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4
votes
1answer
273 views

Unrivaled math classics that would be of practical benefit to the upcoming generation?

I'm often impressed that top mathematicians in a given field seem to have not only a knowledge of the "state of the art" of their subfield, but also a knowledge of the history of the field and thus ...
3
votes
2answers
419 views

Course for self-study

I have basically completed a good deal of Single Variable Calculus from Spivak's Calculus and since I leave school in May next year,I intend to put in some effort to pick up college mathematics.I am a ...
4
votes
1answer
219 views

Are Different Areas of Number Theory That Different

I was wondering if number theorists are "number theorists," or eventually resolve themselves into one of the various branches - i.e., algebraic, analytic, etc. Also out of curiosity, I was wondering ...
2
votes
3answers
459 views

Proof that something is undefined?

How can one tell the difference when the result is undefined or math just doesn't know how to provide a value for that particular equation? (the value still exists however) For example, how could one ...
-1
votes
1answer
134 views

In group theory, what is a complete and incomplete representation?

In group theory, what is a complete representation and what is an incomplete representation?
4
votes
3answers
813 views

What should a PDE/analysis enthusiast know?

What are the cool things someone who likes PDE and functional analysis should know and learn about? What do you think are the fundamentals and the next steps? I was thinking it would be good to know ...
5
votes
1answer
60 views

Can different uniformizations of Riemann surfaces be related somehow

Let $X$ be a hyperbolic compact connected Riemann surface. Let $U\subset X$ be an open subset. Assume that $U\neq X$. We can uniformize $X$ by $\mathbf{H}$ directly to obtain it as a quotient of ...
4
votes
4answers
365 views

How integrals are computed?

I know some integrals can't have undefined integrals, but why? And how, for example, can be proved that the area under the hyperbola $y=\frac{1}{x}$ is $\ln(x)$?
5
votes
4answers
374 views

Would nonmath students be able to understand this?

For a course, I am required to do a presentation. The topic could either be something mundane, like a career strategy report, or something more interesting, such as a controversial topic, or an ...
12
votes
2answers
1k 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
2answers
669 views

Why do mathematicians care so much about zeta functions?

Why is it that so many people care so much about zeta functions? Why do people write books and books specifically about the theory of Riemann Zeta functions? What is its purpose? Is it just to ...
6
votes
1answer
679 views

Topology needed for differential geometry

I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know? I know some basic concepts reading from ...
6
votes
6answers
1k views

Purpose of Linear Algebra

How much emphasizes should be on proof on a first course in Linear Algebra? I sometimes feel that they (proofs) crowd out a coherent vision for linear algebra. However I also think a central theme of ...
39
votes
2answers
6k views

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
4
votes
2answers
374 views

Number of elements vs cardinality vs size

I have been wondered the definition of cardinality and number of elements. One mathematician told me that one can't said that the cardinality or size of the set $\{1\}$ is one, it should be said that ...
10
votes
5answers
2k views

Advice for benefits to directly use analysis textbook to replace calculus

Main purpose: For self-learning performance, neither for exam nor degree courses. Calculus textbook using now[1]: Calculus I, Weinstein&Marsden, UTM, Springer Question Description: I've been ...
24
votes
3answers
2k views

Research in algebraic topology

I have started studying algebraic topology with the help of Armstrong(Basic), Massey, and Hatcher. If I plan to do research in algebraic topology in future: What else should I study after ...
9
votes
3answers
961 views

How do I “learn” more difficult algebra?

I do not really understand where I was suppose to learn this kind of stuff. I am always told that my algebra knowledge is the biggest reason I am so bad at math. But I do not understand where I was ...
10
votes
1answer
3k views

Tips and examples for a poster presentation in pure mathematics

I will be presenting a poster in a few weeks but have no experience with them. I've seen and given plenty of talks, read and written papers, but I have never made or even seen a poster in pure ...
21
votes
5answers
1k views

Elementary results from Algebraic Number Theory

The purpose of this question is to motivate me to study algebraic number theory. Let me explain. My motivation for studying number theory is to learn about beautiful results with simple, accessible ...
19
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 ...
48
votes
24answers
9k views

“Negative” versus “Minus”

As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and not "minus $0.8$" to denote $-0.8$? The so called "textbook answer" regarding this question reads: ...
3
votes
2answers
690 views

Change of Basis vs. Linear Transformation

If i understand it correctly, change of basis is just a specific case of a linear transformation. Specifically given a vector space $V$ over a field $F$ such that $\dim V=n$, change of basis is just ...
2
votes
1answer
431 views

Hard problems in algebraic geometry

People very often say that algebraic geometry is a hard subject and has many challenging problems to solve. I believe the hodge conjecture is the one of the most difficult in the field and you, which ...
13
votes
3answers
1k views

History of elliptic curves

In one sense elliptic curves are a rather modern object as some of its properties have been studied only in the last century or so. But in another sense there are a very classical object for studying ...
1
vote
1answer
411 views

Is classifying one dimensional generalized quasicrystals worthwhile strategy to approach RH?

Works done: After fruitlessly poring over books on zeta functions, it seems Freeman Dyson's sotto voce nudge to classify generalized one-dimensional quasicrystals is a way to go. As he writes: ...
3
votes
0answers
89 views

Importance of Poincare's Conjecture [duplicate]

Possible Duplicate: What is the importance of the Poincaré conjecture? It is rather well known that Poincare conjecture was proved by Perelman in 2003 given the amount of coverage it ...
2
votes
1answer
144 views

Sequences and series and Lebesgue integration

This is a question on a self-study matter. Earlier, I stopped reading W. Rudin's book on "principles of mathematical analysis", at the chapter on Riemann integration. Now for certain other needs, I ...
10
votes
4answers
2k views

Math for computer science?

Math for computer science? I'm a computer science major and just completed linear algebra. Many courses are available to take now. Of particular interest: number theory and abstract algebra (Modern ...
15
votes
3answers
828 views

What are the applications of finite calculus

I'm reading through Concrete Mathematics [Graham, Knuth, Patashnik; 2nd edition], and in the section regarding Summation, they have a sub-section entitled "Finite and Infinite Calculus". In this ...
4
votes
1answer
195 views

Good in-depth books for precalc?

I'm trying to brush up on algebra and precalc to remember stuff I used to know and learn stuff I didn't get to. Trying to pass the accuplacer placement test and test into calculus, so im trying to ...
4
votes
5answers
217 views

Are there methods to well order a finite group in a meaningful way?

Can some finite groups be well ordered in a "meaningful" way? I mean, it is clear that we can trivially find a bijection between $\{1,...,n\}$ and a finite group $G$ with $n$ elements, but I am ...
3
votes
1answer
240 views

Forum for students - your experiences, recommendations, suggestions [closed]

Have you ever used online discussion forum or something similar for students of your course? (Or, if you are a student, have you attended a class, where something like this was used?) If yes, I'd ...
5
votes
6answers
769 views

How to make sure a proof is correct

If you come up with a proof of a mathematical proposition, how do you verify the proof is correct? Put it another way, how do you avoid a wrong proof? I guess there is no definitive answer to this. ...
3
votes
3answers
824 views

Remembering Taylor series

Could anyone suggest a good way of memorizing Taylor series for common functions? I have tried to remember them but never seem to be able to commit them to permanent memory.
13
votes
3answers
686 views

Journals of math history?

In a related question to this one, in what journals do math historians publish their article in? Brian M. Scott provided a link to Judy Grabiner's, who is a math historian, home page and it seems that ...
5
votes
0answers
226 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 ...
13
votes
2answers
895 views

Who is a Math Historian?

In the context of classes, it is very often that discussion on the history of mathematics arises, whether it'd be on who should a lemma be attributed to or a certain event that occurred during the ...
7
votes
3answers
2k views

What is the prerequisite knowledge for learning Galois theory?

What is the prerequisite knowledge for learning Galois theory? I don't know what a ring is.
8
votes
1answer
1k views

Knuth's up-arrow notation - Is there practical use for the numbers involved?

From Wikipedia, Knuth's up-arrow notation begins at exponentiation and continues through the hyperoperations: $a \uparrow b = a^b$ $a \uparrow\uparrow b = {\ ^{b}a} = ...
49
votes
5answers
2k views

How does one give a mathematical talk?

Sometime tomorrow morning I will be presenting a mathematics talk on something related to commutative algebra. The people present there will probably be two mathematicians (an algebraic geometer and a ...
2
votes
1answer
230 views

Reverse mathematics in undergraduate program

This might be different from colleges to colleges, but anyway: Is reverse mathematics covered in usual undergraduate math programs? If so, how far is it covered? Just a curious question, as ...
18
votes
8answers
1k views

When is something “obvious”?

I try to be a good student but I often find it hard to know when something is "obvious" and when it isn't. Obviously (excuse the pun) I understand that it is specific to the level at which the writer ...
0
votes
2answers
145 views

On Reflexive Banach Spaces

My Functional Analysis lecturer gave me a topic for my assignment, the title is "On Reflexive Banach Spaces". I am a looking for several good references to start my work, that is why I brought this ...
10
votes
2answers
1k views

Should one understand a proof of every important theorem in a field of mathematics to research?

I guess not. However, I think I have to understand proofs of some of them, if not all of them. So what is the criterion, if any? What kind of theorem whose proof I can get away with? EDIT For ...
7
votes
0answers
172 views

How to read equations and expressions out loud? [duplicate]

Possible Duplicate: Is there a definitive guide to speaking mathematics? This may be an incredibly stupid question, but I was wondering how would one pronounce simple mathematical equations ...
3
votes
2answers
183 views

How do you explain paradoxes to non-mathematicians?

For example, how do you explain why the perimeter of this staircase does not converge to $\sqrt2$? Or, why isn't $\sqrt{(-1)(-1)}=\sqrt{-1}\sqrt{-1}$? I would say, the reason is simply because they ...
2
votes
1answer
104 views

Intuition on characters of topological groups

I am coming to the end of a series of lecture notes on representations of $S_n$ and $GL(V)$. Near the end, it attempts to introduce the notion of the "character of a topological group", but doesn't ...
4
votes
1answer
193 views

How to get a more intuitive/and motivated understanding of a solution?

Math is a deductive science, that one starts from basic premises to prove more complicated results. From doing questions and reading solutions (at least at the elementary level eg: Olympiad ...
7
votes
1answer
288 views

How to present a paper?

I am going to present several papers to an audience. I have read through all the papers and have a clear idea about them. But this is the first time I have ever presented papers, and I am guessing ...