**0**

votes

**2**answers

441 views

### Correct definition of the sequence of natural numbers with set theory, but without counting or measuring size [closed]

This question may appear banal, but there seems to be more than meets the eye; a common glitch is to explain numbers by the "size" of sets without saying how to measure or compare the size of sets.
...

**5**

votes

**3**answers

598 views

### Higher dimensional Bezout via Hilbert polynomials: a reference

For the purposes of teaching my elementary course in algebraic geometry I am looking for a reference (or notes) that contains a complete proof of a higher-dimensional weak Bezout theorem. I only want ...

**16**

votes

**3**answers

1k views

### Research level applications of “row rank = column rank”?

No less an authority than Gilbert Strang frames "row rank equals column rank" (and a couple of other facts) as "The Fundamental Theorem of Linear Algebra."
I'd simply like to assemble (for teaching ...

**10**

votes

**4**answers

1k 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 ...

**9**

votes

**2**answers

1k 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 ...

**25**

votes

**13**answers

3k views

### Elementary applications of linear algebra over finite fields

I'm teaching axiomatic linear algebra again this semester. Although the textbooks I'm using do everything over the real or complex numbers, for various reasons I prefer to work over an arbitrary ...

**10**

votes

**1**answer

712 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 ...

**0**

votes

**2**answers

281 views

### Lines on degree 2n-3 Fermat hypersufaces

It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their ...

**10**

votes

**4**answers

1k 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) ...

**12**

votes

**2**answers

1k 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 ...

**2**

votes

**3**answers

682 views

### Continuous change of basis (and on the definition of determinant) [closed]

Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...

**11**

votes

**11**answers

4k views

### Textbook for undergraduate course in geometry

I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry ...

**5**

votes

**1**answer

2k views

### Examples of separable ordinary differential equations in economics

I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...

**4**

votes

**2**answers

461 views

### Surface Laplace-Beltrami without coordinates, exterior calculus?

Let $f: M \rightarrow \mathbb{R}^3$ be an immersion of a surface $M$. For pedagogical purposes (i.e., I'm teaching a class!) I am looking for an expression for the scalar Laplace-Beltrami operator ...

**14**

votes

**2**answers

2k views

### The function $\sum_{0}^{\infty} x^n/n^n$

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...

**16**

votes

**4**answers

3k views

### Is $x \, \tan(x)$ integrable in elementary functions?

I'm teaching Calculus and my students asked me to calculate the integral of $x \, \tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be presented in ...

**71**

votes

**20**answers

9k views

### “Mathematics talk” for five year olds

I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels
very tough to prepare. Could the ...

**9**

votes

**9**answers

3k 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 ...

**5**

votes

**4**answers

812 views

### Multivariable Calculus Lecture Ideas

I am teaching a course in multivariable calculus this semester. We are covering the basics about $\mathbb{R}^n$, including dot products and cross products, curves, and quadric surfaces. After that ...

**13**

votes

**2**answers

1k views

### Bitcoin Research

I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...

**3**

votes

**2**answers

348 views

### Integration in several variables and elementary applications

This fall I'm teaching the "second half" of the standard entry-level undergraduate multivariable calculus course: the focus is on double and triple integrals, path integrals, Green's theorem, Stokes' ...

**7**

votes

**5**answers

3k 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 ...

**3**

votes

**4**answers

826 views

### Another Chicken or Egg: Sequence or Series

This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations to "interesting" ...

**31**

votes

**16**answers

3k views

### Justifying/Explaining math research in a public address

I have been chosen by my university to give a 1 hour public research lecture. Every year a researcher is chosen for this honour. Traditionally people explain their own research about designing ...

**2**

votes

**3**answers

357 views

### Pedagogical notes on line bundles on complex projective manifolds

I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems ...

**12**

votes

**10**answers

3k views

### Not especially famous, long-open problems which higher mathematics beginners can understand

This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...

**8**

votes

**5**answers

1k 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 ...

**221**

votes

**99**answers

36k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**11**

votes

**3**answers

2k 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 ...

**10**

votes

**3**answers

1k 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 ...

**11**

votes

**9**answers

1k 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

**12**answers

3k views

### Excellent uses of induction and recursion

Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...

**4**

votes

**1**answer

927 views

### Is Diagonalization worth to be taught? [closed]

When students come to the College (first two years of the University system in most of the developped countries) to train in mathematics, they get a linear algebra / matrix analysis course. After a ...

**10**

votes

**1**answer

919 views

### Teaching Experience for Graduate Students.

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 ...

**19**

votes

**2**answers

837 views

### Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$

Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...

**14**

votes

**6**answers

2k 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 ...

**15**

votes

**17**answers

2k 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 ...

**18**

votes

**9**answers

5k views

### How to motivate and present epsilon-delta proofs to undergraduates?

This would seem to be a common question, but I am surprised not to see it already asked and answered on MO!
I am teaching an undergraduate course, and I want to teach them to construct basic ...

**2**

votes

**2**answers

458 views

### Decomposition of $K_{10}$ in copies of the Petersen graph

It is a well-known and cute exercise in algebraic graph theory to show that $K_{10}$ cannot be written as the edge-disjoint union of three copies of the Petersen graph $P$. Indeed, the graph $G$ whose ...

**10**

votes

**6**answers

3k 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 ...

**7**

votes

**4**answers

2k views

### Help me find good math questions for my students

I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad.
Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...

**15**

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 ...

**9**

votes

**4**answers

2k 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 ...

**1**

vote

**0**answers

392 views

### Professional skills advising for math jobs [closed]

Hi,
I am a postdoc at the University of Nottingham (UK) and I am beginning to apply for Assistant Professor positions in US.
I would like to receive a feedback on the material that I am sending ...

**4**

votes

**2**answers

1k views

### Any suggestions for a course in Mathematical Logic?

I am teaching a topics course for Mathematics majors (at Temple), and am considering Logic as the topic. I was wondering if people (a) have suggestions for an appropriate text and (b) how much might ...

**13**

votes

**17**answers

2k 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 ...

**17**

votes

**8**answers

10k views

### Interesting Applications of the Classical Stokes Theorem?

When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...

**37**

votes

**5**answers

9k views

### Are there any “related rates” calculus problems that don't feel contrived?

I just finished teaching a freshman calculus course (at an American state university), and one standard topic in the curriculum is related rates. I taught my students to answer questions such as the ...

**14**

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 ...

**22**

votes

**17**answers

5k views

### Using slides in math classroom

I am toying with the idea of using slides (Beamer package) in a third year math course I will teach next semester. As this would be my first attempt at this, I would like to gather ideas about the ...