All Questions
367 questions
114
votes
34
answers
86k
views
Why do we teach calculus students the derivative as a limit?
I'm not teaching calculus right now, but I talk to someone who does, and the question that came up is why emphasize the $h \to 0$ definition of a derivative to calculus students?
Something a teacher ...
97
votes
19
answers
38k
views
Collecting proofs that finite multiplicative subgroups of fields are cyclic
I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
69
votes
20
answers
19k
views
Fun applications of representations of finite groups
Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
58
votes
4
answers
5k
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 ...
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
52
votes
22
answers
19k
views
Interesting Calculus Questions/Exercises
I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...
45
votes
10
answers
4k
views
effective teaching
Eric Mazur has a wonderful video describing how physics is taught at many universities and his description applies word for word to the way I learned mathematics and the way it is still being taught, ...
42
votes
4
answers
3k
views
What is the Krull dimension of the ring of holomorphic functions on a complex manifold?
Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$
My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known?
...
- 47.5k
42
votes
16
answers
5k
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 ...
40
votes
4
answers
4k
views
Polynomials on the Unit Circle
I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,…,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...
- 3,626
39
votes
3
answers
6k
views
On linear independence of exponentials
Problem.
Let $\{\lambda_n\}_{n\in\mathbb N}$ be a sequence of complex numbers . Let's call a family of exponential functions $\{\exp (\lambda_n s)\}_{n\in\mathbb N}$ $F$-independent (where $F$ is ...
- 22.3k
38
votes
2
answers
2k
views
Residues in several complex variables
I am trying to educate myself about the basics of the theory of residues in several complex variables. As is usually written in the introduction in the textbooks on the topic, the situation is much ...
- 1,190
37
votes
1
answer
3k
views
Circles and rational functions
Suppose that $\gamma$ is a Jordan analytic curve on the Riemann sphere,
and there exist two rational functions $f$ and $g$ such that
$f$ maps $\gamma$ into a circle, and $g$ maps a circle into $\gamma$...
- 91.8k
36
votes
6
answers
2k
views
When are some products of gamma functions algebraic numbers?
I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...
- 14k
36
votes
2
answers
3k
views
Computing self-intersections with complex analysis
It is possible to find the winding number of a path $C \subset \mathbb{C}$ using complex analysis:
$$n = \oint_C\frac{dz}{z}.$$
You can also count the number of roots of $f(z) = 0$ inside a close ...
- 22.8k
35
votes
7
answers
6k
views
Heuristic argument for the Riemann Hypothesis
Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence? Moreover, what is the strongest theorem that supports the validity of ...
- 3,699
34
votes
13
answers
6k
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 ...
33
votes
15
answers
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 ...
33
votes
1
answer
2k
views
Stone-Weierstrass theorem for holomorphic functions?
The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called
Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth ...
- 7,426
33
votes
11
answers
13k
views
Lecture notes on representations of finite groups
Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck "...
32
votes
20
answers
6k
views
What are your favorite puzzles/toys for introducing new mathematical concepts to students?
We all know that the Rubik's Cube provides a nice concrete introduction to group theory. I'm wondering what other similar gadgets are out there that you've found useful for introducing new math to ...
32
votes
1
answer
1k
views
About a claim by Gromov on proper holomorphic maps
At p. 223 of his paper [G03], Mikhail Gromov makes the following claim:
Let $X$, $Y$ be two complex manifolds (not necessarily compact or Kähler) of the same dimension and having the same even Betti ...
- 66.3k
31
votes
3
answers
2k
views
Polynomials with the same values set on the unit circle
Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
- 109k
30
votes
2
answers
942
views
Why is there no connection between fast-growing functions and complex analysis
I found myself wondering the other day whether the fast-growing functions from natural to naturals that are studied by people like proof theorists are the restriction to the naturals of analytic ...
- 1,625
29
votes
1
answer
4k
views
Zeros of polynomials with real positive coefficients
The following problem arose in collaborative work with Subhro Ghosh:
Question: To any polynomial $P_n(z)=\sum_{i=0}^n a_i z^i =a_n \prod_{i=1}^n(z-z_i)$, attach the empirical measure of zeros
$L_n=n^{...
- 7,559
29
votes
2
answers
2k
views
Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,179
29
votes
3
answers
2k
views
Rational functions with a common iterate
Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are
at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$,
...
- 91.8k
28
votes
2
answers
2k
views
A 14th and 26th-power Dedekind eta function identity?
Given the Dedekind eta function $\eta(\tau)$. Define $m = (p-1)/2$ and a $24$th root of unity $\zeta = e^{2\pi i/24}$.
Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$:
$$\sum_{k=0}^{p-...
- 12.6k
28
votes
2
answers
2k
views
A sum involving roots of unity
Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that
\begin{align*}
\sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}.
\end{align*}
Since $\...
- 2,183
28
votes
5
answers
5k
views
Why are lacunary series so badly behaved?
Hi!
I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...
- 2,358
28
votes
4
answers
3k
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 ...
- 679
27
votes
3
answers
3k
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 ...
27
votes
2
answers
3k
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 ...
27
votes
4
answers
3k
views
Genealogy of the Lagrange inversion theorem
A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...
- 60.6k
27
votes
5
answers
6k
views
The Matrix-Tree Theorem without the matrix
I'm teaching an introductory graph theory course in the Fall, which I'm excited about because it gives me the chance to improve my understanding of graphs (my work is in topology). A highlight for me ...
- 22.1k
27
votes
5
answers
4k
views
What is the naming reason of poles in complex analysis?
A function $f: \textbf{C} \to \textbf{C}$ has a pole of order $k$ if $f(z) = \frac{g(z)}{(z-z_0)^{k}}$ where $g(z)$ is a nonzero analytic function. Why do we call it poles?
- 279
25
votes
1
answer
783
views
Decidability of equality of elementary expressions
In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...
- 6,819
25
votes
19
answers
20k
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 ...
25
votes
5
answers
6k
views
When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
- 923
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
24
votes
9
answers
9k
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 epsilon-...
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, ...
- 156k
24
votes
12
answers
4k
views
2D problems which are easier to solve in 3D
It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...
23
votes
4
answers
5k
views
Is $\ x\! \cdot\!\tan(x)\ $ integrable in elementary functions?
I'm teaching Calculus and my students asked me to calculate the integral of $\ x\! \cdot\!\tan(x)$.
I spent quite a lot of effort to do this, but I'm now even not sure if the integral could be ...
- 1,437
22
votes
1
answer
1k
views
Hadamard factorization of L-functions
I have already asked this question here in a different form, but really need an answer.
Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc...
(Selberg ...
- 13.1k
22
votes
2
answers
3k
views
Reason for studying coherent sheaves on complex manifolds.
Hello everybody! I would be interested in knowing, what the reason is for investigating coherent sheaves on complex manifolds. By definition a sheaf $F$ on a complex manifold $X$ is coherent, when it ...
- 583
22
votes
2
answers
2k
views
Anything special (historical?) about surface $x\cdot y\cdot z\ +\ x+y+z=0$?
QUESTION
I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex logarithm is ...
- 7,500
22
votes
6
answers
2k
views
Elementary solutions to f(z+1)-f(z)=g(z) in entire functions
Let g(z) be an entire function of a complex variable z. Does there exist an entire function f(z) such that f(z+1)-f(z)=g(z)? As I learned several years back, the answer to this is apparently 'yes', ...
- 7,836
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
21
votes
5
answers
7k
views
References for complex analytic geometry?
I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc....