All Questions
262 questions
13
votes
4
answers
2k
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 ...
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
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
3
votes
2
answers
141
views
Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
I am currently wondering whether it is realistically possible to choose the topic "Fractals and fractal dimensions" for a seminar aimed at undergraduate students in the 2nd semester, with ...
- 1,942
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 ...
4
votes
1
answer
183
views
Notation for weak derivatives
I remember that, as a student, I felt a bit uncomfortable because I had to use the same notation (say $f'$, $D^\alpha f$, $\frac{\partial f}{\partial x^j}$, $\nabla \cdot f$ etc...) for classical and ...
- 4,459
3
votes
0
answers
167
views
Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
4
votes
2
answers
287
views
Teaching suggestions for Kleene fixed point theorem
I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
- 41
2
votes
1
answer
628
views
Does some published texbook take a particular approach (described here) to the transition from discrete to continuous probability distributions?
(I posted this question at matheducators.stackexchange.com and it seems to be considered an inappropriate question for that site. I don't understand why.)
Imagine an introductory probability course ...
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 ...
2
votes
1
answer
295
views
Examples of new results found via exams [closed]
I suspect that there have been many instances throughout history where a new proof of an existing result has been discovered by a student while taking an exam. Does anyone have an example of this?
53
votes
7
answers
8k
views
Zorn's lemma: old friend or historical relic?
It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
- 18.7k
7
votes
1
answer
723
views
Alternate algorithms for Chinese remainder theorem
I was teaching Discrete this semester and set the students loose on a system of linear congruences. One of them came up with this solution. Say $$ x \equiv 1 \textrm{ mod } 3 $$ $$ x \equiv 3 \textrm{ ...
- 525
1
vote
1
answer
117
views
Resources on blended teaching and flipped classroom in undergraduate mathematics education [closed]
I'd like to learn about the implementation of "blended teaching" in general and "flipped classroom" in particular for the teaching of undergraduate mathematics. Can anyone ...
- 141
25
votes
2
answers
3k
views
What is the origin/history of the following very short definition of the Lebesgue integral?
Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
- 32.5k
1
vote
0
answers
190
views
what belongs in a first university-level geometry course? [closed]
I know this is not really a research question, but I would like to ask it of research mathematicians, to see if there is a consensus. In a recent discussion on this topic, someone suggested that if ...
- 99
8
votes
2
answers
1k
views
Mathematics of sustainable development and energy sobriety in the classroom
Faculty members are encouraged to highlight the connection between the courses we teach and climate change, and raise awareness of the issue in our lectures, across subjects in my university. I am ...
0
votes
0
answers
303
views
Is Baire's theorem stronger than needed for functional analysis?
Many classic theorems in functional analysis involve using Baire's theorem to prove facts about topology that relate to maps between Banach spaces (or, more generally, F-spaces). The application ...
- 109
22
votes
2
answers
2k
views
Can one deduce the fundamental theorem of algebra from real calculus and linear algebra?
Motivation: let $A\in\mathbf{R}^{n\times n}$ be symmetric. Then by the method of Lagrange multipliers, a maximum of $x\mapsto x^tAx$ on the compact unit sphere $\mathbf{S}^{n-1}$ must be an ...
- 337
0
votes
0
answers
148
views
About the theorem of Weierstrass?
Is $E=Vect\{1,x,x^2,...,x^{2^n},...\}$ dense in $C([0,1])$ for the uniform norm?
While looking for a short proof for Weierstrass' theorem, I came across this justification(*) (which shows this result)...
- 4,074
4
votes
1
answer
1k
views
Chalkboard eraser [closed]
I just started my first year of university and because I'm visually impared I have trouble seeing what's written on the chalkboard.
I've partially solved this problem by purchasing chalk from hagoromo ...
48
votes
8
answers
5k
views
Ideas for introducing Galois theory to advanced high school students
Briefly, I was wondering if someone can suggest an angle for introducing the gist of Galois groups of polynomials to (advanced) high school students who are already familiar with polynomials (...
22
votes
1
answer
3k
views
What is so special about Chern's way of teaching?
First of all sorry for this non-research post.
I was watching Jeffrey Blitz Lucky documentary movie and it was interesting to me that a winner of Lottery was a math Ph.D. from Berkeley.
In the movie ...
- 4,195
5
votes
0
answers
186
views
Examples of partial adjoints
Recall that a functor $$R: D \to C$$ is said to have a partial left adjoint $L$ defined at an object $X \in C$ if the functor
$$D \to Sets, Y \mapsto Hom_C(X, R(Y))$$
is corepresentable by some object ...
- 2,040
25
votes
6
answers
3k
views
What is the standard 2-generating set of the symmetric group good for?
I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
- 4,492
5
votes
1
answer
208
views
Seven Bridges of Königsberg for hypergraphs
I am teaching a course involving hypergraphs. I would like to have a physical analogy/motivating problem for hypergraphs similarly to how the Seven Bridges of Königsberg motivate graphs. Can you help ...
- 51
3
votes
0
answers
873
views
Hard problems solving tricks
This question is motivated by this one that I posted on math.stackexchange.
When I fail to solve a hard math problem (like the ones I presented in the linked post), I read a solution and I noticed ...
- 161
44
votes
10
answers
11k
views
What kid-friendly math riddles are too often spoiled for mathematicians?
Some math riddles tend to be spoiled for mathematicians before they get a chance to solve them. Three examples:
What is $1+2+\cdots+100$?
Is it possible to tile a mutilated chess board with dominoes?...
-4
votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 43.2k
20
votes
4
answers
2k
views
PDF readers for presenting Math online
In the current situation it seems especially important to be able to present your mathematical results online in a way that your audience does not fall asleep in front of their screens. But I am ...
23
votes
14
answers
4k
views
Math talk for all ages
I've been asked to give a talk to the winners of a recent math competition. The talk can be entirely congratulatory, or it can contain a bit of actual mathematics. I'd prefer the latter. I'd also ...
9
votes
3
answers
1k
views
Books on the relationship between the Socratic method and mathematics?
Apart from books on heuristics by George Polya.
When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real ...
- 107
0
votes
1
answer
125
views
Are there search algorithms that are competitive against (gradient based) optimization routines for continuous problems?
Suppose that $f: \mathbb{R}^n \to \mathbb{R}$ is a continuous function for which we want to minimize. We may arbitrarily impose good conditions for $f$, such as Lipschitzness, smoothness, convexity, ...
- 479
8
votes
2
answers
693
views
Seeking a combinatorial proof for a binomial identity
Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means:
$$\sum_{j=0}^m(-1)^j\binom{n+1}j2^{m-j}
=\sum_{j=0}^m(-1)^j\binom{n-m+j}j.$$
QUESTION. Can you provide a ...
- 43.2k
1
vote
0
answers
134
views
What benefits of math can be conveyed to mid/high schoolers? [closed]
I'm teaching mathematical proof writing to a few of math teachers (in the US) this summer. In the beginning of class, I send a survey asking them why they are here. Most of them are here for getting ...
- 5,230
36
votes
3
answers
3k
views
What do we learn from the Wronskian in the theory of linear ODEs?
For a real interval $I$ and a continuous function $A: I \to \mathbb{R}^{d\times d}$, let $(x_1, \dots, x_d)$ denote a basis of the solution space of the non-autonomous ODE
$$
\dot x(t) = A(t) x(t) \...
- 12.6k
6
votes
0
answers
283
views
Interesting things you learned while grading/marking? [closed]
What are some interesting mathematical things you have learned while grading (or marking, if you prefer) student work? For example, clever proofs that students came up with; nice counterexamples or ...
5
votes
1
answer
521
views
How to find eigenvalues following Axler?
Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix ...
- 16.5k
3
votes
1
answer
271
views
Elementary classification of division rings
Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
- 26.5k
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
9
votes
0
answers
887
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
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 ...
2
votes
1
answer
359
views
Defining integrals by residue theorem
I have always been interested in alternative definitions of mathematical objects. I wonder if one can craft an useful definition of definite integral by using the Residue Theorem from complex analysis....
- 341
8
votes
4
answers
788
views
Different derivations of the value of $\prod_{0\leq j<k<n}(\eta^k-\eta^j)$
Let $\eta=e^{\frac{2\pi i}n}$, an $n$-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ...
- 43.2k
7
votes
1
answer
372
views
Theory of surfaces in $\mathbb{R}^3$ as level sets
Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
- 7,197
6
votes
2
answers
1k
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 ...
- 8,842
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!
- 111
17
votes
5
answers
3k
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 ...
- 253
8
votes
2
answers
447
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-...
93
votes
20
answers
10k
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 ...