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2
votes
1answer
100 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
190 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
286 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 ...
11
votes
2answers
782 views

What if your research paper got rejected with few comments given on paper?

I often wonder what if your research paper got rejected? Is it fair enough to resubmit that paper in the same journal? Is there any possibility of publication of that paper in any journal? Thanks
1
vote
1answer
77 views

Good text suggest to abstract algebra and point set topology as well as metric space, or you may consider as introductory to topology

I am going to take courses about those topic. i want some text which is suitable for beginner and with some difficult example or questions.
16
votes
3answers
6k views

How to cite preprints from arXiv?

Obviously when writing a math research paper it is good to cite one's references. However, with the advent of arXiv, oftentimes a paper is only available on arXiv while is awaits the long process of ...
16
votes
13answers
2k views

Creative Thinking Questions?

Math is often intimidating to the average man due to its complex appearance. To show that math requires creative thinking, not just memorization, I was wondering if anyone had any math problems that ...
2
votes
2answers
314 views

Nonlinear Programming and Linear Programming

Why would a nonlinear programming solver come p with a different solution than a linear programming solver if all the constraints are linear? Isn't a linear programming solver basically a "subset" of ...
41
votes
16answers
10k views

Pen, pencils and paper to write math [closed]

The question is inspired by this thread on MO. My father is a mathematician, so I remember seeing very nice pens all around. When I started to do math by myself I've realized how important can be a ...
11
votes
7answers
815 views

What is a Number Theorist

I often find myself facing this problem when asked by non-mathematicians about what I do. When the answer, "I'm a mathematician," doesn't suffice and I have to reply that I am a number theorist, I do ...
1
vote
1answer
266 views

Meaning of the Soul Theorem

The Soul Theorem states that in every complete, connected riemannian manifold $M$ with $\mathrm{sec}(M)\geq 0$, there exists compact, totally convex, totally geodesic submanifold $S$ such that $M$ is ...
5
votes
1answer
595 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
17
votes
6answers
9k views

undergraduate math vs graduate math

It's really a mild, soft question. So far, I am an undergrad student, contemplating on several majors. What will be the major difference if I become a math major and go to a graduate school to study ...
9
votes
2answers
704 views

Variety vs. Manifold

In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am ...
2
votes
2answers
71 views

Isomorphisms that have names (Or: Base-less isomorphisms)

The answer to this question of mine provided me with the fact that every isomorphism $$ \phi: K^{n} \rightarrow V, $$ where $V$ is an arbitrary vector space of finite dimension $n$ over the field ...
-2
votes
1answer
131 views

Fuzzy logical and integer programming

Is there any way of formulating linear/non-linear programming problems in terms of YES, NO, and MAYBE instead of just $0-1$ programming?
5
votes
1answer
319 views

Highest gain mathematical activity

I have this odd dream that online resources like this can serve as a virtual thesis advisor for future mathematicians who are teaching themselves. Here's another question along these lines. You can do ...
0
votes
2answers
48 views

Averages and Team

I have a question: Suppose $5$ players each score an average of $10$ points per game. Then collectively, do they score on average $50$ points per game? So player 1 scores an average of 10 points ...
3
votes
2answers
180 views

Question on mathematical writing

I am now writing my graduate thesis, it includes some basics mathematical theorems/propositions. I got a trouble in writing, more concretely, I do not know when can I state a mathematical claim as a ...
36
votes
18answers
7k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
12
votes
3answers
3k views

Can the Bourbaki series be used profitably by undergraduates?

Can the Bourbaki series be used profitably by undergraduates and high school students?Are we the target audience? I came across the N.Bourbaki texts while surfing the internet(I have not had the ...
101
votes
7answers
3k views

What remains in a student's mind

I'm a first year graduate student of mathematics and I have an important question. I like studying math and when I attend, a course I try to study in the best way possible, with different textbooks ...
13
votes
8answers
1k views

What would be a good outdoor maths puzzle for children?

I have to find an interesting activity for some 11-year-olds moving to high school this year. It is supposed to take about 30-45 minutes, and I thought of having a mathematical theme. I can make a ...
9
votes
1answer
439 views

Stacks in arithmetic geometry [closed]

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
5
votes
3answers
6k views

Main branches of mathematics [closed]

Can anybody please show me the main branches and subbranches of mathematics and the statistical sciences in a hierarchical form? I am not a mathematician and often in my research I see a lot of new ...
7
votes
4answers
400 views

Extracurricular ideas for UK GCSE level maths student

My daughter is 15 years old and enjoys her maths classes (perhaps only because her maths homework takes her the least amount of time). Until now I have managed to introduce her to subject matter ...
2
votes
1answer
253 views

How much connection is there between Commutative Algebra and Algebraic Topology?

How much connection is there between Commutative Algebra and Algebraic Topology? I am looking for general highlights, not complex details.
9
votes
3answers
640 views

How to deal with the temporary nature of my knowledge?

I'm a self-learner trying to learn Math while enrolled in a wrong major (Humanities). I have gone through the many amazing questions and answers here (& elsewhere, including Prof. Tao's blogs) ...
8
votes
3answers
242 views

How come in statistics there is very little justification for the formulas used and proofs are almost nonexistent [closed]

I don't understand why people accept certain formulas in statistics without a mathematical proof style argument. You see this a lot in statistics textbooks and unfortunately this spills over with the ...
21
votes
3answers
1k views

Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$(\forall a,b\in L) x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$ where $\vee$ is the join operation, and $\wedge$ is the meet ...
5
votes
1answer
124 views

What is good about simple Lie algebras?

Recently I've been reading Naive Lie Theory by John Stillwell. In the book our aim usually concerns finding whether Lie algebras or Lie groups are simple. I wonder what beautiful properties does a ...
6
votes
3answers
522 views

The Importance Of Good Teachers and Guidance In the Academics

I'm a first year student for a math degree. I'm very curious on how good students overcome their bad teachers in the journey of learning and grasping the courses material fully, all in the pressure of ...
1
vote
1answer
93 views

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference? And what is other branch of advanced analytic geometry called? in ...
2
votes
1answer
82 views

What is the utility in writing pdfs in terms of their kernel?

Consider the normal distribution. We know that $$p(x| \mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} $$ The kernel is $$ p(x| \mu, \sigma^{2}) \propto ...
25
votes
3answers
1k views

What did Gauss think about infinity?

I have someone who is begging for a conversation with me about infinity. He thinks that Cantor got it wrong, and suggested to me that Gauss did not really believe in infinity, and would not have ...
25
votes
9answers
2k views

Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can't really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one ...
7
votes
1answer
812 views

Independent undergraduate research — what to do?

I hope this question is within the scope of this website. I am currently a rising senior, and need to decide on a topic for my independent undergraduate research/thesis. I was hoping to get some ...
14
votes
8answers
2k views

What math should a programmer know?

I am an application programmer focussing on Line Of Business (LOB) applications. I am from non-mathematics and non-CS background. What mathematics should I learn which help me improve my programming ...
5
votes
3answers
330 views

Is there a branch of mathematics that requires the existence of sets that contain themselves?

I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking ...
2
votes
1answer
126 views

Is there a theory that extend real analysis to functions maps into other algebraic structure?

I am studying real analysis now, reading Rudin's book Real and Complex Analysis. One thing confused me is when talking about measurable functions, we assume the function to be, from an abstract space ...
5
votes
1answer
140 views

Status of PL topology

I'm starting to learn about geometric topology and manifold theory. I know that there are three big important categories of manifolds: topological, smooth and PL. But I'm seeing that while topological ...
23
votes
5answers
1k views

Importance of rigor

I always have a hard time explaining the importance of rigor to my friends who are not mathematically minded. A lot of past mathematicians develop the foundations of today's mathematics without going ...
17
votes
2answers
465 views

Which results depend on the irrationality of $\pi$?

Recently the following uninteresting clock picture was posted by one of my non-mathematically inclined friends to my facebook wall, saying that it was funny and possibly thinking that I would find it ...
3
votes
2answers
490 views

How much algebra is there in Noncommutative Geometry?

My Professor of Homological Algebra got me into some Hochschild (co)homology and then suggested to continue with formally smooth algebras, noncommutative differential forms and so forth. Now, my ...
4
votes
7answers
3k views

Simple applications of complex numbers

I've been helping a high school student with his complex number homework (algebra, de Moivre's formula, etc.), and we came across the question of the "usefulness" of "imaginary" numbers - If there ...
3
votes
1answer
117 views

Varieties given by non-algebraic equations

In algebraic geometry one (mostly) studies varieties given by polynomial equations. Such equations define algebraic varieties and there are many "dictionaries" available. For example, the category ...
24
votes
4answers
5k views

What does it really mean for something to be “trivial”?

I see this word a lot when I read about mathematics. Is this meant to be another way of saying "obvious" or "easy"? What if it's actually wrong? It's like when I see "the rest is left as an exercise ...
1
vote
0answers
84 views

I'm interested in different meanings of “normal”~ [duplicate]

Possible Duplicate: What is it to be normal? I've learned in algebra class that "normal" means a linear operator is commutative with its adjoint; also we say that $H$ is a normal ...
9
votes
5answers
782 views

What are special functions for?

If you read enough mathematics, you eventually come across several so-called "special functions". I'm always left wondering what on Earth these things are actually for. We have the Euler Gamma ...
9
votes
5answers
3k views

Why do we consider prime numbers important, and what are their applications other than number theory in pure math?

Why do we consider prime numbers important, and what are their applications other than number theory in pure math? I know that Number theory is devoted to studying prime numbers, but there must be ...