**15**

votes

**4**answers

2k views

### Languages beyond enumerable

A language is a set of finite-length strings from some finite alphabet $\Sigma$.
It is no loss of generality (for my purposes) to take $\Sigma=\{0,1\}$; so a language is a set of bit-strings.
...

**43**

votes

**5**answers

3k views

### How do you mentor undergraduate research?

Lets say you had an undergraduate who wanted to do some advanced work and some research, possibly for a thesis, or things like that.
There are two slightly more specific groups of questions I have ...

**1**

vote

**0**answers

40 views

### Endpoints as relative extrema (differences in definition in a freshman calculus course)

I'm currently teaching an applied calculus course for freshmen. The textbook I'm using (Lial, Greenwell, and Ritchey's "Calculus with Applications") defines relative maxima and minima in the usual way ...

**35**

votes

**13**answers

3k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...

**9**

votes

**11**answers

1k 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 ...

**22**

votes

**10**answers

1k views

### What (fun) results in graph theory should undergraduates learn?

I have the task of creating a 3rd year undergraduate course in graph theory (in the UK). Essentially the students will have seen minimal discrete math/combinatorics before this course. Since graph ...

**21**

votes

**4**answers

906 views

### Technical issue in the approach to Lie groups taken in a book

I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...

**10**

votes

**1**answer

458 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 ...

**7**

votes

**0**answers

248 views

### Pedagogical question on Lie groups vs. matrix Lie groups

There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...

**42**

votes

**3**answers

3k views

### Advice for PhD Supervisors

My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...

**4**

votes

**1**answer

249 views

### How to teach generalizing the induction hypothesis? [closed]

I just finished teaching a class on using proof assistants (in this case, Agda) to write provably correct programs. Reflecting on how it went, the biggest difficulty I noticed the students having was ...

**27**

votes

**12**answers

2k 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 ...

**52**

votes

**2**answers

1k views

### History of $\frac d{dt}\tan^{-1}(t)=\frac 1{1+t^2}$

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught:
1º that
$$
\frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2},
\tag1
$$
2º that, via the fundamental theorem of calculus, this is ...

**11**

votes

**3**answers

479 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 ...

**7**

votes

**2**answers

2k views

### Which universities teach true infinitesimal calculus? [closed]

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...

**14**

votes

**2**answers

1k views

### Teaching stochastic calculus to students who know no measure theory (or PDE, or…)

I've got quite a challenge as my teaching assignment for the next Fall (not that I want to get rid of it, quite the contrary, but I still feel like asking for advice won't hurt :-)).
I'm to teach the ...

**12**

votes

**1**answer

472 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 $\mathbb{P}^3$ ...

**6**

votes

**0**answers

351 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 ...

**18**

votes

**19**answers

7k views

### Math books for advanced high school students

I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way ...

**20**

votes

**2**answers

1k views

### Teaching the fundamental group via everyday examples

This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...

**6**

votes

**0**answers

295 views

### Lower semicontinuity of naive fiber size

I would like to present the following result in my algebraic geometry class, but it is seeming much harder than I would expect. Since my class is working with closed points over an algebraically ...

**1**

vote

**2**answers

384 views

### Simple yet interesting applications of Calculus or Linear Algebra to Economics [closed]

This is essentially a vast generalization of my previous question: Examples of separable ordinary differential equations in economics
I'm giving a talk to college-level math teachers on some ...

**21**

votes

**7**answers

2k views

### Conceptual algebraic proof that Grassmannian is closed in Plucker embedding

I'm planning lectures for my intro algebraic geometry course, and I noted something awkward that is coming up. We're starting projective varieties soon. Of course, we'll prove that projective maps are ...

**6**

votes

**1**answer

1k views

### Can one live without actual infinity? [closed]

The title of this question is the exact title of one of the sections of a book written by Alexandre Borovik: Mathematics under the Microscope. Under the title, we read:
How should we approach the ...

**5**

votes

**1**answer

294 views

### Where can I find resources for creating a mathematics “bridge course”?

My department is in the very early stages of developing a "bridge course" or "introduction to proofs" course, motivated by our lower-level courses not currently doing a good job of preparing our ...

**6**

votes

**3**answers

940 views

### An application of Maschke's theorem

I've been teaching some elementary representation theory to undergraduates, and want to provide applications of Maschke's theorem to complex group algebras to present in class. In particular, I'd like ...

**3**

votes

**0**answers

78 views

### Are injective modules flabby on basic open sets?

In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds:
Statement: If $A$ is a commutative ring and ...

**3**

votes

**4**answers

532 views

### Understanding reasons for best constants in inequalities

Why, in functional analysis, is so important to calculate best constant in an embedding inequality?
Cross-posted from ...

**0**

votes

**1**answer

418 views

### Teaching profession:Differential Equations and Mean Value Theorems

Usually I teach Algebra,Algebra and Geometyry, Topology, at various University levels. This semester (Spring 2014) I have to teach Differential Equations to University second year students (4th ...

**3**

votes

**3**answers

342 views

### undergraduate handle decomposition. Reference

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.

**15**

votes

**2**answers

977 views

### How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...

**7**

votes

**5**answers

634 views

### Easy to state applications of dimension theory in algebraic geometry

Dimension theory is quite a sophisticated topic (at least for me), it is fully settled in Shafarevich's book on the first 100 pages.
Shafarevich gives two nice applications of the theory. 1) A proof ...

**3**

votes

**3**answers

1k 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 ...

**5**

votes

**1**answer

314 views

### Not quite adjoint functors

What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...

**50**

votes

**15**answers

4k views

### Teaching homology via everyday examples

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory?
To be more precise, I am teaching a short course on homology, from ...

**58**

votes

**10**answers

12k views

### Is Euclid dead? [closed]

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" ...

**3**

votes

**2**answers

553 views

### Can this informal argument (for the fact that almost all reals in the unit interval are irrational) be saved?

In the textbook from which I am teaching a Discrete Math course, the authors propose randomly generating an infinite sequence of decimal digits $d_1, d_2, \dots$. We are to think of this as the ...

**27**

votes

**6**answers

2k views

### Does seeing beyond the course you teach matter? The case of linear algebra and matrices

This question is indeed very important for me. Thus I hope you bear with my subjective explanations for a few minutes. I am an "excellent" lecturer, at least according to course evaluation forms ...

**4**

votes

**4**answers

800 views

### Variation on the Sobolev space $H^1_0$

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let
$$
C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\},
$$
and let $C^1_c(\Omega)$ be the space of ...

**1**

vote

**0**answers

611 views

### Linear Algebra Text Book

In our department we do not like our current linear algebra book and so we would want to find a better book. This is for the first course in linear algebra and the title of the course is
Elementary ...

**7**

votes

**2**answers

925 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 ...

**10**

votes

**3**answers

608 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.
...

**25**

votes

**3**answers

2k views

### Is “problem solving” a subject to be taught?

I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...

**2**

votes

**4**answers

523 views

### Eigenvalues of powers of linear mappings

Let $\tau$ be a linear map on a finite dimensional complex vector space. Clearly, if $\lambda$ is an eigenvalue of $\tau$ then $\lambda^n$ is an eigenvalue of $\tau^n$, for any natural (integer, on ...

**24**

votes

**3**answers

2k views

### Nearly all math classes are lecture+problem set based; this seems particularly true at the graduate level. What are some concrete examples of techniques other than the “standard math class” used at the *Graduate* level?

In the fall, I am teaching one undergraduate and one graduate course, and in planning these courses I have been thinking about alternatives to the "standard math class". I have found it much easier ...

**22**

votes

**7**answers

2k views

### Pros and cons of math teaching using smartboards

Currently, there is some talk in my university concerning a change in our lecture rooms from blackboards to smartboards (or other alternatives, such as a smart podium). For that reason, I'm interested ...

**24**

votes

**2**answers

1k views

### Can the unsolvability of quintics be seen in the geometry of the icosahedron?

Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...

**5**

votes

**2**answers

1k views

### What is the best *general triangle*?

During courses on geometry it is sometimes necessary to draw a triangle on the blackboard that can easily be recognized as a general triangle. It must not be rectangular and must not have two or more ...

**0**

votes

**1**answer

479 views

### Sierpinski Triangle and the Chaos Game

The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...

**4**

votes

**4**answers

649 views

### Lecture on Fractals for Middle School Students

I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject.
I'll show some fractal images and a few ...