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For questions related to teaching mathematics. For questions in Mathematics Education as a scientific discipline there is also the tag mathematics-education. Note you may also ask your question on http://matheducators.stackexchange.com/.

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2answers
460 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 ...
-1
votes
1answer
74 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!
13
votes
3answers
2k views

Teaching Prime Number Theorem in a Complex Analysis Class for Physicists

This is a question about pedagogy. I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis ...
8
votes
2answers
265 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-...
61
votes
16answers
5k views

Short papers for undergraduate course on reading scholarly math

(I know this is perhaps only tangentially related to mathematics research, but I'm hoping it is worthy of consideration as a community wiki question.) Today, I was reminded of the existence of this ...
31
votes
2answers
786 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 ...
5
votes
2answers
660 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 ...
16
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1answer
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 ...
16
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2answers
593 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. ...
10
votes
3answers
305 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 ...
6
votes
3answers
419 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 ...
33
votes
4answers
1k 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 ...
219
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29answers
80k 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 ...
7
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0answers
404 views

How necessary is the knowledge of Lebesgue integral for non-analysts? [closed]

Recently I have learned that at some math department the introductory course to Lebesgue integration not obligatory. Thus in another course on introduction to Hilbert spaces the $L^2(0,1)$ space is ...
7
votes
2answers
444 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....
6
votes
2answers
261 views

Applications of isotropic quadratic forms

I will soon be teaching an introductory course on bilinear algebra and quadratic forms. I will likely spend most of the time and effort on positive definite quadratic forms and euclidean spaces. These ...
13
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3answers
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+\...
2
votes
1answer
245 views

Defining negation

I'm currently coauthoring a book intended to teach first-year students basic proof techniques. One of the chapters, written by my coauthor, is about basic logic. In that chapter the negation of a ...
6
votes
1answer
424 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 \...
5
votes
0answers
656 views

A course on modern algebraic geometry from “The Stacks Project”

I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't. For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...
41
votes
16answers
8k views

Why do we need random variables?

In this MathStackExchange post the question in the title was asked without much outcome, I feel. Edit: As Douglas Zare kindly observes, there is one more answer in MathStackExchange now. I am not ...
-1
votes
1answer
704 views

Why isn't mathematics education treated like software development? [closed]

When the assumptions of the Curry-Howard correspondence hold, then mathematical proofs literally are computer programs in a rigorous sense. However, even when that is not the case, it still seems ...
1
vote
1answer
188 views

proof without words for logarithms [closed]

Does anyone know of any PROOF WITHOUT WORDS for logarithmic functions? The only one I've seen in calculus based and I need one for high school math kids in MATH 1,2,3. Any suggestions would be ...
17
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4answers
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. ...
42
votes
5answers
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 ...
40
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14answers
7k 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 ...
13
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12answers
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....
27
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10answers
2k 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 ...
22
votes
4answers
1k 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 ...
11
votes
2answers
870 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 ...
8
votes
0answers
289 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 ...
52
votes
4answers
4k 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 ...
5
votes
1answer
302 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 ...
28
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12answers
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 ...
76
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2answers
2k 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
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3answers
557 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 $\...
9
votes
2answers
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 ...
16
votes
2answers
2k 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 ...
13
votes
1answer
512 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 $\mathbb{P}^3$ ...
9
votes
0answers
763 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 ...
23
votes
19answers
13k 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 ...
22
votes
2answers
2k 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 ...
7
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0answers
393 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
2answers
623 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 ...
23
votes
7answers
3k 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
2answers
414 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
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3answers
1k 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 ...
4
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0answers
99 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
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4answers
729 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 "https://math.stackexchange.com/questions/727690/understanding-reasons-for-best-...
0
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1answer
467 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 ...