Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [soft-question]

For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.

113
votes
0answers
6k views

Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
20
votes
0answers
566 views

Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
14
votes
0answers
208 views

Why is 2 so troublesome a prime?

I have been asking to myself for a while now why $2$ has such an exceptional behaviour in algebraic number theory. For example, the Kronecker-Weber Theorem proof was completed for all cases but that ...
14
votes
0answers
441 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) \...
13
votes
0answers
202 views

How far from research frontier is Hatcher's book

After a student masters the entire Hatcher's book on Algebraic Topology, possibly including the additional chapter on Spectral Sequences, I am curious how far is he/she from the research frontier to ...
13
votes
0answers
392 views

Can you build metric space theory without the real numbers?

Every metric space $M$ has a completion $\widehat M$, that is, it can be embedded as a dense subset in a complete metric space. When I first came across this theorem, I thought "well, that's amazing, ...
13
votes
0answers
469 views

Approximating a blackboard with long-distance communication

From time to time I need to talk about mathematics with my advisor remotely. I would like to approximate writing on a blackboard together as closely as is reasonable. What are some technological ...
12
votes
0answers
103 views

$\frac{1}{p}+\frac{1}{q}=1$ vs $\sum_{n=0}^\infty \frac{1}{p^n}=q$

It just occurred to me that conjugate exponents, i.e. $p,q\in(1,+\infty)$ such that $$\frac{1}{p}+\frac{1}{q} =1$$ also satisfy the relations: $\sum_{n=0}^\infty \frac{1}{p^n}=q;$ $\sum_{n=0}^\infty \...
12
votes
0answers
254 views

Write Occam's razor rigorously

What does Occam's razor look like written rigorously? Motivated by the desire to develop this idea further: https://philosophy.stackexchange.com/questions/41728/ I'm curious to see how a formal ...
11
votes
0answers
405 views

What differentiates algebraic geometry over $\mathbb{C}$ from complex analysis?

I have just begun learning algebraic geometry, and there are a lot more connections to complex analysis than I expected. For example: $\mathbb{CP}^1$ is the Riemann sphere Elliptic curves are ...
11
votes
0answers
152 views

Can we characterize all infinite PID s whose group of units is singleton?

I am looking for a way to characterize all infinite PID s having exactly one unit i.e. invertible element ( finite PID s are not interesting , they are all fields ) . The only such example I know of ...
11
votes
0answers
366 views

Advanced stochastic process book (a bit flavor from real analysis)

I am looking for the book about advanced stochastic process. It may cover the following content: Stochastic matrices. Ex: $A(k)$, where $k$ is the time index. Stochastic process in space (...
11
votes
0answers
294 views

What are some arguments/counterarguments for Zeilberger's “proof certificates”?

Here is the quote I wish to ask about: "I speculate that similar developments will occur elsewhere in mathematics, and will 'trivialize' large parts of mathematics, by reducing mathematical ...
11
votes
0answers
2k views

Publication date of the book of Michael Spivak - Physics for Mathematicians II?

I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
10
votes
0answers
111 views

Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
10
votes
0answers
124 views

Peculiar pictures in advanced maths books

I have recently started reading Introduction to Symplectic Topology by McDuff and Salamon and I came across this picture: I find it very funny and really interesting. I read on Wikipedia that Ian ...
10
votes
0answers
289 views

Uses of category-theory

I am a graduate student currently working my way through an introductory course in category theory. A question I have for this theory is why it is usefull? Now i know this is a rich theory that is ...
10
votes
0answers
963 views

What is “Field with One Element”?

I was reading the Wikipedia article about The Field with One Element and I came across the following quotes: "...F1 refers to the idea that there should be a way to replace sets and operations, the ...
9
votes
0answers
115 views

Roadmap to Modern Research in Partial Differential Equations

Let me start off with a description of my background. I am an undergraduate student, with some background in real analysis (Rudin, Principles of Mathematical Analysis), measure theory (Royden, Real ...
9
votes
0answers
83 views

What is the precise definition of the prefix “co” in mathematics?

Given a notion "A" in mathematics, in many cases "coA" is also defined. Here are some common examples: sine and cosine; tangent and cotangent; secant and cosecant; function and cofunction; morphism ...
9
votes
0answers
164 views

Topological Features and Graph Spectra

I was just thinking recently about if there are any possible meaningful connections between tools such as persistent homology used for things like topological data analysis and tools used in spectral ...
9
votes
0answers
129 views

When was “co-” first used to mean duality in mathematics?

As one can see on Wiktionary, the meaning of the prefix is "co-" as used in mathematics is different from its meaning as used in the rest of the English language, and does not seem to be a natural ...
9
votes
0answers
487 views

A question on popularization of math: inspiring the beauty of mathematics while making New Year's wishes

Many fellow students of mine today shared by various means the following picture: . I was told that this picture is supposed to communicate the beauty of math in a funny way, but I really can't ...
9
votes
0answers
511 views

Is there really anything wrong with Bourbaki's Set Theory?

Recently I have started reading Bourbaki's Theory of Sets on my own. Regarding one of the explanations of a concept when I went to a Professor of our college, he asked me why I was wasting my time ...
9
votes
0answers
2k views

How to take limit *along a path*

So in multivariable calculus for a limit of a function to exist, the limits of the function along all possible paths must exist and equal the same value. But how does one calculate the limit along a ...
9
votes
0answers
442 views

(Mathematical) Applications of Quantum Group

What are some mathematical applications of Quantum Groups? I have tried researching and found that it is used to find solutions to Yang-Baxter equation. Any other applications? Thanks a lot.
8
votes
0answers
91 views

Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
8
votes
0answers
130 views

How can we formalize Jorge Luis Borges' Aleph?

Background. Jorge Luis Borges was a post-modern short-story writer of the 20th century, whose stories often invoke a healthy dose of surrealism. One of his works is called The Aleph. In this book, ...
8
votes
0answers
163 views

Legitimate papers refuting the significance of the golden ratio in art?

I'm not sure this is the right place to ask about this, but is there any legitimate peer-reviewed paper refuting the significance of the golden ratio in art? I can find numerous websites and blogs ...
8
votes
0answers
669 views

Comparing the U.S. undergraduate math education to the French “classes préparatoires”

Could anyone comment on how the math track of the "classes préparatoires" compares to the U.S. undergraduate major? I took a look at some of the French entrance exams and was rather intimidated. How ...
8
votes
0answers
279 views

Why are injective $\mathscr{O}$-modules flasque?

Let $X$ be a topological space, and let $\mathscr{O}$ be a sheaf of rings on $X$. It is easy to verify that the functor $\Gamma (U, -) : \textbf{Mod}(\mathscr{O}) \to \textbf{Ab}$ is representable, ...
8
votes
0answers
1k views

How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?

I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
7
votes
0answers
61 views

Heuristic on Sobolev and BV functions

Let $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a Sobolev or BV vector field. A heuristic that I've heard frequently is the following: $f$ is almost Lipschitz on a large "good" set but ...
7
votes
0answers
124 views

Where to publish expository mathematics/personal notes?

I have a short expository paper that is more or less my personal take/exploration of a topic that interests me. I submitted to Mathematics Magazine and it was rejected for non-mathematical reasons - ...
7
votes
0answers
68 views

Conjectures for which there are strong heuristic arguments both for and against

I find the conjecture that there are infinitely many Fermat primes to be very interesting, because there is a very reasonable-seeming and semi-quantitative heuristic argument for the conjecture, but ...
7
votes
0answers
95 views

What are some present day examples of mathematics that needs to be made rigorous?

Hundreds of years ago, calculus, while still very useful, lacked rigor. It was later made rigorous and put on a sound basis by Dedekind and others. Another example is intuitive set theory, which was ...
7
votes
0answers
468 views

What's there to do in category theory?

I'm sure anyone who's heard of categories has also heard the classical "Well obviously there aren't any real theorems in category theory, it's much too general", or something in the likes of it. Now ...
7
votes
0answers
123 views

“Smaller” math proofs discovered as a means to prove a more substantial theorem

I know the title may not be fully descriptive, but please bear with me. When I do math, whether I have formed a conjecture and seek to prove it or am simply proving nontrivial theorems for homework ...
7
votes
0answers
165 views

“many numbers to describe the proof of the fact that A is isomorphic to itself”

In "conceptual mathematics" by Lawvere et al., it says on page 107: The expression 'each set is number' refers to the attitude toward sets in general whereby we consider relationships like $\...
7
votes
0answers
761 views

Good video lectures on complex analysis?

I would like to find a complete series of video lectures on complex analysis, preferably with the following conditions: The videos are in English and clearly recorded. (Using English as a second ...
7
votes
0answers
281 views

Is the Risch algorithm useful for calculating antiderivatives by hand?

In a German forum, a user asked how the "Feynman"-trick works. The example was $$f(x)=xe^x$$ Another user mentioned that the Risch algorithm should be taught. Therefore, I wonder whether the Risch ...
7
votes
0answers
123 views

Intuition on the external Zappa–Szép product

$\newcommand{\Aut}{\operatorname{Aut}}$A classmate of mine recently posted an interesting question on Facebook. It didn't get an answer, and I couldn't get anywhere myself, so I'm hoping that someone ...
7
votes
0answers
163 views

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

Recently I was reading the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found, An anecdote survives about Grothendieck's arrival in Nancy: ...
7
votes
0answers
129 views

Can we express the roots of all polynomials in terms of roots of some special polynomials?

We can describe the roots of quadratic equations in terms of addition, subtraction, multiplication, division, and the square-root function $\sqrt a$ which computes a root of the special polynomial $x^...
7
votes
0answers
356 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
7
votes
0answers
147 views

Decorating eggs

I came across this question as I have been helping someone decorate styrofoam eggs by gluing wrapping paper onto it. The question is quite simple but I found myself stumped. Given an egg what is the ...
7
votes
0answers
154 views

Why is topological group not a popular topic?

In Japan, there are many universities with a formal course about topological group using the classic by Pontryagin. Yet topological group is not studied in a formal course in many other countries, ...
7
votes
0answers
184 views

The Existence of “Simple” Prime Generating Functions

Obviously, we do not know an explicit and easily manipulable formula for finding every prime - that is, a function $f(n)$ which yields the $n^{th}$ prime. I've seen plenty of formulas that "cheat" in ...
7
votes
0answers
118 views

Strategy for self-studying after M.S.

For someone who has finished their M.S. degree in pure mathematics, what is a good way to keep learning mathematics within your specialization? Would you suggest reading research articles from ...
6
votes
0answers
98 views

Consequences of finite many twin primes?

Suppose , it turns out that the number of twin primes is finite (this is very unlikely, but let us assume it). Which consequences would such a result have for number theory ? To be more concrete :...