All Questions
3,560 questions
24
votes
7
answers
8k
views
How do professional mathematicians learn new things? [closed]
How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues?
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
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 ...
24
votes
11
answers
8k
views
The role of the mean value theorem (MVT) in first-year calculus
Should the mean value theorem be taught in first-year calculus?
Most calculus textbooks present the MVT just before the section that says that if $f'>0$ on an interval then $f$ increases on that ...
24
votes
3
answers
3k
views
How to find a conformal map of the unit disk on a given simply-connected domain
By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
- 42k
24
votes
3
answers
9k
views
Functions of several complex variables: book recommendations?
Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...
24
votes
7
answers
4k
views
Why are two notions of Gaussian curvature are the same - what is the simplest & most didactic proof?
This question is still wide open - all of the answers so far rely on magical calculations. I've only accepted an answer because, by bounty rules, otherwise one would be accepted automatically. I can't ...
- 3,203
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
1
answer
4k
views
analytic vs. algebraic Gauss-Manin connection
There are the following two notions of "Gauss-Manin connection":
The complex-analytic one: let $f:X\to S$ be a smooth family of complex manifolds. Then we obtain a local system $R^nf_{\ast}\...
- 1,298
23
votes
13
answers
7k
views
Pedagogical question about linear algebra
Last semester I taught a linear algebra class that is intended to introduce young students (at a sophmore-junior level) to "abstract mathematics". It seems that a major conceptual hurdle for many of ...
23
votes
5
answers
11k
views
Example of continuous function that is analytic on the interior but cannot be analytically continued?
I am looking for an example of a function $f$ that is 1) continuous on the closed unit disk, 2) analytic in the interior and 3) cannot be extended analytically to any larger set. A concrete example ...
- 757
23
votes
12
answers
15k
views
Textbook for undergraduate course in geometry
I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (...
23
votes
4
answers
2k
views
Lower bounding $|1+z+\cdots + z^{n-1}|$ when $z\approx 1$
I am trying to lower bound $|1+z+\cdots + z^{n-1}|$ when $z$ is a complex number close to $1$ (and $n$ is sufficiently large). My main concern occurs in the case $z = 1 + it$, where $t$ is small. In ...
- 347
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
23
votes
1
answer
5k
views
On equation $f(z+1)-f(z)=f'(z)$
Original Problem
If $f$ is an entire function such that
$$ f(z+1)-f(z)=f'(z) $$
for all $z$.
Is there a non-trivial solution? ($f(z)=az+b$ is trivial)
And here is something uncertainty
If we use ...
- 1,551
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 ...
23
votes
2
answers
975
views
Does Rademacher's convergent series for p(n) define an analytic function?
Let $p(n)$ be the number of partitions of $n\geq 0$. We can let $n$ be
any complex number in Rademacher's convergent infinite series for
$p(n)$. (See e.g. equation (24) here.)
For what $n$ does it ...
- 50.8k
23
votes
1
answer
5k
views
Analogue of the Chebyshev polynomials over C?
I've been driven up a wall by the following question: let p be a complex polynomial of degree d. Suppose that |p(z)|≤1 for all z such that |z|=1 and |z-1|≥δ (for some small δ>0). Then what's the ...
- 9,833
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 ...
23
votes
2
answers
2k
views
Which smooth compactly supported functions are convolutions?
If $f,g$ are smooth functions with support in the interval $[-r,r]$ for some $r>0$, then their convolution $f*g$ is smooth with support in $[-2r,2r]$. My question is about the converse: Given ...
- 1,017
23
votes
2
answers
1k
views
Theta functions on an elliptic curve and Serre duality
Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ ...
- 411
23
votes
1
answer
3k
views
More mysteries about the zeros of the Riemann zeta function
Update on 12/26/2020: I added the Appendix at the bottom: simplified formula for $|\zeta(s)|^2$, when $\frac{1}{2}<\Re(s)<1$.
Update on 1/5/2020: I added the section "more interesting ...
- 3,098
23
votes
0
answers
720
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essentially different"?
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize ...
- 118k
23
votes
0
answers
1k
views
Is analytic capacity inner regular?
For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by
$$\gamma(K) := \sup |f'(\infty)|$$
where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...
- 2,154
22
votes
3
answers
3k
views
Cardioid-looking curve, does it have a name?
The curve, given in polar coordinates as $r(\theta)=\sin(\theta)/\theta$
is plotted below.
This is similar to the classical cardioid, but it is not the same curve (the curve above is not even ...
- 15.8k
22
votes
13
answers
8k
views
Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
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
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
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
22
votes
1
answer
3k
views
Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?
Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
- 393
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
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
4
answers
5k
views
What is the best way explain to undergraduates that all 1-dimensional manifolds are orientable?
Let's suppose that $M$ is a connected $1$-dimensional smooth manifold (Haussdorf and paracompact). We know that there are exactly two types, up to diffeomorphism (even up to homeomorphism), namely $\...
- 3,540
22
votes
4
answers
2k
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 ...
- 28.1k
22
votes
1
answer
3k
views
Why is Oka's coherence theorem a deep result?
This is a very naive question.
Let $X$ be a complex manifold. Let $\mathcal{O}_X$ be the structure sheaf of $X$, a sheaf of rings whose sections over opens $U\subset X$ are just the holomorphic ...
- 10.7k
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
21
votes
10
answers
6k
views
Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
21
votes
7
answers
3k
views
What should be taught in a 1st course on Riemann Surfaces?
I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
- 3,284
21
votes
4
answers
3k
views
Is the Euler product formula always divergent for 0<Re(s)<1?
It is known that the Euler product formula converges for $\Re(s)>1$
(and there it represents the Riemann zeta function).
My question: Is the Euler product always divergent for
$0 < \Re(s) < ...
- 255
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 1,889
21
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 ...
21
votes
6
answers
1k
views
What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?
Background
Taking a relatively arbitrary combination of exponential and polynomial terms, for instance
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
- 1,730
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
21
votes
5
answers
7k
views
References for complex analytic geometry?
I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc....
21
votes
4
answers
2k
views
Holomorphic vector fields acting on Dolbeault cohomology
The question.
Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) ...
- 6,247
21
votes
0
answers
2k
views
Cartan–Oka vanishing in one variable without $\overline{\partial}$?
This is a literature question, about possible proofs of some very basic results in complex analysis.
Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
- 21.3k
20
votes
2
answers
9k
views
Does module Hom commute with tensor product in the second variable?
Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...
- 1,906
20
votes
4
answers
6k
views
Conformal maps in higher dimensions
In dimension 2 we know by the Riemann mapping theorem that any simply connected domain ( $\neq \mathbb{R}^{2}$) can be mapped bijectively to the unit disk with a function that preserves angles between ...
- 757
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 ...
20
votes
2
answers
2k
views
Bitcoin Research
I have recently been assigned to advise a student on a senior thesis. She has taken linear algebra, introductory real analysis, and abstract algebra. Her interest is in cryptography. And she has a ...
- 822