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.

2,328 questions with no upvoted or accepted answers
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203
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9k 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 ...
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812 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 ...
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673 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) \...
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596 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, ...
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298 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 ...
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498 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 ...
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318 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 ...
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696 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 ...
13
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1answer
764 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 ...
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338 views

Techniques for doing “handwritten” math with current 2020 technology?

A bit of a soft question here. But what techniques are people using to efficiently do analytical math that would have typically been handwritten in the past? For example, if you're working with a ...
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184 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 ...
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529 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 (...
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332 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 ...
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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 ...
12
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1answer
327 views

How do we know we get the right answer?

The problem of ontology is one much discussed in mathematical philosophy with much categorization into different schools of thought, but the problem of epistemology seems to be less discussed; ...
12
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1answer
879 views

Balancing Coursework with original research

I am a first-year graduate student in a US university pursuing a PhD in mathematics. I am a bit frustrated in trying to balance coursework with original research. I saw students who spend most of ...
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226 views

Examples of hard but accessible problems in mathematics

There are some very old problems in mathematics, whose solutions involve reasonably accessible tools, by which I mean that (assuming knowledge of the foundations of the subject) it does take one or ...
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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 ...
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176 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 ...
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352 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 ...
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452 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 ...
11
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1answer
540 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
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522 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.
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129 views

What objects can be turned into a category?

A poset $(P,\le)$ can be turned into a category in a standard way. A group $(G,\cdot)$ can also be turned into a category in a standard way. Can we topological space $(X,\tau)$ into a category in a ...
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104 views

Exemples of applications of “groupoidification” to linear algebra

I just read Baez's very nice blog notes about groupoidification, and around the beginning, he states : "From all this, you should begin to vaguely see that starting from any sort of incidence ...
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230 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 ...
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1k 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 ...
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436 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 ...
10
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1answer
740 views

Infinitely many axioms of ZFC vs. finitely many axioms of NBG

It is known that ZFC needs infinitely many axioms, but NBG (Neuman-Bernays-Gödel set theory) is finitely axiomatizable (as first-order theories of course). But both theories agree completely on the ...
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222 views

Ars longa, vita brevis.

There's little use studying mathematics without actually doing mathematics. There is a plethora of exercises in any textbook worth its salt. I suppose those with some business in looking up something ...
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288 views

Notations in Functional Analysis: $L^p$, $L_p$, $\mathscr{L}^p$, $\mathscr{L}_p$, $\mathcal{L}^p$, and $\mathcal{L}_p$

If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$...
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Matrix performing local differintegral analysis being its own inverse. Coincidence?

I found a curious matrix $$T = \begin{bmatrix}1&2&1\\1&0&-1\\1&-2&1\end{bmatrix}$$ This matrix (or actually $\frac 1 2 T$) performs Local mean value (integral) estimation. ...
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76 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 ...
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153 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 ...
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1k views

Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
9
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1answer
240 views

Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006. In his talk the first slide he shows has the following written on it: ...
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140 views

How are the topos-theoretic programs of Caramello and Lurie related?

It is clear that we are living in very exciting time for topos theory, with many exciting developments from different directions. Of course most notable are Lurie's work on higher categories and ...
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178 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 ...
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228 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, ...
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145 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 :...
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484 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 ...
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574 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 ...
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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 ...
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90 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
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267 views

How strong is the analogy between spectra and abelian groups?

I am led to understand that spectra are some kind of $\infty$-analogue of (discrete) abelian groups, or perhaps more accurately, some kind of generalisation of chain complexes of abelian groups. How ...
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908 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 ...
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316 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, ...
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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 ...
8
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1answer
521 views

logic lectures on youtube

Currently I am reading Logic an Structure by Dirk van Dalen (2008). As I am missing some basics I try to find related lectures on youtube. I frequently watch MIT, Stanford, and University of ...
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Has anyone studied the set $\Big\{\sum_{k<t}e^{ik^2}\ |\ \forall t\in\mathbb{N}\Big\}$

As a curiosity and a test of the pseudo-random nature of trigonometric functions evalutated at integer-squares, I inquired to see what it would look like to graph the subset of $\mathbb{C}$ descibed ...

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