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Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
1
vote
1
answer
201
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real-valued functions on the modular surface
How does one write down $\mathbb{R}$-valued functions on the modular surface? I am considering taking an arbitrary function on the upper half plane $f:\mathbb{H} \to \mathbb{R}$ and averaging over th …
1
vote
0
answers
163
views
kostant partition function vs Haar measure
I am trying to understand the relationship between the Kostant partition function and the Haar measure. Both seem to involve the Vandermonde determinant:
$$ \Delta(\theta) = \prod_{i< j} |e^{i\theta …
2
votes
1
answer
184
views
quadrature domains from circles?
If $h(z)$ is analytic on the disk centered at 0 of radius r, by the Cauchy Residue formula
\[ \int \int_D h(z)\, dx dy = \pi r^2 h(0) \]
The disk is the simplest example of a quadrature domain since …
2
votes
1
answer
280
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pick interpolation -- why is it symmetric? $\left[\frac{1 - w_i \overline{w_j}}{1 - z_i \ove... [closed]
I am reading notes on a complex interpolation problem:
Let $z_1, \dots, z_n \in \mathbb{D}$ and $w_1, \dots, w_n \in \mathbb{C}$. There exists (bounded holomorphic?) $f \in H^\infty(\mathbb{D})$ w …
6
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0
answers
142
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Evaluate $\sum_{\sigma} (2\pi i)^{-n}\oint \frac{f_{\sigma(1)}(u)\dots f_{\sigma_n(1)}(u)}{(...
In a probability theory paper I found this rather pleasant result:
Theorem 4.1 Let $n \geq 2$ and $f_1, \dots, f_n : \mathbb{C} \to \mathbb{C}$ be meromorphic with possible poles at $\{ \mathfrak{ …
2
votes
0
answers
241
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Plane Curve invariants via Contour Integrals
We learn in complex analysis class how to find the winding number of the contour $\Gamma$ around the origin.
\[ n = \frac{1}{2\pi i} \oint \frac{dz}{z} = \frac{1}{2\pi i} \oint d(\log z)
= \fra …
2
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0
answers
220
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why is this result about Gaussian analytic functions equivalent to the Crofton formula
I am reading Zeros of Gaussian Analytic Functions by Mikhail Sodin and he gives an much-too-easy proof of density of zeros of a Gaussian Analytic function.
Definition A Gaussian analytic function …
11
votes
6
answers
3k
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Explicit Spin Structures on the Torus
Basically, I am trying to build explicit examples of Dirac operators. To this end, I'm looking at the surface E = C/(Z + λZ) - for some λ in H \ SL(2,Z) - with the Euclidean metric and the flat conne …
4
votes
2
answers
916
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Self-similarity of a dendrite fractal
The Julia set of the map $z \mapsto z^2+i$ is a dendrite fractal. I would like to know which affine maps (other than identity) map this region to a subset of itself. I imagine there are two three ge …
36
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2
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3k
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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 cur …
0
votes
1
answer
2k
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What is the orthonormal basis for the Bergman space on the disk?
[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]
In arXiv:0310.5297, Yu …
6
votes
0
answers
397
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semiclassical proof of Wigner semicircle
In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int …
4
votes
3
answers
636
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Traceless GUE : Four Centered Fermions
The proof of the Wigner Semicircle Law comes from studying the GUE Kernel
$$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} …