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5
votes
0answers
194 views

Recommendation textbooks on D-module

I am going to take part in a seminar on D-module and applications, the textbooks that will be used are : D-modules, Perverse Sheaves, and Representation Theory, and A Primer of Algebraic D-Modules ...
66
votes
9answers
3k views

How to make notes when learning a new topic [closed]

At the moment I'm trying to teach myself Riemannian geometry by reading some books. I take notes by hand but I find that I forget it all and worse, either lose the sheets of paper or don't want to ...
12
votes
3answers
522 views

Elementary Geometry Nomenclature: why so bad?

A long-ish wall of text, and I apologize. Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A ...
3
votes
3answers
382 views

For Maths Major, advice for perfect book to learn Algorithms and Date Structures

Purpose: Self-Learning, NOT for course or exam Prerequisite: Done a course in basic data structures and algorithms, but too basic, not many things. Major: Bachelor, Mathematics My Opinion: Prefer ...
12
votes
6answers
1k views

Algebraic Geometry Text Recommendation

I need to learn about Algebraic Geometry (perhaps from in the context of finite fields) and am looking for a recommendation for a text. Now, I've already done a search and checked out what was ...
2
votes
2answers
106 views

Projective representations of loop groups

If $G$ is a Lie group and we take its loop group $LG$ why do we deal with projective representations of $LG$ and central extensions thereof? Where does the extra complexity come in to require us to ...
5
votes
1answer
122 views

2x2 Matrices and Differences of Fractions

Consider the difference of two arbitrary fractions, $\frac{a}{b}$ and $\frac{c}{d}$. $$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$$ The numerator is the determinant of the 2x2 matrix $$ \left( ...
10
votes
4answers
498 views

What are the reasons for not supporting constructive mathematics

It is obvious that in constructive mathematics, you cannot use the law of excluded middle. What else would be the reasons for not adopting constructive stance in mathematics?
4
votes
6answers
910 views

Book suggestion for linear algebra “2”

I am almost finishing Gilbert Strang's book "An introduction to linear algebra" (plus video lectures at MIT OCW). First and foremost, I would like to suggest this course for everyone. It has been ...
5
votes
1answer
649 views

How to select an field of study?

As my question implies, I am a young, developing mathematician, with concerns about my future math career. In particular, I'd like to know how to select a future field of study. Through my courses ...
11
votes
4answers
611 views

What makes elementary row operations “special”?

This is probably a stupid question, but what makes the three magical elementary row operations, as taught in elementary linear algebra courses, special? In other words, in what way are they "natural" ...
3
votes
1answer
547 views

Books like Grundlagen der Analysis in French

I am looking for some recommendations for a mathematics (text)book written in French. I am hoping to learn to read and write mathematics in French since I expect to take some mathematics courses that ...
3
votes
2answers
1k views

History of Quadratic Formula

My wife is planning a lesson on the quadratic formula for high school students, who have previously learned how to complete the square. It would be nice to open the lesson with some historical ...
21
votes
3answers
795 views

Swatting flies with a sledgehammer

Prompted by a recent exchange with Gerry Myerson, I was wondering if anyone has a favorite example of a relatively simple problem with a rather elementary (though perhaps complicated) answer for which ...
23
votes
8answers
3k views

I'm teaching a college geometry course. What should I cover?

I've been asked to teach "Foundations of Geometry" at the University of South Carolina. Apparently, professors in the past have all done very different things, and I have a lot of choice in the ...
12
votes
3answers
805 views

The “need” for cohomology theories

In many surveys or introductions, one can see sentences such as "there was a need for this type of cohomology" or "X succeeded in inventing the cohomology of...". My question is: why is there a need ...
2
votes
3answers
540 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
13
votes
3answers
982 views

Category Theory usage in Algebraic Topology

First my question: How much category theory should someone studying algebraic topology generally know? Motivation: I am taking my first graduate course in algebraic topology next semester, and, ...
2
votes
2answers
514 views

How to get better at algebraic manipulations?

I've always been able to manipulate equations found in school homework easily. But when tackling more challenging questions from puzzle books - where I might need three quarters of a page to ...
5
votes
1answer
518 views

Difficulty of functional analysis exam

I've just written the final exam in my introductory course to functional analysis (2nd year bachelor degree). I felt quite well prepared but nevertheless found the exam pretty challenging in the ...
22
votes
2answers
918 views

Why algebraic closures?

Let me begin by summarizing the question: Why do we care about fields closed under rational exponentiation, and less about fields closed under other operations? Historically the solution for ...
5
votes
1answer
940 views

Is trying to prove a theorem without Axiom of Choice useless?

Suppose there is a well-known theorem whose usual proof uses Axiom of Choice. Is trying to prove it without Axiom of Choice useless? What merits can such a proof have?
12
votes
5answers
678 views

Can the order of learning be changed?

I have been advised by many people to learn from scratch. So I decided to learn it. But I have the following questions in my mind. Can we skip "the Trinity" and learn something else directly? ("the ...
17
votes
7answers
2k views

Using mathematics in theoretical physics

I'm a non-mathematician who is self-studying mathematics. Although I'm very interested in mathematics, my main purpose is to apply math in theoretical physics. The problem is that when I read a ...
2
votes
1answer
1k views

knowledge needed to understand Fermat's last theorem proof

It's a very soft question. So, Wiles proved Fermat's last theorem, and what knowledge of which area would be required to understand the theorem?
6
votes
3answers
2k views

Are male and female body topologically equivalent [closed]

Roughly speaking male and female body are topologically equivalent. But doubts arise when we compare sexual organs systems. For example, internal organs heart or liver of male compared to female ...
2
votes
2answers
97 views

Indexing notation after $i$, $j$ and $k$ have been used?

If I have already used $i,j,k$ as indices, and need two more, where should I look? $l,m$?, $m$? I know from my previous questions that I can use anything I want as long as I define it, but I use $n$ ...
8
votes
4answers
462 views

Should one imagine diagrams/figures when working?

I'm working through Baby Rudin and find it exceedingly difficult to understand what's happening without drawing a small figure. For instance when proving properties of compactness, I would often draw ...
48
votes
6answers
4k views

Why don't analysts do category theory?

I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects. Recently, ...
10
votes
4answers
662 views

Motivation for Topology study in Real Analysis

I'm an engineering student trying to work out some Real Analysis to learn how to write proofs (Needed for my PhD thesis) and just to rekindle my Calculus fires. From what I see, Real Analysis is the ...
6
votes
2answers
318 views

To what extent does the distributive axiom sculpt the face of ring theory?

How much of ring theory arises simply from applying group-theoretic results to the additive structure, or semigroup/monoid-theoretic results to the multiplicative structure? To be more precise, can ...
5
votes
3answers
956 views

How popular and used were logarithm tables?

I've heard that, for a time, logarithm tables "sold more than the Bible". Can someone produce some reliable documentation about how prevalent they were ? Would a common shopkeep have one ? Would a ...
3
votes
2answers
319 views

Documentaries about mathematics and mathematicians [duplicate]

Possible Duplicate: List of Interesting Math Videos/ Documentaries I have watched "Fermat's last theorem" a documentary about Andrew Wiles proof of the theorem, it was a great show. and i ...
9
votes
3answers
3k views

What does a “convention” mean in mathematics?

We all know that $0!=1$, the degree of the zero polynomial equals $-\infty$, the interval$[a,a)=(a,a]=(a,a)=\emptyset$ ... and so on, are conventions in mathematics. So is a convention something that ...
3
votes
0answers
95 views

Fixed marginals of joint distribution: status

One of the well-known problems of classical probability theory is the determination of the set of all extreme points in the convex set of all probability distributions in a product Borel space $\left ...
37
votes
6answers
3k views

How to learn from proofs?

Recently I finished my 4-year undergraduate studies in mathematics. During the four years, I've met all kinds of proofs. Some of them are friendly: they either show you a basic skill in one field or ...
19
votes
5answers
5k views

Graduate school self-doubt (currently an undergraduate)

I don't know if this is the right place for such a question, so please redirect me to a more appropriate place if necessary. My main question is, if you go to a small school, are you doomed to fail ...
2
votes
2answers
239 views

How many significant digits should be retained mid-calculation?

I always feel paranoid when dealing with long series of calculations on paper. How many significant digits should one use to reduce to negligible the probability of getting a wrong final result in a ...
1
vote
0answers
92 views

Historical relation between computer science and the theory of dynamical systems

I wonder if there is any historical relation between the fields of Dynamical systems (and related fields such as Optimal control) and (theoretical) Computer science. The reason for which I ask this ...
50
votes
7answers
2k views

Am I too young to learn more advanced math and get a teacher?

I am still 15 years old, but I am very interested in pure math. I have been teaching myself though books, from the internet and from others for the past year or so. I haven't mastered all the topics ...
3
votes
1answer
108 views

Techniques for speedproving

This is not a math problem per se, but I figured it wouldn't hurt asking for some advice in here, since a large part of the community consists of students. I've recently taken several courses in ...
6
votes
0answers
436 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 ...
2
votes
1answer
61 views

Can ≈ be used to express an arbitrarily rounded rational number?

Is it a formally acceptable use of ≈ to express a rational number rounded to an arbitrary number of significant digits? For example, $\frac{4}{7}\approx0.57.$ If formally acceptable, is it expected? ...
2
votes
2answers
406 views

Why is full- & faithful- functor defined in terms of Set properties?

Wikipedia entry or Roman's "Lattices and Ordered Sets" p.286, or Bergman's General Algebra and Universal Constructions, p.177 and in fact every definition of full and/or faithful functor is defined in ...
2
votes
1answer
185 views

Why should coordinate transformations be reversible?

Intuitively I understand why coordinate transformation should be reversible. New coordinates should cover the same area covered by the initial coordinates, i.e. there should be one-to-one mapping. ...
7
votes
1answer
303 views

Spectrum of a real number

Whilst reading Concrete Mathematics, the authors mention something which they refer to as the "spectrum" of a real number (pg. 77): We define the spectrum of a real number $\alpha$ to be an ...
4
votes
3answers
1k views

Help in self-studying mathematics.

Is this a reasonable list for who seek to learn mathematics by "self learning" program? and is it a well sorted list to follow? http://www.math.niu.edu/~rusin/known-math/index/index.html
44
votes
10answers
2k views

Have there been efforts to introduce non Greek or Latin alphabets into mathematics?

As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic ...
37
votes
7answers
4k views

Open math problems which high school students can understand

I request people to list some moderately and/or very famous open problems which high school students,perhaps with enough contest math background, can understand, classified by categories as on ...
4
votes
2answers
179 views

What things can be defined in terms of universal properties?

We can define some mathematical objects using universal properties, for example the tensor product, the free group over a set or the Stone–Čech compactification. I'm wondering about how to develop my ...