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212 votes
52 answers
82k views

Ways to prove the fundamental theorem of algebra

This seems to be a favorite question everywhere, including Princeton quals. How many ways are there? Please give a new way in each answer, and if possible give reference. I start by giving two: ...
31 votes
11 answers
13k views

Uniformization theorem for Riemann surfaces

How does one prove that every simply connected Riemann surface is conformally equivalent to the open unit disk, the complex plane, or the Riemann sphere, and these are not conformally equivalent to ...
31 votes
0 answers
1k views

"Three great cocycles" in Complex Analysis as cohomology generators

In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$ and the Schwarzian ...
Kostya_I's user avatar
  • 8,992
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 ...
Mohammad Ghomi's user avatar
29 votes
2 answers
560 views

A strange infinite fraction, and a functional equation

The following curious-looking fraction, with numerical value approximately $1.7302267782385217$, appears in this Reddit question: $$1+\cfrac{2+\cfrac{4+\cfrac{8+\cdots}{9+\cdots}}{5+\cfrac{10+\cdots}{...
chronondecay's user avatar
27 votes
3 answers
2k views

Kasteleyn's formula for domino tilings generalized?

It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$. Kasteleyn's ...
T. Amdeberhan's user avatar
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 ...
Per Alexandersson's user avatar
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
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 $\...
Peter Scholze's user avatar
20 votes
2 answers
1k views

Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles

Some background on (compact) Belyi surfaces $\newcommand{\Ch}{\hat{\mathbb{C}}}$ A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that $...
Lasse Rempe's user avatar
  • 6,548
18 votes
2 answers
1k views

Vanishing of Dolbeault cohomologies and Steinness

That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
hassan's user avatar
  • 243
17 votes
2 answers
1k views

Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman: A connected open subset $D$ of the plane $\mathbb C$ is simply connected if and only if its complement $\widetilde D = \mathbb C \setminus D$ ...
Mirko's user avatar
  • 1,375
15 votes
5 answers
2k views

Zeros of the derivative of Riemann's $\xi$-function

The Riemann xi function $\xi(s)$ is defined as $$ \xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s). $$ It is an entire function whose zeros are precisely those of $\zeta(s)$. Since $\xi$ is real ...
Stopple's user avatar
  • 11.1k
15 votes
1 answer
2k views

Dirichlet series expansion of an analytic function

Let $F(s)=\sum_{n\geq 1}\frac{a_n}{n^s}$ be a Dirichlet series with (finite) abscissa of absolute convergence $\sigma_a$. It can be shown that $\forall \sigma >\sigma_a:$ $$\lim_{T\to\infty}\frac{1}...
M.G.'s user avatar
  • 7,127
14 votes
1 answer
3k views

How is the "conformal prediction" conformal?

The question is clarified by Prof.V.Vovk. See his answer below for discussion. Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
Henry.L's user avatar
  • 8,071
13 votes
2 answers
1k views

Is the exponential function the sole solution to these equations?

Let us take the exponential function $\lambda^z$ where $0 < \lambda < 1$. There are many great uniqueness conditions this holomorphic function satisfies. For example, it is the only function ...
user avatar
13 votes
4 answers
1k views

"Simple" Kahler manifolds

I have some lecture notes from Demailly on Kahler geometry where he talks about "variétés Kahleriennes simples", which are defined as Kahler manifolds $X$ such that for very generic points $x_0$ in $X$...
Gunnar Þór Magnússon's user avatar
13 votes
2 answers
725 views

Special values of $\zeta$ outside the real line and the critical strip

The values of Riemann's function at the integers have been extensively studied. I was wondering, is there anything interesting known (or conjectured) to happen arithmetically outside the real line (...
Myshkin's user avatar
  • 17.6k
12 votes
3 answers
784 views

Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?

The question is stated in the title of this post. It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
Iosif Pinelis's user avatar
12 votes
2 answers
1k views

Short research articles

I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article. For example; One of the best known ...
12 votes
2 answers
1k views

Has there been further work on Bender-Brody-Müller approach to RH?

Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I ...
Sylvain JULIEN's user avatar
11 votes
3 answers
899 views

How many isolated roots can a polynomial in $z$ and $\overline{z}$ have?

This is probably well-known by I can't find it now online. My guess is that if the degree is $n$ then it's $2n$ but it's just a hunch. EDIT: This is an edited version. Before I asked about roots ...
Felix Goldberg's user avatar
11 votes
1 answer
1k views

Extending an assignment property from Q to R (or C)

Property of any odd number of nonnegative integers: Given $x_1 \leq \cdots \leq x_{2n + 1}$ with each $x_i \in \mathbb{Z}_{\geq 0}$, suppose that for any $x_i$ we remove, the remaining numbers can be ...
Benjamin Dickman's user avatar
11 votes
2 answers
997 views

Is there a holomorphic Morse-Bott lemma?

It asks for a generalization of the question in the post Normal form for a holomorphic Morse function Suppose $f$ is a holomorphic function on a complex manifold $M$ which has Bott type critical ...
UVIR's user avatar
  • 803
11 votes
1 answer
485 views

Resources for divergent / asymptotic series

This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside [Edit (1/14/21) from the answer by Count Iblis to a recent MO-Q on math vids: An enthusiastic intro is ...
Tom Copeland's user avatar
  • 10.5k
11 votes
1 answer
676 views

Analysis and finitely generated groups

Dear all, this is perhaps a bit a vague question, but some references would already be very helpfull. So let $G$ be a finitely generated group and choose some finite set of generators. This allows to ...
Stefan Waldmann's user avatar
10 votes
1 answer
705 views

On entire functions with polynomial Schwarzian derivative

The Schwarzian derivative of an entire holomorphic function $f$ is defined as $$Sf:=\left(\frac{f^{''}}{f'}\right)'-\frac{1}{2}\left(\frac{f^{''}}{f'}\right)^2.$$ In the following, we only consider ...
student's user avatar
  • 1,350
10 votes
3 answers
2k views

Origin of term Ahlfors-David regular

Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
mdr's user avatar
  • 565
10 votes
4 answers
1k views

Analytic function avoiding elements of the modular group

A friend recently told me the following two facts, for which he cannot recall a proof or a reference (but he remembers seeing them in the literature): Let $f$ be a holomorphic function mapping the ...
Alexandre Eremenko's user avatar
10 votes
2 answers
417 views

zeros on the circle of convergence

In this question some experiments were used to conjecture that the zeros of partial sums of a series converging to a function with natural boundary on the unit circle were (weakly) converging to the ...
Igor Rivin's user avatar
  • 96.4k
10 votes
2 answers
538 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to {\...
Misha's user avatar
  • 31.2k
10 votes
1 answer
432 views

Modern version of an inequality of R. M. Gabriel for contour integrals

I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm: Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following ...
Ian Morris's user avatar
  • 6,206
10 votes
0 answers
656 views

“Taylor series” is to “Volterra series” as “Laurent series” is to _________?

Preamble My question is similar to an earlier MathOverflow question: “Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
Nike Dattani's user avatar
9 votes
2 answers
8k views

The Paley-Wiener theorem and exponential decay.

Consider a function whose Fourier transform is supported on a half-ray: $$ A(t)=\int_0^\infty \omega(E) e^{-iEt}d E, $$ where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on $...
Emilio Pisanty's user avatar
9 votes
2 answers
873 views

Implicit Function Theorem on Singular Varieties

Let $X$ and $Y$ be two complex reduced affine algebraic or analytic varieties, possibly singular. Take a regular proper function $$f\colon X \to Y $$ and assume that it is bijective at the level of $...
Giulio's user avatar
  • 2,384
9 votes
1 answer
321 views

Cauchy path integral as a linear operator: kernel and image?

Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
Loïc Teyssier's user avatar
9 votes
1 answer
3k views

Complex geometry text/research introduction for the analyst

To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...
MBM's user avatar
  • 141
9 votes
2 answers
2k views

References on Taylor series expansion of Riemann xi function

I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$. $$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$ where $$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/...
mike's user avatar
  • 603
9 votes
1 answer
657 views

Is anything known about the power series $\sum x^p$ for $p$ prime?

I'm interested in information about the power series $$\sum_{\text{$p$ prime}} x^p$$ and the related power series $$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$ where $p(n)$ is the nth prime. Immediately, the ...
Caleb Briggs's user avatar
  • 1,730
9 votes
0 answers
461 views

$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$

Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
Damian Rössler's user avatar
8 votes
4 answers
1k views

Monge Ampere equations

I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
Hammerhead's user avatar
  • 1,201
8 votes
2 answers
2k views

Applications of the Small and Great Theorems of Picard

I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications. List of applications (rather ...
smyrlis's user avatar
  • 2,933
8 votes
1 answer
734 views

Local polynomial form of holomorphic functions

It is well-known that a germ of a holomorphic function of $n\geq 2$ variables, with at most an isolated singularity (i.e. the singularity of the analytic variety defined by the zero locus of the ...
Loïc Teyssier's user avatar
8 votes
3 answers
2k views

Harmonic level sets and boundary data

This is probably a classic problem, so a good reference book or paper to get me started on this type of question would be great: Let $\mathbb{D} \subset \mathbb{C}$ be the unit disk with boundary $\...
partition_of_unity's user avatar
8 votes
2 answers
495 views

Literature on non-Archimedean analogues of basic complex analysis results

It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and ...
Very Forgetful Functor's user avatar
8 votes
1 answer
594 views

Reference for flatness in complex-analytic geometry

What is a good reference for flat morphisms of complex-analytic spaces? (The book by Grauert and Remmert doesn't treat them). Topics I'm interested in: openness of flat maps, descent for coherent ...
user avatar
8 votes
2 answers
2k views

Pair correlation for the Riemann zeros and $(\zeta^\prime(s)/\zeta(s))^\prime$

Added Background: The pair correlation of the zeros of the Riemann zeta function is influenced by the the derivative of the logarithmic derivative $(\zeta^\prime(s)/\zeta(s))^\prime$; see for example ...
Stopple's user avatar
  • 11.1k
8 votes
1 answer
638 views

Composite residues with determinant denominators

I am looking for a good reference on composite residues of multi-variable contour integrals (something better and more explicit than Griffiths and Harris or Tsikh). This means I want to evaluate $\...
Jared Kaplan's user avatar
8 votes
0 answers
304 views

On the remainder of a power series evaluated on the boundary of its convergence disk

Background This question is related to this one, in the sense that, as the previous one, it originates from my efforts to extend an estimate on the remainder of a power series on a non necessarily ...
Daniele Tampieri's user avatar
8 votes
0 answers
277 views

Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
asv's user avatar
  • 21.8k

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