All Questions
262 questions
17
votes
6
answers
7k
views
Explaining the concept of projective space: notes for students
This is a question on teaching.
I am teaching at this moment a course in algebraic geometry for master students on a very basic level. Today (this was the fourth lecture) I discovered that only four ...
16
votes
12
answers
10k
views
How seriously should a graduate student take teaching evaluations? [closed]
Pretty much the question in the title. If a grad student gets bad reviews as a TA, how much does that hurt them later? How much do good reviews help? What if the situation is more complex? (For ...
16
votes
9
answers
4k
views
How to motivate the skein relations?
I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these ...
- 30.5k
16
votes
5
answers
3k
views
Integrating powers without much calculus
I'll jump into the question and then back off into qualifications and context
Using the definition of a definite integral as the limit of Riemann sums, what is the best way (or the very good ways) to ...
- 30.1k
16
votes
6
answers
3k
views
How to mentor an exceptional high school student?
I have a unique and, quite truthfully, humbling opportunity. The parents of an exceptionally talented high school freshman have reached out to me and asked if I might be able to help.
This kid is ...
16
votes
5
answers
1k
views
Permission to use Online Notes
I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by ...
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 ...
- 2,437
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. ...
- 161
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
16
votes
1
answer
977
views
Pedagogically intuitive reformulation of Zorn's Lemma for functional analysis
While teaching an applied functional analysis class, I’ve noticed that students often struggle to develop an intuitive understanding of Zorn’s lemma. It’s relatively straightforward to explain why ...
- 5,824
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 $\mathbb{R}$ ...
- 23.3k
15
votes
4
answers
3k
views
How does one motivates the method of separation of variables when teaching PDE's?
I'm not sure if this question is appropriate for MO. Add comments if it is not. Thanks.
How to explain/motivate the method of separation of variables for PDEs to undergraduates? What's the real math ...
- 5,052
15
votes
5
answers
2k
views
"Classical" consequences of Bezout's theorem in dimensions $>2$
By Classical I mean something that could have been found before 1900 (say).
A well known consequence of Bezout's theorem for plane curves is Pascal's theorem http://en.wikipedia.org/wiki/Pascal'...
- 14.3k
15
votes
1
answer
757
views
Teaching cohomology via everyday examples
This question is a "sequel" to my similar questions about the fundamental group and homology. All of these questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics ...
14
votes
1
answer
3k
views
An elementary proof that the degree of a map of spheres determines its homotopy type
I'm helping to teach an undergraduate algebraic topology course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...
- 7,338
14
votes
3
answers
3k
views
Open source LaTeX lecture notes/slides/books [closed]
In the mathematics community it's quite common for professors to write their own notes for the classes they are teaching. The notes are then usually published in both PDF and PS form on the course ...
14
votes
2
answers
7k
views
What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
14
votes
9
answers
2k
views
math circles video lectures for school children?
Hello,
I am from India. I find the mathoverflow amazing.
I have a question: Are there any good quality video lectures on school math topics?
There are a lot of high quality lectures available on ...
14
votes
1
answer
961
views
Founding of homological without quite involving derived categories
I am looking at the foundations of homological algebra, e.g. the introduction
of Ext and Tor, and am unsatisfied. The references I look at start with
"this is called a projective module, this is ...
- 27.9k
13
votes
7
answers
35k
views
Real analysis has no applications?
I'm teaching an undergrad course in real analysis this Fall and we are using the text "Real Mathematical Analysis" by Charles Pugh. On the back it states that real analysis involves no "applications ...
13
votes
3
answers
1k
views
Teaching polarisation formula
When teaching about Hilbert spaces, one begins with a polarisation formula, which allows us to reconstruct the scalar product from the norm:
$$\langle u,v\rangle=\frac14(\|u+v\|^2-\|u-v\|^2+\imath\|u+\...
- 52.3k
13
votes
17
answers
3k
views
Short Course Suggestions For High School Students
I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...
13
votes
5
answers
2k
views
How to make a lecture series useful
I have been to a number of advanced lecture courses (of between 3 and 10 lectures) over the years, given (in principle) by experts to graduate students and experts in neighbouring fields. Examples of ...
13
votes
4
answers
1k
views
The ten most fundamental topics in geometric group theory
What are the ten most fundamental topics in geometric group theory?
This is a pedagogical question prompted by the fact that I am teaching geometric group theory to undergraduates. They are expected ...
13
votes
3
answers
2k
views
History surrounding Gauss Theorema Egregium and differential geometry
I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Gauss Theorema Egregium, that is the Gaussian ...
- 193
13
votes
1
answer
605
views
A funny factorization of the Jacobian coming from the lines on the Fermat cubic
Here is something which came up in my algebraic geometry class, and I'm wondering if it has a deeper explanation. Let $F(w,x,y,z) = w^3+x^3+y^3+z^3$ and let $X$ be the cubic surface in $\mathbb{P}^3$ ...
- 156k
13
votes
1
answer
2k
views
conditional equality symbol
Is there a standard notation (perhaps $A \stackrel{\leftarrow}{=} B$) meaning "in all situations where $B$ is defined, $A$ is defined and equals $B$"?
The kind of situation in which such a notation ...
- 19.7k
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 ...
- 534
12
votes
12
answers
2k
views
What are fun elementary subjects in probability?
I have to read several lectures on probability or applications of probability for high school students (of high level). There is no necessary part I must lecture, that is, my aim is just advertisement....
12
votes
11
answers
2k
views
Giving a math talk with no blackboard or projector
I need to give a math talk to a group of undergraduates. I am asking for advice because this talk will take place at a department picnic and there will be no blackboard or projector. I would like to ...
12
votes
9
answers
6k
views
Topics for an Undergraduate Expository Paper in Number Theory
I am teaching an undergraduate course in number theory and am looking for topics that students could take on to write an expository paper (~10 pages). No new results are expected of them. Many of the ...
12
votes
1
answer
521
views
Source of a quote by Ferdinand Rudio
I am looking for the source and context of this quote, found e.g. at St Andrews:
Only with the greatest difficulty is one able to follow the writings of any author preceding Euler, because it was ...
- 31.5k
12
votes
4
answers
929
views
Interesting examples of systems of linear differential equations with constant coefficients
In this paper, Gian-Carlo Rota wrote:
A lot of interesting systems with constant coefficients have been discovered in the last thirty years: in control, in economics, in signal
processing, even in ...
12
votes
5
answers
2k
views
Introducing Cryptology to Undergraduates
This summer I am going to give some lectures to some REU students. I am still tossing around ideas for what I am going to talk about, but one thing I would at least like to give one or two lectures on,...
- 4,842
12
votes
1
answer
775
views
Teaching Methods and Evaluating them
Hey,
As a lowly graduate student, I'm on a committee (I'm not sure how important my role really is) trying to evaluate how effective different approaches teaching undergraduates. We are looking at ...
11
votes
5
answers
4k
views
Applications of Liouville's theorem
I'm looking for "nice" applications of Liouville's theorem (every bounded entire map is constant) outside the area of complex analysis.
An example of what I'm not looking for : a non-constant entire ...
11
votes
6
answers
2k
views
Hard problems with an easy-to-understand answer
I am very interested by problem in mathematics which are difficult (go at least 10 years without a resolution, say) but which have a solution that is short and elementary.
In this video Launay gave an ...
11
votes
4
answers
3k
views
Topological examples of profinite groups
I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first ...
- 113
11
votes
3
answers
729
views
Why does inconstructibility of $\sqrt[3]{2}$ imply impossibility of cube doubling? [closed]
In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.
On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\...
- 802
11
votes
2
answers
3k
views
Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
- 2,201
11
votes
3
answers
729
views
Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?
I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover.
...
11
votes
1
answer
1k
views
Teaching Experience for Graduate Students. [closed]
I am currently a graduate student, who will (hopefully!) graduate in the next year (or two..). I have slowly come to realize that I enjoy teaching, and consequently want to do more of it! My main ...
11
votes
2
answers
1k
views
Social Reading Platform for Math or LaTeX texts
Social reading is considered to be one of the big trends that could be catalysing learning by reading. Features could include:
Highlighting or annotating paragraphs or single steps in a proof for ...
11
votes
3
answers
447
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 ...
- 156k
11
votes
1
answer
1k
views
Teaching stacks to differential geometry students
Does anyone have any experience teaching stacks over the category of manifolds to students whose background is, say, a semester-long course on manifolds? Does anyone know of any publicly available ...
- 1,710
11
votes
1
answer
2k
views
Good chalk in the UK
Sometime ago it was asked in Mathoverflow about good chalk in the US Where to buy premium white chalk in the U.S., like they have at RIMS?. I will be grateful for any recommendations on good chalk in ...
11
votes
0
answers
2k
views
Total spaces of tangent/cotangent bundles in a course where all varieties are quasi-projective
$\def\PP{\mathbb{P}}$In a course where all varieties are quasi-projective (as in Shafarevich Volume I), I am trying to figure out whether I can justify talking about the total spaces of the tangent ...
- 156k
10
votes
7
answers
2k
views
Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
10
votes
8
answers
2k
views
Undergraduate Probability Topics
I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or ...
10
votes
3
answers
1k
views
About the classification of commutative and of cocommutative, fin. dim. Hopf algebras
I want to prove that the cocommutative finite dimensional Hopf algebras over an algebraically closed field of characteristic zero are group algebras (for some finite group) and that the commutative f....
- 7,746