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19 votes
3 answers
2k 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 ...
7 votes
2 answers
767 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 ...
Greg Friedman's user avatar
7 votes
3 answers
1k 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 ...
aglearner's user avatar
  • 14.3k
9 votes
4 answers
10k views

Applications of Euler-Cauchy ODEs

The Euler-Cauchy ODE (2nd order, homogeneous version) is: $$ x^2 y'' + a x y' + b y = 0 $$ Looking in various books on ODEs and a random walk on a web search (i.e. I didn't click on every link, but ...
Andrew Stacey's user avatar
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 ...
Giuseppe's user avatar
  • 193
6 votes
3 answers
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 ...
David Hill's user avatar
  • 1,472
3 votes
2 answers
432 views

A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality: $$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$ I have just ...
Włodzimierz Holsztyński's user avatar
14 votes
5 answers
17k views

Reference letters for graduate school after a couple years in the industry

How does one return to graduate school after spending a couple years in the industry? In particular, what are ways of getting good recommendations? I'm not concerned about the "adjustment" to the grad ...
12 votes
4 answers
5k views

A learning roadmap for Additive combinatorics.

Hello, I'd love to learn more about the field of additive combinatorics. From what I've understand, there's a book by Tao and Vu out on the subject, and it looks fun, but I think I lack the ...
czikszentmihalyi's user avatar
7 votes
3 answers
3k views

The etale fundamental group of a field

Background and motivation: I am teaching the "covering space" section in an introductory algebraic topology course. I thought that, in the last five minutes of my last lecture, I might briefly sketch ...
Charles Staats's user avatar
7 votes
4 answers
2k views

Help me find good math questions for my students [closed]

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, ...
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 ...
Uwe Franz's user avatar
  • 2,201
4 votes
4 answers
4k 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 ...
timur's user avatar
  • 3,322
7 votes
5 answers
2k views

Commutative algebra final project

I'm looking for a topic for a final project in commutative/homological algebra, for first year master's students (in a decent European university). During the course, they will cover the following ...
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 ...
5 votes
0 answers
2k 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 ...
TavukKaghul's user avatar
15 votes
5 answers
3k views

Why is a topology made up of 'open' sets? Part II [closed]

Because the display was getting quite cluttered, I thought I'd post a second part to this question separately. I hope the Gods of Math Overflow don't take too much offense. I'll go now into some ...
3 votes
2 answers
598 views

Math and social commitment [closed]

I am a master's student and am looking for ways that link a certain social commitment with serious math. Since I have not found such an overview yet and in order to raise public awareness of such ...
Leroy's user avatar
  • 129
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 ...
Yuhao Huang's user avatar
  • 5,052
23 votes
4 answers
4k views

Curriculum reform success stories at an "average" research university

Greetings all, There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar ...
3 votes
4 answers
2k views

Less-known conjectures of significant influence and the contrary

In mathematics, it is common that theorems/results and problems appearing dull in one generation get revitalized and become the center of research in another one. Sometimes conjectures that are ...
5 votes
5 answers
2k views

Topics for a matrix analysis course

I recently taught a new (to my department) course titled "Matrix Analysis". For various reasons that I won't go into here, I was dissatisfied with the textbook I (loosely) followed, and with every ...
8 votes
4 answers
2k views

Differential Equation Examples for Calculus Students

I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for ...
21 votes
3 answers
1k views

Do rational numbers admit a categorification which respects the following "duality"?

I need to give a lot of quite basic background to this question because I think (at least from conversing with fellow graduate students) that most mathematicians have not really thought about ...
Steven Gubkin's user avatar
6 votes
7 answers
5k views

Best way to teach concept of real numbers using a hands-on activity?

I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.
mshafrir's user avatar
  • 163
8 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 ...
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 \...
Izaak Meckler's user avatar
16 votes
2 answers
952 views

Where and when did "transition to abstraction" courses start?

I often find myself debating the content and structure of such courses and I would find it useful to know the basic history. I don't remember any such offerings during my own undergraduate days in ...
David Feldman's user avatar
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!
BayesianMonk's user avatar
3 votes
6 answers
2k views

Teach a course in 1 month

I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?
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 ...
7 votes
1 answer
243 views

Five cubes, Hadamard and Shklyarskiy

Here is my(=bad) translation of from the paper about Shklyarskiy by Golovina: ... in 1937/38 Dodik presented to school students a complete proof of Abel's theorem about equations of degree 5. He ...
Anton Petrunin's user avatar
5 votes
7 answers
12k views

Undergraduate approach to learning math [closed]

I am going into my sophomore year as an undergraduate and I would like to ask the more experienced folks a couple questions about learning math and related things. What are your experiences and advice ...
8 votes
6 answers
1k views

Seemingly emergent structures in mathematics

I rather suspect that this must have come up here on MO already, but my handful of searches didn't turn up the thread, so... I'm curious about examples of mathematical structure that seems to arise "...
12 votes
2 answers
2k views

Can formally differentiating give a derivative of a discrete function?

When I teach calculus, I really try to stress the importance of knowing the domain of a function. One example that I sometimes like to use to show students the importance of inspecting the domain is ...
Steven Gubkin's user avatar
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,...
B. Bischof's user avatar
  • 4,842
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 ...
M.G.'s user avatar
  • 7,127
2 votes
4 answers
4k views

Best way to introduce the Chinese Remainder Theorem (to a high school student)

What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and ...
Maurizio Monge's user avatar
0 votes
1 answer
2k views

Everyday, real-life applications of mathematical concepts, and human intuition vs mathematical analysis [closed]

I'm working on an educational project about the applications of reasonably 'lofty', high-ish-level mathematical concepts in the real world. I've already scoured these links (1) (2) (3) after ...
Krister Janmore's user avatar
6 votes
0 answers
622 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 ...
asv's user avatar
  • 21.8k
8 votes
3 answers
2k views

The harmonic (series) beetle: live illustrations of mathematical theorems

In my analysis class I use the following problem to illustrate the divergence of the harmonic series (consider this as a hint for solving it). Exercise. A beetle creeps along a 1-meter infinitely ...
13 votes
1 answer
1k views

Classroom platonism

I'd like to know whether any form a certain hypothesis about the learning of higher mathematics has entered the mathematical or educational literature. I'll frame the hypothesis here but not defend ...
David Feldman's user avatar
0 votes
5 answers
2k views

How to teach addition of negative numbers? [closed]

I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are). She ...
Tommi's user avatar
  • 648
8 votes
3 answers
9k views

Applications of Group Theory Which Motivate Theoretical Questions?

I'm going to be a teaching assistant for an undergraduate class in abstract algebra next semester, for students who have not taken abstract algebra before. It will deal with group theory and linear ...
24 votes
2 answers
2k 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, ...
David E Speyer's user avatar
13 votes
4 answers
1k views

Simple groups with the same cardinality as PSL_2(Z/p)

In an undergrad honors algebra course it's sometimes shown that when $p$ is prime and $>3$ then $PSL_2(Z/p)$ is simple of of order $p(p-1)(p+1)/2$. But that this is the "only" simple group having ...
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. ...
8 votes
4 answers
2k views

Choice of adviser

Not sure how to tag this one so feel free to edit and add tags. When I initially started graduate school my choice for an area of study was quite nebulous. I had only figured out enough to know that ...
8 votes
4 answers
4k views

How to teach introductory statistic course to students with little math background?

Next semester I will teach an elementary statistic course for the first time (which I am actually quite excited about). A brief description can be found here. I am told to expect very little math ...
8 votes
4 answers
1k 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 ...
Joe Johnson's user avatar

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