All Questions
3,560 questions
1
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278
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Localization in analytic geometry
Let $X$ be a Stein complex analytic space, and let $Z$ be a closed complex analytic subspace. Set $U=X-Z$.
I was wandering if there is any relationship between $A_1:=\mathcal{O}_X(U)$
and the ...
- 23.4k
7
votes
3
answers
1k
views
What information do the roots of the generating function of the nontrivial zeroes of the Riemann zeta function encode.
Let $a_{m}$ be the imaginary part of the nontrivial roots of the Riemann zeta function $\zeta(s)$. Suppose we have their generating function $u(x)=\sum_{m=1}^{\infty} a_{m}x^{m}=14.134725\ldots{}x^{1}+...
- 975
7
votes
2
answers
414
views
Can curves induced by analytic maps wiggle infinitely across a line?
Let $f$ be a function analytic on an open subset $D\subset \mathbb{C}$, and let $\gamma:[0,1] \to D$ be a line segment. $g = f\circ\gamma$ is another curve in the complex plane; is it possible to for $...
- 2,019
7
votes
3
answers
3k
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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 ...
- 7,338
3
votes
1
answer
1k
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Amazing examples in complex Algebraic Geometry
Good example teaches sometimes more than couple of theorems. I wonder what are your favourite examples in complex algebraic geometry, the ones that were astonishing for you, the simpler (at least ...
- 161
42
votes
13
answers
20k
views
How to draw knots with LaTeX?
I am writing an exam for my students, and the topic is intro knots theory. I have no idea how to put knots into the file, but I know many MO users who can draw amazing diagrams in their papers.
Can ...
- 30.6k
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
12
votes
1
answer
1k
views
How to best distribute points on two concentric circles?
An N-subset $\{x_1,\dots,x_N\}$ of a compact set $X\subset \mathbb R^d$ is called a set of Fekete points (named after Michael Fekete) if it maximizes the product $$\prod_{1\le k<j\le N}|x_k-x_j|\...
2
votes
0
answers
354
views
What is this effect in Fourier/additive synthesis called?
Hi, I have re-synthesized a cyclic function additively, and I added a fixed offset to the frequency of each partial. So if the function was $\sum a_{n} sin(2 \pi x * n)$ and its frequencies were $n*f_{...
- 165
6
votes
2
answers
2k
views
Does there exist a holomorphic function which takes given values on the positive integers?
Inspired of course by What's a natural candidate for an analytic function that interpolates the tower function?
I am minded to ask what looks to me like a more natural question: given a sequence $...
- 41.4k
9
votes
1
answer
629
views
conformally embedding complex tori into R^3
Let $L$ be a lattice in $\mathbb{C}$ with two fundamental periods, so that $\mathbb{C}/L$ is topologically a torus. Let $p:\mathbb{C}/L \mapsto \mathbb{R}^3$ be an embedding ($C^1$, say). Call $p$ ...
- 1,961
16
votes
9
answers
4k
views
How to motivate the skein relations?
I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these ...
- 30.6k
6
votes
0
answers
490
views
Lacunar series with an interesting (in-formula) symmetry.
So, I wrote out a table of functions like so:
$\sum_{n=1}^{\infty} (-1)^{n+1}q^{n}=$ $+q^{1}$ $-q^{2}$ $+q^{3}$ $-q^{4}$ $+q^{5}$ + $\ldots$
$\sum_{n=1}^{\infty} (-1)^{n}q^{n^{2}}=$ $-q^{1}$ $+q^{4}...
8
votes
3
answers
2k
views
what is the formal definition of multi-valued holomorphic function?
It seems that there exists ring structure on all multi-valued holomorphic functions on a punctured disc.
Can someone explain the formal definition of multi-valued holomorphic function?
I only know ...
- 1,457
5
votes
3
answers
3k
views
Branched coverings of Riemann surfaces with specified branch points.
Today I showed, using some ad hoc algebraic topology, that if $\Sigma$ is a Riemann surface and $\mathfrak{p} \subset \Sigma$ is a finite set of points, then there is another Riemann surface $S$ and a ...
6
votes
1
answer
5k
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How would You encourage graduate students to learn algebraic geometry and/or complex analysis? [closed]
Hello,
I am the 3rd year undegraduate student of mathematics.
After I obtain a bachelor degree I want to study maths at graduate level, especially algebraic geometry and complex analysis.
This fields ...
- 1,042
16
votes
3
answers
3k
views
Infinite projective space
Is infinite (say complex) projective space a scheme? More generally, can schemes have infinite cardinal dimension? It seems that infinite dimensional projective space is not a manifold, since it is ...
user2529
6
votes
3
answers
678
views
Approximately holomorphic functions
In real analysis one can define something known as the approximative derivative of a function. See here eg Roughly speaking one asks that the limit of the difference quotient exists as long as h goes ...
- 757
4
votes
2
answers
2k
views
Upper half plane quotient by a discrete group
I was reading Mehta and Seshadri's paper "Moduli of vector bundles on curves with parabolic structures".
In the second paragraph, they wrote:
"Suppose that $H$ mod $\Gamma$ has finite measure ($H$ ...
4
votes
1
answer
313
views
Maximally symmetric smooth projective varieties in CP^2
Let P(X,Y,Z) be a homogeneous polynomial in ℂ[X,Y,Z] whose locus M in ℂℙ2 is a nonsingular curve of genus ≥ 2.
Define M to be maximally symmetric if the following is not true:
...
- 2,858
1
vote
0
answers
1k
views
Bessel function in polar coordinates
I want to write the Bessel function of the first kind in polar coordinates
$J_\alpha(z)=|J_\alpha(z)|e^{i\varphi_\alpha(z)}$
Is anything known about $\varphi_\alpha(z)$?
In particular, I'm ...
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
2
votes
1
answer
475
views
Finding Functional form for a given Scaling Condition
Dear all
While studying the overlap distribution for two random Cantor sets (long story made short), I came across the following problem.
$G(k)$ is a complex valued function, and satisfy the ...
- 317
40
votes
16
answers
11k
views
"Homotopy-first" courses in algebraic topology
A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...
6
votes
2
answers
433
views
Triangles, squares, and discontinuous complex functions
Is there some onto function $f:$ $\mathbb{C}$ $\rightarrow$ $\mathbb{C}$
such that for each triangle $T$ (with its interior), $f(T)$ is a
square (with interior, too) ?
I would have the same question ...
- 63
1
vote
1
answer
338
views
Power series for meromorphic differentials on compact Riemann surfaces
Suppose I have a compact Riemann surface of $g>1$ given by the quotient $H/\Gamma$ where I do know $\Gamma$ explicit. Is there a way to write down the power series of meromorphic functions, ...
- 6,825
2
votes
1
answer
897
views
Text/structure for an analysis course for students with pre-existing understanding of some applied aspects of analysis
Greetings,
I'm teaching a one-off course (perhaps never to be repeated) in a curriculum that's in transition, and I'm looking for advice on a textbook, or stories from people who have taught similar ...
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 ...
2
votes
1
answer
2k
views
Definition of a complex structure on a vector bundle
Given a Riemann surface $S$, e.g. $\mathbb{P}^1(\mathbb{C})$, with complex conjugation on the coordinates and a holomorphic vector bundle $E$ over $S$.
The complex conjugation $f$ is not holomorphic, ...
- 1,391
2
votes
1
answer
251
views
Help determining the asymptotic behavior of an integral involving rational functions.
Let $\phi:\mathbb{P}^1\to\mathbb{P}^1$ be a rational function of degree $d\geq2$. How can one prove, using the normalized spherical measure, that
$$\int_{\mathbb{P}^1(\mathbb{C})}|(\phi^n)'(z)|\ d\mu (...
user4324
0
votes
1
answer
1k
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How to prove that rational functions satisfy a Lipschitz condition in the *chordal metric*?
How can one show that rational functions satisfy a Lipschitz condition in the chordal metric on the Riemann sphere?
user4324
1
vote
1
answer
359
views
the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n
we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the ...
- 399
12
votes
1
answer
775
views
Teaching Methods and Evaluating them
Hey,
As a lowly graduate student, I'm on a committee (I'm not sure how important my role really is) trying to evaluate how effective different approaches teaching undergraduates. We are looking at ...
2
votes
4
answers
6k
views
Undergraduate Derivation of Fundamental Solution to Heat Equation
It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...
- 5,935
28
votes
2
answers
3k
views
Restriction of a complex polynomial to the unit circle
I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).
Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there ...
- 1,304
3
votes
0
answers
131
views
Slicing the fibres of a meromorphic function with the zero set of a section of an ample line bundle
I'm going through a proof of a vanishing theorem by Sommese ($H^{p,q}(X,L) = 0$ for $p+q > n+k$ if $L$ is $k$-ample) and have hit the following brick wall:
I've got a complex projective manifold $...
- 4,343
7
votes
2
answers
6k
views
Using Weierstrass’s Factorization Theorem
I am trying to factorize $\sin(x)\over x$ which by Taylor series expansion and using the roots is $$a \cdot \left(1 - \frac{x}{\pi} \right) \left(1 + \frac{x}{\pi} \right) \left(1 - \frac{x}{2\pi} \...
- 5,935
5
votes
0
answers
694
views
Has the Weierstass transform been used to give Hermite series representations of the Riemann zeta function?
The inverse of the Weierstrass transform
expands a function as a series of Hermite polynomials $H_{n}$. There are several ways to invert the Weierstrass transform which led me to the following ...
0
votes
1
answer
412
views
About vertex algebra, mode expansion
A vertex operator is a linear map associating every state to a operator-valued distributions (quantum field) on a algebra curve, which is also called operator-state correspondence.
Chose a local ...
- 787
1
vote
2
answers
3k
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bounding roots of a polynomial with Rouche's Theorem
Suppose f(z) = z^n - k [ z^(n-1) + ... + z + 1 ] where n is a positive integer and k is a real constant such that nk<1.
I have shown that a root of this ...
- 27
51
votes
6
answers
5k
views
What does it take to run a good learning seminar?
I'm thinking about running a graduate student seminar in the summer. Having both organized and participated in such seminars in the past, I have witnessed first-hand that, contrary to what one might ...
10
votes
2
answers
629
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What do the numbers G_4 and G_6 of a lattice actually measure?
If you have a lattice $L \subset \mathbb{C}$, you can compute the following numbers:
$
G_4(L) = \sum_{\omega \in L, \omega \neq 0} \frac{1}{\omega^4}, \quad G_6(L) = \sum_{\omega \in L, \omega \neq 0}...
- 1,821
3
votes
1
answer
473
views
Is a function which is finitely multiple-valued in each variable separately, also finitely multiple-valued in all its variables jointly?
It is well known that under suitable conditions, a function which is:
a polynomial in each variable separately is a polynomial in all its variables jointly.
a rational function in each variable ...
- 371
18
votes
12
answers
10k
views
Looking for an introductory textbook on algebraic geometry for an undergraduate lecture course
I am now supposed to organize a tiny lecture course on algebraic geometry for undergraduate students who have an interest in this subject.
I wonder whether there are some basic algebraic geometry ...
0
votes
1
answer
198
views
An integral arising in statistics(2)
The integral I am interested in is:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 one can use contour integration.
So for K>1 we have :
$$\pi/2-\...
- 267
11
votes
1
answer
813
views
Approximation to divergent integral
Hi everyone,
I'm a physicist working on stochastic processes and I've come up against an integral that I'm not able to approximate using steepest descent (I don't have a large or small parameter), ...
- 111
6
votes
3
answers
3k
views
Zeros of the Weierstrass $\wp$-function
This question was prompted by the post here, and I asked this earlier, deleted it, and due to pressure exerted by Ilya Nikokoshev, I am asking it again. Apologies to Pavel Etingof.
Q1. Let $\Lambda$ ...
- 7,442
2
votes
2
answers
242
views
Simultaneous convergence of powers of unit complex numbers
Let $z_1,\ldots,z_n$ be complex numbers of modulus one. Does it exist an increasing sequence $k_j\in\mathbb{N}$ such that $\lim_{j\to\infty}z_i^{k_j}=1$ for all i?
- 971
13
votes
1
answer
860
views
What does the incidence algebra of the lattices in C tell us about modular forms?
I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of ...
- 118k
0
votes
1
answer
412
views
An integral arising in statistics
The integral I need:
$$t(x)=\int_{-K}^{K}\frac{\exp(ixy)}{1+y^{2q}}dy$$
$K<\infty$, q natural number
For q=1 this integral is
$$\pi/2-\int_{Arc}\frac{\exp(ixy)}{1+y^{2}}dy $$
Where Arc ...
- 267