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8
votes
3answers
477 views

Explanation of a Phrase from Prof. Ravi Vakil's Website

I was just browsing through Ravi Vakil's website when i found a nice article written on what he demands from his students. Here is the webpage. (For those interested!) ...
13
votes
3answers
894 views

Guidelines for learning about Ramanujan's work?

It is well known that one of the first books Ramanujan studied was "Synopsis of Pure and Applied Mathematics" and that it shaped the way Ramanujan thought and wrote about mathematics. Being interested ...
5
votes
10answers
3k views

Does a negative number really exist?

Second Update: I see that some answers that reference my image are more closely answering my question. Here is a second image to clarify my point. Take this image representing a checkerboard like ...
11
votes
9answers
949 views

Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
5
votes
2answers
396 views

On the Möbius $\mu$ function

A search on wikipedia shows: $$\mu(n) = \sum_{k=1,gcd(k,n)=1}^{n} e^{2\pi i \frac{k}{n}}$$ But that uses complex numbers... and requires finding out the gcd... How useful will be a method, if that ...
26
votes
10answers
2k views

Contributions of Galois Theory to Mathematics

What are the major and minor contributions of Galois Theory to Mathematics? I mean direct contributions (like being aplied as it appears in Algebra) or simply by serving as a model to other theories. ...
5
votes
1answer
246 views

Are there other pseudo-random distributions like the prime-numbers?

Does there exists other structures in math, which are seemingly random, but deterministic, and follow rules similar to the prime numbers, by rules I mean there must be statements similar to goldbach's ...
29
votes
8answers
3k views

Why should we care about groups at all?

Someone asked me today, "Why we should care about groups at all?" I realized that I have absolutely no idea how to respond. One way to treat this might be to reduce "why should we care about groups" ...
2
votes
4answers
226 views

Choosing subjects at high school

For a high school student that has the options of taking one of Statistics, Algebra or Calculus as subjects, which subject would be the best choice if the person is still undecided on a specific ...
135
votes
28answers
16k views

List of Interesting Math Videos/ Documentaries

This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list. Story of ...
3
votes
2answers
312 views

For teaching: Combinatorial Construction Riddles

Can you give examples of combinatorial construction riddles, approachable by gifted high school students? Examples: Find a finite set $A$ and $B \subset 2^A$ such that any element of $A$ is covered ...
3
votes
1answer
493 views

How to check the veracity of arXiv.org papers?

Suppose I found a preprint on arXiv.org (like this one: http://arxiv.org/abs/math/0309146 ), and I need to check if what's in there is actually true. Any tips? I don't want to study an article only to ...
7
votes
3answers
471 views

Why are they called “Isothermal” Coordinates?

If I understand correctly, Gauss proved that given any oriented Riemannian surface, one can find a complex structure on the surface so that the metric on the charts is just $f|dz|$, where $f>0$. ...
19
votes
3answers
689 views

Why isn't the gamma function defined so that $\Gamma(n) = n! $?

As a physics student, I have occasionally run across the gamma function $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$ when we want to generalize the concept of a factorial. ...
5
votes
2answers
357 views

Map of Mathematical Logic

My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a ...
8
votes
4answers
592 views

What material is good for extra-studying

I am an undergraduate Math student, i find the courses that I am in are too easy (calculus, linear algebra, Discrete math, logic). I want to become a successful mathematician. I want to know what ...
32
votes
17answers
3k views

Non-Scientific questions solved by mathematics

I have a general question about the applications of mathematics. What are some applications of mathematics that are not scientific, perhaps maybe literary or philosophical, or political. I am ...
3
votes
1answer
1k views

Math for “Electrical Engineering & Communication Technology” recommendation

I have the feeling that an engineer should be more comfortable with math, than I'm felling I am right now. I also have a tendency toward learning math not required by the math classes I took, and was ...
2
votes
2answers
253 views

Where can I get the OLD (1940-1970) high school scholarship question papers?

Today matematics are moving away from understanding basics & core methodology. Level of the difficulties are reducing. Questions are becoming more simpler. Hence I would like to see the difficulty ...
10
votes
2answers
530 views

Category Theory with and without Objects

Slight Motivation: In Mac Lane and Freyd's books (the latter being a reprint of an older book called "Abelian Categories") they note that instead of defining any Objects in a category we may define an ...
3
votes
0answers
77 views

Function Shape Reference

I'm wondering if their exists a visual/behavioral reference for the fundamental families of functions. I'm not a mathematician so excuse my language if I'm being overly vague. I would like to have a ...
10
votes
3answers
858 views

What is the key to success for a mathematician?

What is the most recommended quality a mathematician should have? Extremely high IQ levels?Passion for what he does?Patience and "stuborness"?Something else? Of course all of them are necessary ,but ...
10
votes
7answers
3k views

Topic for a high school-level math elective?

I'm looking for ideas for a 15-hour mathematical enrichment course in a Chinese high school. What (fairly) elementary subject would you suggest as a topic for such a course? ...
11
votes
3answers
1k views

Nice application of the Cauchy?-Frobenius?-Burnside?-Pólya? formula

Burnside's Lemma, whose list of names is longer than the proof, says that the number of orbits of a permutation group is the average number of fixed points of its elements. It's a very elegant result, ...
57
votes
7answers
4k views

Why do books titled “Abstract Algebra” mostly deal with groups/rings/fields?

As a computer science graduate who had only a basic course in abstract algebra, I want to study some abstract algebra in my free time. I've been looking through some books on the topic, and most seem ...
14
votes
2answers
2k views

Summer programs to gain research experience as an undergraduate, outside the United States?

There are many summer programs in the United States, targeted at good, motivated undergraduate students majoring in mathematics. Most programs, however (if not all), demand that the applicant is ...
0
votes
2answers
106 views

how can I present an idea about the difference between two functions clearly?

I have a presentation in which I want to point out a difference between two functions. Instead of putting the two functions on a slide and pointing to the differences, I want to do something simple. ...
42
votes
8answers
2k views

Designing an Irrational Numbers Wall Clock

A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock." There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational. ...
6
votes
2answers
296 views

which job to find to increase teaching experience

If you don't have enough teaching experience, what kind of job you should try to find to increase the teaching experience. Now many jobs have set requirements for teaching experience. It seems ...
14
votes
6answers
2k views

Books to learn physics, being a math major

What books would you recommend to learn physics, being a a Math major, from classical mechanics, electricity, etc. to modern physics?
4
votes
2answers
264 views

How to analyze triangles in Lobachevsky geometry?

I got an assignment to prove certain things about right triangles in Lobachevsky geometry, but so far I don't know where to start. What model is the best for studying these objects? What is the ...
5
votes
3answers
287 views

Fundamental Math Theory Resources?

I became interested in how numbers actually work recently. I want a source that covers number theory. Explaining how and why numbers react the way they do. A friend recommended Mathematical Proofs: A ...
14
votes
3answers
901 views

learning algebra and harmonic analysis

I've revised my question a bit in response to the (very helpful) advice so far-- I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather ...
4
votes
3answers
792 views

Suggest me the path to learn Maths

I am a Software Engineer. With interest on computers I have chosen this path. But I don't know anything about great maths. I want to learn from basic to some level(which would help me in my ...
2
votes
1answer
128 views

Mathematical Results from Counting Points in Lattices

I'm preparing a talk on lattice point enumeration in polytopes (Ehrhart-Macdonald Theory), and I'd like to have an introduction with a few motivational problems/results which arise from the ...
3
votes
0answers
157 views

notes on properties of chern character of coherent sheaves

I'm trying to understand a bit about chern classes/chern characters (in the algebraic setting, for varieties over C say) and was hoping to find some notes describing some well-known properties of ch ...
4
votes
1answer
121 views

property of a space

A two dimensional space (eg. $\mathbb{R}^2$ ) could be a flat or it could be the surface of sphere (or of any shape) in a 3-dimensional space. How do we distinguish between such spaces without ...
3
votes
3answers
873 views

Intuition for Surface Integrals

Is there any intuition behind surface integrals and their applications? I get that there's some stuff with fluid flow but it doesn't really stick that well, especially when it comes to surface ...
49
votes
7answers
3k views

Intuition in algebra?

My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) ...
4
votes
2answers
509 views

Advice: How can one prepare for a maths entrance exam? How can one develop mathematical thinking?

I am Computer science student so pardon me if I am asking this in a wrong place! My inspiration of learning maths is mainly due to algorithms and its feels to me that without mathematics I am not ...
6
votes
1answer
615 views

Geometry or Topology

So, I am a graduate student who is certain that he does not want to do analysis (I think...). What are the most exciting fields in mathematics right now? It seems to me that very generally, they are ...
7
votes
2answers
810 views

Mathematical Telescoping

Bill Dubuque has answered several questions by indicating that some form of "telescoping" is taking place. See this post and the links provided by Bill for more information. I have never heard of ...
27
votes
3answers
3k views

how to read a mathematical paper?

I hope that this question is on-topic, though it is not quite technical. I am curious to hear from people how they approach reading a mathematical paper. I am not asking specific questions on ...
7
votes
2answers
828 views

What are Diophantine equations REALLY?

Sometimes when you want to solve an equation you can just use algebra and rearrange it then you are done. But sometimes no amount of algebra can ever prove the equation, and then you need an idea, ...
15
votes
2answers
551 views

Why study “curves” instead of 1-manifolds?

In most undergraduate differential geometry courses -- I am thinking of do Carmo's "Differential Geometry of Curves and Surfaces" -- the topic of study is curves and surfaces in $\mathbb{R}^3$. ...
2
votes
0answers
247 views

Invariant Subspace Problem

Louis de Branges had a paper on his homepage claiming a solution for the Invariant Subspace Problem. But I don't see that paper anymore, though he still has a "proof" of Riemann Hypothesis on his ...
5
votes
2answers
518 views

deadline in math jobs application

In some math job advertisments, it shows "the deadeline is...", but to surprised some schools review applications before that, does that mean we submit application as earlier as possible could be in a ...
6
votes
2answers
2k views

Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
36
votes
6answers
3k views

How hard is the proof of $\pi$ or e being transcendental?

I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
5
votes
3answers
389 views

Soft Question Hilbert Space Geometry

Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like ...