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40
votes
2answers
2k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
44
votes
5answers
3k views

How to write a good mathematical paper?

I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar. A good time always goes too fast! Two years are fled. In the third year of ...
39
votes
6answers
1k views

Has any previously unknown result been proven by an automated theorem prover?

The Wikipedia page on automated theorem proving states: Despite these theoretical limits, in practice, theorem provers can solve many hard problems... However it is not clear whether these 'hard ...
10
votes
2answers
603 views

Varieties as schemes

Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne. Firstly, my main question. I understood that Grothendiecks introduction of schemes ...
3
votes
1answer
157 views

Reference request for examples of probabilistic heuristics, help put some examples in a broader context.

I was thinking about how probability is used in heuristic arguments, an example being the argument that there are an infinite number of twin primes: the probability that $n$ is the first of two twin ...
256
votes
23answers
30k views

Proofs that every mathematician should know?

There are mathematical proofs that have that "wow" factor in being elegant, simplifying one's view of mathematics, lifting one's perception into the light of knowledge etc. So I'd like to know what ...
11
votes
3answers
7k views

I want to start mathematics from scratch. What should I begin with?

I've just finished high school but I don't feel my knowledge of mathematics is good enough. I'd like to start again from scratch, possibly adding a bit (a lot) of problem solving to it. What is the ...
3
votes
1answer
518 views

What does the continuum hypothesis imply?

Are there any fundamental/interesting results that are a consequence of assuming the continuum hypothesis as an additional axiom? I'm sorry if this question was already asked. I'm also sorry if there ...
12
votes
7answers
4k views

Why we need to know how to solve a quadratic?

Five years ago I was tutoring orphans in a local hospital. One of them asked me the following question when I tried to ask him to solve a quadratic: Why do I need how to solve a quadratic? I am ...
10
votes
2answers
9k views

Tips for an adult to learn math — from the beginning.

First let me start with I am an adult and I can't do simple maths. I some how got through all of my math courses in University (after several attempts) but I honestly couldn't tell you how... I ...
6
votes
1answer
179 views

What is a good communication platform, which supports something like TeX-code?

I moved away and am now planning to communicate with a friend over the internet and talk about mathematical subjects (e.g. via Skype). However the text blocks like "\int_G\omega..." in a chat window ...
2
votes
3answers
286 views

How to say the notation $ x_{i,j}^{ k}$

$ x_{i,j}^{ k}$ May I say the following to say the above notation? x sub i j to k ? x sub i j sup(prounced like "soup") k ? (Can I say "soup" for "superscript"? Could you please show me a ...
27
votes
4answers
2k views

Why is the Daniell integral not so popular?

The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
27
votes
3answers
1k views

Good ways to help people learn math.

I have a student position at my university, which involves helping people who are struggling with basic calculus, precalculus, and discrete math. In general, the students that I help are hostile ...
9
votes
5answers
1k views

The definition of metric space,topological space

I have read some books in analysis,all of them define metric space,topological space or vector space directly,without any reason. Therefore, I want to know the background of the definition, the ...
5
votes
3answers
134 views

How do I get insight into equations with the $\sum$ notation without actually expanding it for a specific n every time?

I am reading Goldstein's Classical Mechanics and I've noticed there is copious use of the $\sum$ notation. He even writes the chain rule as a sum! I am having a real hard time following his arguments ...
1
vote
0answers
157 views

Intuition for PDE Change of Variables

The algebraic manipulations for changing variables in PDE/ODE problems are often very simple once you know the transformation to use (at least at my level it's just applying the chain rule carefully). ...
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
499 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
914 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
654 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
549 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
2k 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
556 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
988 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
516 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
519 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
921 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
1k 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
464 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 ...
49
votes
6answers
3k 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
663 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
319 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
320 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 ...