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15
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0answers
850 views

Efficient ways to read and learn a new topic

I started reading the book "Topology without tears" by Sidney A Morris and lecture notes on "Elementary Number Theory" by WWL.Chen. To get the maximum out of the book and understand the material ...
13
votes
0answers
374 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 ...
13
votes
0answers
1k views

What are some strong algebraic number theory PhD programs?

I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ...
13
votes
0answers
679 views

What is magical about Cartan's magic formula?

Why is Cartan's magic formula $$\mathscr{L}_X\omega = i_Xd\omega + d(i_X\omega)$$ called "magic"? Should it be considered a highly surprising result? Does it "magically" prove several other ...
12
votes
0answers
359 views

Irrationality of e

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
11
votes
0answers
259 views

Anecdote about mathematicians leaping to tops of problems and then building a staircase down?

I've run across this cute little story before, and now for the life of me I can't find it anywhere. It goes something like: Two people are looking out onto a mathematical landscape, and there are ...
11
votes
0answers
1k views

Which is better strategy to learn and read books, traditionally one by one OR re-read carefully on perfect books

(Just focus on how to learn and master the stuff pretty well, not involve the aspect of courses or exam) Because recently I always feel that the time and energy are pretty limited, I want to try ...
10
votes
0answers
126 views

Soft question: How does basic differential geometry “fit together”?

I'm self-studying diff geom from Lee's Introduction to Smooth Manifolds. He warns the reader that there's a lot of machinery to construct, which is fine, and he explains things with wonderful clarity. ...
10
votes
0answers
163 views

How much practice is too much practice?

I have an exam in single-variable calculus tomorrow. Seeing as its my first exam at the university level, I have been spending some time doing old exams in order to get used to the format. While doing ...
10
votes
0answers
301 views

What Do Mathematicians Do?

The American Mathematical Society maintains a web page entitled "What Do Mathematicians Do?" which references two interesting surveys. (One of the reference links is broken, but this one works: What ...
10
votes
0answers
513 views

Are there “mental calculator” persons that can recognize closed-form expressions?

There are people, sometimes called mental calculators, who is capable of performing fast calculations in their head, involving multiplication, factorization, finding logarithms and roots of a high ...
9
votes
0answers
132 views

How much math is there?

Among other things I teach high school-level math, and one question that often comes up is: "How long would I have to study math in order to know all of it?" I usually tell them that it's like ...
9
votes
0answers
181 views

Big geometry grad schools - for an average applicant

What are some schools that have a lot of geometry going on, but that might accept some middle-of-the-range applicants? Let me add some context... I left grad school (UC Davis) with an MS in 2012 ...
8
votes
0answers
264 views

How much time is reasonable to complete baby Rudin?

I've been teaching myself math for more than a year. My current aim is towards algebraic topology and differential geometry. Apart from a messy (by which i mean some rigorous and some not) ...
8
votes
0answers
334 views

Is it possible to learn mathematics right from the source instead of reading textbooks. By studying the masters and not their pupils

i was wondering if mathematics learning process require the use of textbooks. When i was a high school student, i read as a preparation for university, Legendre book on Elements of geometry and ...
8
votes
0answers
242 views

The status of $\mathbb{R}$ in homotopy theory.

The definition of a path as a continuous map $I \rightarrow X$ is a completely natural one. But this raises two questions in my mind. First, what properties of the interval give rise to useful ...
7
votes
0answers
87 views

Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the ...
7
votes
0answers
1k views

Using distance formula to find slope, any reason to use the concluding equation?

So, today I was observing a class that I will be a TA for this semester and the professor started to talk about the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Well, my mind wandered a little ...
7
votes
0answers
165 views

Reference suggestion: eigenvalues of tridiagonal matrices

I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals). I have seen authors use continued ...
7
votes
0answers
213 views

Graham's Number : Why so big?

Can someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !
6
votes
0answers
63 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
6
votes
0answers
76 views

Intuition behind Erdős proof of the infinitude of prime numbers

Suppose by contradiction that there are finitely many primes, namely $p_1, p_2,...,p_k$, where $k$ is a natural number. Now consider another natural number $n$, and all natural numbers $m \leq n$. ...
6
votes
0answers
69 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) ...
6
votes
0answers
148 views

Math Behind the Dragon Illusion!

Dragon illusion has been one of the items presented in the 3rd "Gathering for Gardner". This video shows the illusion. What does it have to do with mathematics?
6
votes
0answers
109 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 ...
6
votes
0answers
293 views

formulating theories in math

One of my profs mentioned that sometimes people formulate theories about some type of object, but then later realize that those objects do not exist. Can someone given me an example of such a theory? ...
6
votes
0answers
351 views

Why didn't Cartan-Eilenberg develop homological algebra on sheaf theory?

Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at ...
6
votes
0answers
137 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, ...
6
votes
0answers
509 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 ...
5
votes
0answers
44 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 ...
5
votes
0answers
77 views

Reference Request: Prereqs for Lecture Notes on “Abstract Linear Algebra”

I just found this set of lecture notes on linear algebra which seems to go over several things I've been wondering about as I study linear algebra. Unfortunately there are very few exercises in the ...
5
votes
0answers
69 views

Relations between definite integrals not having a known closed form

Are there any known cases, when there are two (or more) definite integrals, none of them having any known closed-form expression on its own, but there is still a non-trivial$^\dagger$ elementary ...
5
votes
0answers
161 views

Propper algorithm to integrate ODE's numerically.

I have studied in a course several algorithms to integrate ODE's numerical: Runge-Kutta, Predictor-Corrector methods, Taylor... However the teacher failed to show which is the best for every ...
5
votes
0answers
114 views

Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site. Several years ago, I ...
5
votes
0answers
65 views

Note-taking software which is capable of representing mathematical relation

After quite a few studying experiences (both alone and following a class, studying topology, algebra, analysis, measure theory, differential geometry) I have acquired a lot of notes that I have either ...
5
votes
0answers
55 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
5
votes
0answers
103 views

Help needed with Masters' Thesis

My brother is at the very end of his Masters Program at a well-known University (Math, of course, hence my inclusion of this question on this site) and he is totally done with course work, but is ...
5
votes
0answers
98 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
5
votes
0answers
52 views

Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
5
votes
0answers
67 views

Is there a proper etiquette for asking an author for their (mathematical) software?

This may not be the correct place to ask such a question. I have read a mathematical paper on multiclass total variation clustering. I wish to use the algorithm in the contents to compare with another ...
5
votes
0answers
85 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
5
votes
0answers
165 views

Difference between dynamic system and complex system

I know this might not be an easy question, I've already read the wikipedia page, and there is an interesting view: Therefore, the main difference between chaotic systems and complex systems is ...
5
votes
0answers
116 views

Proof for '$AB = I$ then $BA = I$' without Motivation?

I have read this question page (If $AB = I$ then $BA = I$) by Dilawar and saw that most of proofs are using the fact that the algebra of matrices and linear operators are isomorphic. But from a ...
5
votes
0answers
252 views

Paul Erdős Joke.

I was watching the great documentary "$N$ is a Number" and in it Erdős tells a joke where he writes: PGOM LD AD LD CD Which means poor great old man, living dead, archeological discovery, legally ...
5
votes
0answers
130 views

which branch of maths studies Standard Logical Matrices

In classical logic you have truthtables like: & | T | F ---|---|--- T | T | F F | F | F In many valued logic you have tables like: (this one is ...
5
votes
0answers
268 views

Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
5
votes
0answers
146 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 ...
5
votes
0answers
359 views

Top graduate schools in dynamical systems.

I will be applying to graduate school next fall and I want to start my list of where to apply. I have looked on some of the top graduate schools to see how large of a dynamical systems group they ...
5
votes
0answers
147 views

The high road to learn algebraic geometry

Suppose that a student has a basic knowledge in commmutative algebra at the level of the Atiyah-MacDonald. What do you think about the following steps to learn algebraic geometry? step 1: read ...
5
votes
0answers
170 views

Convex Hulls vs Shrink Wrap

I was recently explaining to a friend what the convex hull of a set of points is using the analogy of an elastic band around a set of nails hammered into a board. I was about to say that we can ...