-top journals?

There has already been a question about important papers that were initially rejected. Many of the answers were very interesting. The question is here. My concern in this question is slightly ...