All Questions
495 questions
51
votes
22
answers
19k
views
Why linear algebra is fun!(or ?)
Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.
I'm doing an introductory talk on linear algebra with the ...
- 878
7
votes
1
answer
372
views
Theory of surfaces in $\mathbb{R}^3$ as level sets
Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
- 4,713
1
vote
1
answer
378
views
Why is $n_{n^2-1}$ the smallest graph that clearly shows the structure of multiplication by $n$?
Initially, I wanted to ask this question as a puzzle.
Consider a regular $m$-gon. Let $0$ be the lower corner and count the corners clockwise.
Let $n_m$ be the multiplication-by-$n$-graph of $...
- 10.9k
114
votes
1
answer
10k
views
What happened to Suren Arakelov? [closed]
I heard that Professor Suren Arakelov got mental disorder and ceased research. However, a brief search on the Russian wikipedia page showed he was placed in a psychiatric hospital because of political ...
- 6,259
33
votes
15
answers
3k
views
Historical (personal) examples of teaching-based research
The phrase "teaching-based research" brings to mind research about teaching, though important, it is not what I mean. Unfortunately, I couldn't come up with a better phrase, thus please bear with me ...
- 4,786
6
votes
2
answers
1k
views
Pages from a known textbook on Euclidean geometry?
Do you recall having seen the attached pages in a textbook once? If so, would you be so kind as to share its bibliographic record (or the main items in it) with me below?
A teacher provided us xerox ...
- 6,700
60
votes
8
answers
13k
views
Why the Killing form?
I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
- 23.4k
69
votes
20
answers
19k
views
Fun applications of representations of finite groups
Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
- 13.6k
29
votes
2
answers
2k
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Why did Dedekind claim that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ hadn't been proved before?
In a letter to Lipschitz (1876) Dedekind doubts that $\sqrt{2}\cdot\sqrt{3}=\sqrt{6}$ had been proved before:
quoted from Leo Corry, Modern algebra, German original:
Why did Dedekind doubt that $(\...
CommunityBot
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0
votes
1
answer
114
views
Name of a matrix with one column and row removed [closed]
I am looking for the exact name of a matrix where the i-th column and rows have been removed.
I cannot remember how it is called in linear algebra, does anyone got an idea?
Thanks!
- 20.2k
16
votes
7
answers
2k
views
Uppercase Point Labels in High-School Diagrams: from Euclid?
I wonder if the convention of labeling points in geometric
diagrams with uppercase symbols ultimately derives from
Greek mathematics, which was originally written in
"majuscule" (uppercase) Greek ...
- 2,185
81
votes
22
answers
15k
views
Are there proofs that you feel you did not "understand" for a long time?
Perhaps the "proofs" of ABC conjecture or newly released weak version of twin prime conjecture or alike readily come to your mind. These are not the proofs I am looking for. Indeed my question was ...
- 6,039
7
votes
2
answers
1k
views
How should you respond to a student who asks whether a very nice physical example constitutes a proof? [closed]
"Is this really a proof?" is the exact question e-mailed to me today from an undergraduate mathematics student whom I know as a highly competent student. The one sentence question was accompanied with ...
- 6,039
9
votes
7
answers
7k
views
Review papers in mathematics
Are there review papers, literature reviews in mathematics that describe the recent developments in various fields for a newcomer? Or is the prerequisite knowledge always provided in research ...
- 558
24
votes
15
answers
5k
views
Applications of connectedness
In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea of ...
CommunityBot
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15
votes
7
answers
6k
views
Freshman's definition of sin(x)?
I would like to know how you would rigorously introduce the trigonometric functions ($\sin(x)$ and relatives) to first year calculus students. Suppose they have a reasonable definition of $\mathbb{R}$ ...
- 4,713
8
votes
2
answers
447
views
Big ideas and big ways of thinking in statistics?
I'm moving to a new university for the fall semester, and I'll be teaching a statistics class for the first time. I'm familiar enough with doing statistics (my dissertation in math ed was a mixed-...
- 6,237
17
votes
17
answers
3k
views
Readings for an honors liberal art math course
Our university has an Honors section of our "liberal arts mathematics" course. Typically 10-20 students enroll each Fall, with most of them extremely bright, but lacking the interest and/or ...
CommunityBot
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14
votes
3
answers
1k
views
Where can I read reviews of mathematical theories? [closed]
I'm really enjoying the AMS column "What is ..." (http://arminstraub.com/math/what-is-column) and The Princeton Companion to Mathematics.
I am looking for something similar. I'd like to acquire some ...
- 82.7k
9
votes
1
answer
635
views
De-Nesting Absolute Value Function into Linear Combination of Absolute Value Functions
Context: In formulating problems for secondary school mathematics teachers (and students) about absolute value functions, which we define as functions $\mathbb{R} \rightarrow \mathbb{R}$ that send $x \...
CommunityBot
- 1
263
votes
29
answers
89k
views
Mathematical games interesting to both you and a 5+-year-old child
Background: My daughter is 6 years old now, once I wanted to think on some math (about some Young diagrams), but she wanted to play with me...
How to make both of us to do what they want ? I guess ...
CommunityBot
- 1
43
votes
9
answers
29k
views
Applications of knot theory
An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.
I regularly teach a knot theory class. ...
- 23
18
votes
12
answers
10k
views
Theorems in Euclidean geometry with attractive proofs using more advanced methods
The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The ...
- 2,185
33
votes
11
answers
13k
views
Lecture notes on representations of finite groups
Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck "...
- 111
6
votes
0
answers
167
views
Is there Cauchy-Goursat for $1$-cycles without invoking winding numbers?
Depending on one's approach to Complex Analysis in One Variable, Cauchy's Integral Theorem is one of the first interesting results about holomorphic functions in any course. There are several related ...
- 7,127
14
votes
4
answers
5k
views
Which edition of Philosophiae Naturalis Principia Mathematica of Isaac Newton would you recommend to me?
I'm searching for a good edition of Philosophiae Naturalis Principia Mathematica of Isaac Newton in English. Which edition of the Principia can you suggest me? If it's possible, cheap and similar to ...
- 2,821
46
votes
15
answers
11k
views
Strong induction without a base case
Strong induction proves a sequence of statements $P(0)$, $P(1)$, $\ldots$ by proving the implication
"If $P(m)$ is true for all nonnegative integers $m$ less than $n$, then $P(n)$ is true."
for ...
- 8,842
8
votes
2
answers
2k
views
Examples of analytic functions to motivate a first course in complex variables
[Changed title as a plea to re-open the question.]
If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an ...
- 485
35
votes
2
answers
2k
views
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
16
votes
2
answers
1k
views
Teaching Steenrod Operations
I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. ...
- 3,439
20
votes
4
answers
2k
views
Problems for developing mathematical visualization expertise
Einstein stated that he often explored and reasoned visually and spatially, and only after achieving understanding cast his insights into algebraic form. He could just "see" the answer. There are ...
22
votes
13
answers
8k
views
Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
- 1,228
30
votes
15
answers
5k
views
Making sure that you have comprehended a concept
I have a question that I've been thinking about for a long time.
How can you assure yourself that you've fully comprehended a concept or the true meaning of a theorem in mathematics?
I mean how can ...
- 15.8k
19
votes
10
answers
6k
views
Research Experience for Undergraduates: Summer Programs
Some time ago, I found this list of REU programs held in 2009.
The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; and, admittedly more importantly,...
CommunityBot
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24
votes
11
answers
8k
views
The role of the mean value theorem (MVT) in first-year calculus
Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
CommunityBot
- 1
9
votes
5
answers
3k
views
Assessing effectiveness of (epsilon, delta) definitions [closed]
There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in calculus and the student reception of them. The ...
- 16.6k
16
votes
1
answer
2k
views
A conjecture in which both "if" and "only if" are near misses
[Migrated from Math Stack Exchange]
More than a year ago, I posted the following on the Math Stack Exchange.
Consider $2^n-1$. Based on checking a few small numbers for $n$ (in
fact, the first ...
- 32.5k
13
votes
2
answers
2k
views
teaching higher algebra
Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The kind of ...
CommunityBot
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7
votes
3
answers
3k
views
Problems reducing to a graph-theory algorithm
This is essentially a question in pedagogy -- the answers could be useful to teach (or rather, motivate) graph theory, and especially the algorithmic side of it.
I have been very impressed with this ...
CommunityBot
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11
votes
3
answers
448
views
Easy proof that reflections generate $N(T)/T$ for connected compact group?
I'm teaching a course on Coxeter groups and I'd like to provide an overview of the connection to compact Lie groups. Let $G$ be a compact connected Lie group, $T$ a maximal torus and $N(T)$ the ...
- 878
39
votes
4
answers
2k
views
Important open exposition problems?
Timothy Chow, in his article A beginner's guide to forcing, defines an open exposition problem as a certain concept or topic in mathematics that has yet to be explained "in a way that renders it ...
CommunityBot
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67
votes
9
answers
7k
views
Taking "Zooming in on a point of a graph" seriously
In calculus classes it is sometimes said that the tangent line to a curve at a point is the line that we get by "zooming in" on that point with an infinitely powerful microscope. This explanation ...
- 151k
7
votes
5
answers
6k
views
Advantages of the sequence definition of limits
I will be teaching an introductory analysis course in the coming semester. In it the students will learn about limits of real sequences, and then will learn about limits of functions in terms of ...
- 16.6k
15
votes
13
answers
23k
views
Math journal for high school students?
I recently discovered The College Mathematics Journal and enjoyed reading through some of the articles on fun applications of mathematics. I'd like to send some of the articles to my younger sister, a ...
- 1,500
14
votes
2
answers
5k
views
A certain mathematical competition in the UK
There is a foreword, written by professor Snow, to the book A mathematician's apology.
In the foreword, it is written some thing like the following:
"Hardy was opposed to a certain mathematical ...
CommunityBot
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21
votes
7
answers
3k
views
What should be taught in a 1st course on Riemann Surfaces?
I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
CommunityBot
- 1
37
votes
18
answers
5k
views
Insightful books about elementary mathematics
What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic non-abstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way?...
- 1,181
0
votes
3
answers
1k
views
How to be a Great mathematician in prison/without a master? [closed]
Is it possible to be a great mathematician in our home with a laptop+poor internet+electronic books+some books+a little food +a little money or not? without having a constant job
without studying P.H....
- 131
2
votes
3
answers
274
views
learning sources about Ihara Coefficient
Do we have any good sources(lecture notes or books) for learning about $Ihara$ Coefficient?
Is there any relation between $Ihara$ Coefficient and the eigenvalues of graphs?
Thanks for any help.
- 457
6
votes
1
answer
462
views
How to talk about certain "free" categories?
Given two categories $\mathcal{C}$ and $\mathcal{D}$, we can describe the following category $\mathcal{E}$. It is the initial category whose object set contains $\mathrm{Obj}(\mathcal{C}) \times \...
CommunityBot
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