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9
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4answers
447 views

Beside transcendental or uncomputable numbers what other types of numbers are there?

What other types/categories of numbers are there that we know of today (i.e. some one has done some work on them, like Chaitin's uncomputable $\Omega$ number)? Of course there are uncountably many ...
8
votes
1answer
449 views

Studying Deformation Theory of Schemes

Versal Property Local Deformation Space Mini-versal deformation space I came across these words while studying these papers a) Desingularization of moduli varities for vector bundles on curves, ...
8
votes
1answer
2k views

Very interesting graph!

I found a VERY interesting graph on http://www.xamuel.com/graphs-of-implicit-equations/. It looked very, very cool, but the equation of the graph is so simple! Here is the image of the graph: My ...
8
votes
4answers
462 views

What is the difference between analytic combinatorics and the theory of combinatorial species?

Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't ...
8
votes
5answers
3k views

How do you rebuild your Math skills after college? [closed]

I minored in Math in college (B.S. Software Engineering), and went through advanced Calc, Differential Equations, etc. But about 6 months after I graduated I had lost all my formulas. Now, about 5 ...
7
votes
3answers
846 views

Exercise books in analysis

I'm studying Rudin's Principles of mathematical analysis and I was wondering if there are some exercise books (that is, books with solved problems and exercises) that I can use as a companion to ...
7
votes
2answers
499 views

What is the best approach when things seem hopeless?

I'm finding that I am getting to the point of being hopelessly behind in one of my courses. What is the best thing to do when it feels impossible to get caught up in the literal sense. Being "caught ...
5
votes
2answers
175 views

The two distinct cultures in mathematics

I once read an article about two distinct culture within mathematics, "analysts" and "algebraists" when I was in high school. I am not still a graduate student, but I really want to hear what it feels ...
5
votes
3answers
474 views

Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
5
votes
4answers
420 views

Intuitive Explanation of Morphism Theorem

Is there an intuitive explanation for the morphism theorem from introductory abstract algebra? First Morphism Theorem: Let $K$ be the kernel of the group morphism $f: G \to H$. Then $G/K$ is ...
19
votes
5answers
1k views

Results that were widely believed to be false but were later shown to be true

What are some results which were widely believed to be false, but were later to be shown to be true, or vice-versa?
14
votes
5answers
3k views

Opposite of Fermat's Last Theorem?

So Wiles' proof showed that no three positive integers $a$, $b$, and $c$ can solve the equation $a^n+b^n=c^n$ for any integer value of n greater than $2$. Now what about the opposite? What does this ...
14
votes
5answers
2k views

What does “formal” mean?

I know the definition of formal power series, power series and polynomials. But what does the adjective "formal" mean? In google English dictionary, does it mean "9. Of or relating to linguistic or ...
13
votes
3answers
643 views

Where are the geometric figures in those “advanced geometry” textbooks?

It is me again to bother you. Since my last post I started to look at some seemingly "serious" mathematics for background study. Accidentally I went into a university bookshop and came across some ...
13
votes
2answers
943 views

What is the smallest constant that has explicitly appeared in a published paper? [closed]

You can read a lot about what large number in mathematics are, like Skewes' number, Moser's number or even Graham's number. So just for the sake of non-discrimination, I ask, what is the smallest ...
12
votes
2answers
2k views

High school mathematical research

I am a grade 12 student. I am interested in number theory and I am looking for topics to research on. Can you suggest some topics in number theory and in general that would make for a good research ...
11
votes
0answers
204 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
11
votes
2answers
372 views

Is there a geometric interpretation of $F_p,\ F_{p^n}$ and $\overline{F_p}?$

I have been doing some exercises about finite fields lately and I think I've obtained some understanding of what they are. What seems to be missing though is some kind of picture. Learning to work ...
10
votes
2answers
699 views

Applications of Geometry to Computer Science

How is differential geometry (or any type of theoretical math) being used in computer science? Any research I have done on this topic leads me to some sort of applied math concept. I know that there ...
10
votes
4answers
463 views

Application of computers in higher mathematics

Currently the main application of computers in mathematics seems to be to compute things, i.e. to solve equations, evaluate integrals, etc. It is at all possible to delegate the thinking of a ...
10
votes
1answer
32k views

Why is a full circle 360° degrees? [duplicate]

What's the reason we agreed to setting the number of degrees of a full circle to 360? Does that make any more sense than 100, 1000 or any other number? Is there any logic involved in that particular ...
10
votes
5answers
725 views

Book Recommendations and Proofs for a First Course in Real Analysis

I am taking real analysis in university. I find that it is difficult to prove some certain questions. What I want to ask is: How do we come out with a proof? Do we use some intuitive idea first and ...
10
votes
1answer
1k views

What distinguishes the Measure Theory and Probability Theory?

It is clear that the Theory of Probability works primarily with limited measures on measurable spaces. On the other hand there is a folklore that says that what distinguishes Measure Theory and ...
10
votes
3answers
1k views

What is the key to success for a mathematician? [closed]

What is the most recommended quality a mathematician should have? Extremely high IQ levels? Passion for what they do? Patience and "stubbornness"? Something else? Of course all of them are necessary, ...
9
votes
3answers
944 views

Prerequisite for Petersen's Riemannian Geometry

A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. ...
9
votes
1answer
396 views

Good references for stacks

I have seen stacks come up in various settings recently. I understand that, at least heuristically, they are some sort of generalization of a scheme, but I don't actually know anything about them. ...
8
votes
4answers
674 views

What are some elementary results (number theory) using theorems that went long-unproven?

Firstly, I do not mind if there are examples from fields other than number theory! This was just the first field, and where I think the richest examples, may come from. Now to elaborate on the title, ...
7
votes
5answers
500 views

Examples of open problems solved through short proof

Are there good examples of reasonable open problems in mathematics that had an 'obvious' solution via application of a theorem already known/not yet found in mathematics but could have been found with ...
7
votes
3answers
12k views

Abstract Algebra book with exercise solutions recommendations.

I am new to studying abstract algebra (and math in general). I've been reading Gilligan and Pinter's books. I am trying to improve my understanding by doing exercises. However none of the books I am ...
7
votes
1answer
2k views

Symmetric vs. Positive semidefinite

This may be a trivial or irrelevant question. I'm very sorry if so. Why do mathematicians use only symmetric matrices when they want positive semi/definite matrices? I mean haven't seen using ...
7
votes
2answers
347 views

self studying advice on analysis

I am trying to learn analysis on my own but there are times when I can't solve the problem or I get the solution wrong after looking it up, but I will only look up the problems online after I am ...
7
votes
2answers
716 views

What are Free Objects?

I've read the wikipedia article, but couldn't grasp the concept. Is there an informal definition? Are there examples of uses of free objects in calculus? Are free objects somehow connected to ...
6
votes
1answer
184 views

Recommendations for an “illuminating” (explained in the post) group theory/abstract algebra resource?

I recently asked a question regarding why homomorphisms and isomorphisms are important. The best answer to that question was actually a comment, which referred me to Brian M. Scott's answer here: ...
6
votes
2answers
218 views

Are there any theories being developed which study structures with many operations and many distributive laws?

"Algebraic structure" will mean a set with some n-ary operations defined on it. This does not include vector spaces for example. During my study of algebra I have encountered mostly algebraic ...
6
votes
2answers
607 views

Organization of the Learning Process

Sorry for off topic. I'll delete this topic immediately when community decides it's useless, however if anyone finds it's interesting, share your opinion with us. I just want to know your opinion ...
6
votes
3answers
490 views

Visual representation of groups

I vaguely remember seeing something like a "picture" of various groups a while back. It was as if the elements of the group were each associated with a point and many points had segments connecting ...
6
votes
6answers
3k views

Best Intermediate/Advanced Computer Science book

I'm very interested in Computer Science (computational complexity, etc.). I've already finished a University course in the subject (using Sipser's "Introduction to the Theory of Computation"). I know ...
5
votes
1answer
447 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
20
votes
2answers
2k views

Spotting crankery

Underwood Dudley published a book called mathematical cranks that talks about faux proofs throughout history. While it seems to be mostly for entertainment than anything else, I feel it has become ...
19
votes
2answers
763 views

History of the theory of equations: John Colson

This is an EDIT version of my original question: Recently I've been interested in the history of the Theory of Equations. The thing is that I learned about this mathematician named John Colson, he ...
17
votes
11answers
1k views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
15
votes
2answers
660 views

H0w have group theory and fractal geometry been combined?

Has there been a significant tie made between group theory and fractal geometry? What are some ways that they have been tied together? I've been inspired to ask this question by this image of a free ...
14
votes
2answers
186 views

Bellard's exotic formula for $\pi$

In his webpage, Fabrice Bellard mentions an exotic formula for $\pi$ as follows $$\pi = \frac{1}{740025}\left(\sum_{n = 1}^{\infty}\dfrac{3P(n)}{{\displaystyle \binom{7n}{2n}2^{n - 1}}} - ...
14
votes
6answers
534 views

Book with novel approaches to analysis

Now I'm studying Rudin's Principles of mathematical analysis, but I'm searching for a book that offers geometric, physical or otherwise non-standard approaches to topics in analysis. Also, I'm looking ...
14
votes
3answers
6k views

Importance of Linear Algebra

In one of his online lectures Benedict Gross comments that one can never have too much Linear Algebra. Also, looking around it seems like I can find comments to the effect that Linear Algebra has ...
13
votes
1answer
320 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
13
votes
7answers
1k views

How do we know whether certain mathematical theorems are circular?

There are countless mathematical theorems and lemmata, some of which, obviously, depend on others. My question is: how do we know that, say, Theorem $A_1$- which uses a result proved in Theorem $A_2$ ...
12
votes
1answer
1k views

Teaching engineers mathematical thinking skills

In my experience, many introductory engineering mathematics textbooks these days tend to skip proofs and discuss logic only in the context of digital electronics. On the other hand, I can imagine that ...
12
votes
3answers
3k views

*Recursive* vs. *inductive* definition

I once had an argument with a professor of mine, if the following definition was a recursive or inductive definition: Suppose you have sequence of real numbers. Define $a_0:=2$ and ...
12
votes
2answers
794 views

Recalling Proofs

When I am able to follow a proof presented in class or in a textbook, I usually can prove the same corollary or theorem a couple days later using the same arguments. But after a week of seeing the ...