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content/Math401/Extending_thesis/Math401_S3.md
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# Math 401, Fall 2025: Thesis notes, S3, Coherent states and POVMs # Math 401, Fall 2025: Thesis notes, S3, Coherent states and POVMs
> This section should extends on the reading for
>
> [Holomorphic methods in analysis and mathematical physics]()
## Bargmann space (original)
Also known as Segal-Bargmann space or Bargmann-Fock space.
It is the space of [holomorphic functions](../../Math416/Math416_L3#definition-28-holomorphic-functions) that is square-integrable over the complex plane.
> Section belows use [Remarks on a Hilbert Space of Analytic Functions](https://www.jstor.org/stable/71180) as the reference.
A family of Hilbert spaces, $\mathfrak{F}_n(n=1,2,3,\cdots)$, is defined as follows:
The element of $\mathfrak{F}_n$ are [entire](../../Math416/Math416_L13#definition-711) [analytic functions](../../Math416/Math416_L9#definition-analytic) in complex Euclidean space $\mathbb{C}^n$. $f:\mathbb{C}^n\to \mathbb{C}\in \mathfrak{F}_n$
Let $f,g\in \mathfrak{F}_n$. The inner product is defined by
$$
\langle f,g\rangle=\int_{\mathbb{C}^n} \overline{f(z)}g(z) d\mu_n(z)
$$
Let $z_k=x_k+iy_k$ be the complex coordinates of $z\in \mathbb{C}^n$.
The measure $\mu_n$ is the defined by
$$
d\mu_n(z)=\pi^{-n}\exp(-\sum_{i=1}^n |z_i|^2)\prod_{k=1}^n dx_k dy_k
$$
<details>
<summary>Example</summary>
For $n=2$,
$$
\mathfrak{F}_2=\text{ space of entire analytic functions on } \mathbb{C}^2\to \mathbb{C}
$$
$$
\langle f,g\rangle=\int_{\mathbb{C}^2} \overline{f(z)}g(z) d\mu(z),z=(z_1,z_2)
$$
$$
d\mu_2(z)=\frac{1}{\pi^2}\exp(-|z|^2)dx_1 dy_1 dx_2 dy_2
$$
</details>
so that $f$ belongs to $\mathfrak{F}_n$ if and only if $\langle f,f\rangle<\infty$.
This is absolutely terrible early texts, we will try to formulate it in a more modern way.
> The section belows are from the lecture notes [Holomorphic method in analysis and mathematical physics](https://arxiv.org/pdf/quant-ph/9912054)
## Complex function spaces
### Holomorphic spaces
Let $U$ be a non-empty open set in $\mathbb{C}^d$. Let $\mathcal{H}(U)$ be the space of holomorphic (or analytic) functions on $U$.
Let $f\in \mathcal{H}(U)$, note that by definition of holomorphic on several complex variables, $f$ is continuous and holomorphic in each variable with the other variables fixed.
Let $\alpha$ be a continuous, strictly positive function on $U$.
$$
\mathcal{H}L^2(U,\alpha)=\left\{F\in \mathcal{H}(U): \int_U |F(z)|^2 \alpha(z) d\mu(z)<\infty\right\},
$$
where $\mu$ is the Lebesgue measure on $\mathbb{C}^d=\mathbb{R}^{2d}$.
#### Theorem of holomorphic spaces
1. For all $z\in U$, there exists a constant $c_z$ such that
$$
|F(z)|^2\le c_z \|F\|^2_{L^2(U,\alpha)}
$$
for all $F\in \mathcal{H}L^2(U,\alpha)$.
2. $\mathcal{H}L^2(U,\alpha)$ is a closed subspace of $L^2(U,\alpha)$, and therefore a Hilbert space.
<details>
<summary>Proof</summary>
First we check part 1.
Let $z=(z_1,z_2,\cdots,z_d)\in U, z_k\in \mathbb{C}$. Let $P_s(z)$ be the "polydisk"of radius $s$ centered at $z$ defined as
$$
P_s(z)=\{v\in \mathbb{C}^d: |v_k-z_k|<s, k=1,2,\cdots,d\}
$$
If $z\in U$, we cha choose $s$ small enough such that $\overline{P_s(z)}\subset U$ so that we can claim that $F(z)=(\pi s^2)^{-d}\int_{P_s(z)}F(v)d\mu(v)$ is well-defined.
If $d=1$. Then by Taylor series at $v=z$, since $F$ is analytic in $U$ we have
$$
F(v)=F(z)+\sum_{k=1}^{\infty}a_n(v-z)^n
$$
Since the series converges uniformly to $F$ on the compact set $\overline{P_s(z)}$, we can interchange the integral and the sum.
Using polar coordinates with origin at $z$, $(v-z)^n=r^n e^{in\theta}$ where $r=|v-z|, \theta=\arg(v-z)$.
For $n\geq 1$, the integral over $P_s(z)$ (open disk) is zero (by Cauchy's theorem).
So,
$$
\begin{aligned}
F(z)&=(\pi s^2)^{-1}\int_{P_s(z)}F(z)+\sum_{k=1}^{\infty}a_n(v-z)^n d\mu(v)\\
&=(\pi s^2)^{-1}F(z)+(\pi s^2)^{-1}\sum_{k=1}^{\infty}a_n\int_{P_s(z)}r^n e^{in\theta} d\mu(v)\\
&=(\pi s^2)^{-1}F(z)
\end{aligned}
$$
For $d>1$, we can use the same argument to show that
Let $\mathbb{I}_{P_s(z)}(v)=\begin{cases}1 & v\in P_s(z) \\0 & v\notin P_s(z)\end{cases}$ be the indicator function of $P_s(z)$.
$$
\begin{aligned}
F(z)&=(\pi s^2)^{-d}\int_{U}\mathbb{I}_{P_s(z)}(v)\frac{1}{\alpha(v)}F(v)\alpha(v) d\mu(v)\\
&=(\pi s^2)^{-d}\langle \mathbb{I}_{P_s(z)}\frac{1}{\alpha},F\rangle_{L^2(U,\alpha)}
\end{aligned}
$$
By definition of inner product.
So $\|F(z)\|^2\leq (\pi s^2)^{-2d}\|\mathbb{I}_{P_s(z)}\frac{1}{\alpha}\|^2_{L^2(U,\alpha)} \|F\|^2_{L^2(U,\alpha)}$.
All the terms are bounded and finite.
For part 2, we need to show that $\forall z\in U$, we can find a neighborhood $V$ of $z$ and a constant $d_z$ such that
$$
|F(z)|^2\leq d_z \|F\|^2_{L^2(U,\alpha)}
$$
Suppose we have a sequence $F_n\in \mathcal{H}L^2(U,\alpha)$ such that $F_n\to F$, $F\in L^2(U,\alpha)$.
Then $F_n$ is a cauchy sequence in $L^2(U,\alpha)$. So,
$$
\sup_{v\in V}|F_n(v)-F_m(v)|\leq \sqrt{d_z}\|F_n-F_m\|_{L^2(U,\alpha)}\to 0\text{ as }n,m\to \infty
$$
So the sequence $F_m$ converges locally uniformly to some limit function which must be $F$ ($\mathbb{C}^d$ is Hausdorff, unique limit point).
Locally uniform limit of holomorphic functions is holomorphic. (Use Morera's Theorem to show that the limit is still holomorphic in each variable.) So the limit function $F$ is actually in $\mathcal{H}L^2(U,\alpha)$, which shows that $\mathcal{H}L^2(U,\alpha)$ is closed.
which shows that $\mathcal{H}L^2(U,\alpha)$ is closed.
</details>
> [!TIP]
>
> [1.] states that point-wise evaluation of $F$ on $U$ is continuous. That is, for each $z\in U$, the map $\varphi: \mathcal{H}L^2(U,\alpha)\to \mathbb{C}$ that takes $F\in \mathcal{H}L^2(U,\alpha)$ to $F(z)$ is a continuous linear functional on $\mathcal{H}L^2(U,\alpha)$. This is false for ordinary non-holomorphic functions, e.g. $L^2$ spaces.
#### Reproducing kernel
Let $\mathcal{H}L^2(U,\alpha)$ be a holomorphic space. The reproducing kernel of $\mathcal{H}L^2(U,\alpha)$ is a function $K:U\times U\to \mathbb{C}$, $K(z,w),z,w\in U$ with the following properties:
1. $K(z,w)$ is holomorphic in $z$ and anti-holomorphic in $w$.
$$
K(w,z)=\overline{K(z,w)}
$$
2. For each fixed $z\in U$, $K(z,w)$ is a square integrable $d\alpha(w)$. For all $F\in \mathcal{H}L^2(U,\alpha)$,
$$
F(z)=\int_U K(z,w)F(w) \alpha(w) dw
$$
3. If $F\in L^2(U,\alpha)$, let $PF$ denote the orthogonal projection of $F$ onto closed subspace $\mathcal{H}L^2(U,\alpha)$. Then
$$
PF(z)=\int_U K(z,w)F(w) \alpha(w) dw
$$
4. For all $z,u\in U$,
$$
\int_U K(z,w)K(w,u) \alpha(w) dw=K(z,u)
$$
5. For all $z\in U$,
$$
|F(z)|^2\leq K(z,z) \|F\|^2_{L^2(U,\alpha)}
$$
<details>
<summary>Proof</summary>
For part 1, By [Riesz Theorem](../../Math429/Math429_L27#theorem-642-riesz-representation-theorem), the linear functional evaluation at $z\in U$ on $\mathcal{H}L^2(U,\alpha)$ can be represented uniquely as inner product with some $\phi_z\in \mathcal{H}L^2(U,\alpha)$.
$$
F(z)=\langle F,\phi_z\rangle_{L^2(U,\alpha)}=\int_U F(w)\overline{\phi_z(w)} \alpha(w) dw
$$
And assume part 2 is true, then we have
$K(z,w)=\overline{\phi_z(w)}$
So part 1 is true.
For part 2, we can use the same argument
$$
\phi_z(w)=\langle \phi_z,\phi_w\rangle_{L^2(U,\alpha)}=\overline{\langle \phi_w,\phi_z\rangle_{L^2(U,\alpha)}}=\overline{\phi_w(z)}
$$
... continue if needed.
</details>
#### Construction of reproducing kernel
Let $\{e_j\}$ be any orthonormal basis of $\mathcal{H}L^2(U,\alpha)$. Then for all $z,w\in U$,
$$
\sum_{j=1}^{\infty} |e_j(z)\overline{e_j(w)}|<\infty
$$
and
$$
K(z,w)=\sum_{j=1}^{\infty} e_j(z)\overline{e_j(w)}
$$
### Bargmann space
The Bargmann spaces are the holomorphic spaces
$$
\mathcal{H}L^2(\mathbb{C}^d,\mu_t)
$$
where
$$
\mu_t(z)=(\pi t)^{-d}\exp(-|z|^2/t)
$$
> For this research, we can tentatively set $t=1$ and $d=2$ for simplicity so that you can continue to read the next section.
#### Reproducing kernel for Bargmann space
For all $d\geq 1$, the reproducing kernel of the space $\mathcal{H}L^2(\mathbb{C}^d,\mu_t)$ is given by
$$
K(z,w)=\exp(z\cdot \overline{w}/t)
$$
where $z\cdot \overline{w}=\sum_{k=1}^d z_k\overline{w_k}$.
This gives the pointwise bounds
$$
|F(z)|^2\leq \exp(\|z\|^2/t) \|F\|^2_{L^2(\mathbb{C}^d,\mu_t)}
$$
For all $F\in \mathcal{H}L^2(\mathbb{C}^d,\mu_t)$, and $z\in \mathbb{C}^d$.
> Proofs are intentionally skipped, you can refer to the lecture notes for details.
#### Lie bracket of vector fields
Let $X,Y$ be two vector fields on a smooth manifold $M$. The Lie bracket of $X$ and $Y$ is an operator $[X,Y]:C^\infty(M)\to C^\infty(M)$ defined by
$$
[X,Y](f)=X(Y(f))-Y(X(f))
$$
This operator is a vector field.
> Continue here for quantization of Coherent states and POVMs

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# Math 401, Fall 2025: Thesis notes, S4, Complex function spaces and complex manifold # Math 401, Fall 2025: Thesis notes, S4, Complex manifolds
## Bargmann space (original)
Also known as Segal-Bargmann space or Bargmann-Fock space.
It is the space of [holomorphic functions](../../Math416/Math416_L3#definition-28-holomorphic-functions) that is square-integrable over the complex plane.
> Section belows use [Remarks on a Hilbert Space of Analytic Functions](https://www.jstor.org/stable/71180) as the reference.
A family of Hilbert spaces, $\mathfrak{F}_n(n=1,2,3,\cdots)$, is defined as follows:
The element of $\mathfrak{F}_n$ are [entire](../../Math416/Math416_L13#definition-711) [analytic functions](../../Math416/Math416_L9#definition-analytic) in complex Euclidean space $\mathbb{C}^n$. $f:\mathbb{C}^n\to \mathbb{C}\in \mathfrak{F}_n$
Let $f,g\in \mathfrak{F}_n$. The inner product is defined by
$$
\langle f,g\rangle=\int_{\mathbb{C}^n} \overline{f(z)}g(z) d\mu_n(z)
$$
Let $z_k=x_k+iy_k$ be the complex coordinates of $z\in \mathbb{C}^n$.
The measure $\mu_n$ is the defined by
$$
d\mu_n(z)=\pi^{-n}\exp(-\sum_{i=1}^n |z_i|^2)\prod_{k=1}^n dx_k dy_k
$$
<details>
<summary>Example</summary>
For $n=2$,
$$
\mathfrak{F}_2=\text{ space of entire analytic functions on } \mathbb{C}^2\to \mathbb{C}
$$
$$
\langle f,g\rangle=\int_{\mathbb{C}^2} \overline{f(z)}g(z) d\mu(z),z=(z_1,z_2)
$$
$$
d\mu_2(z)=\frac{1}{\pi^2}\exp(-|z|^2)dx_1 dy_1 dx_2 dy_2
$$
</details>
so that $f$ belongs to $\mathfrak{F}_n$ if and only if $\langle f,f\rangle<\infty$.
This is absolutely terrible early texts, we will try to formulate it in a more modern way.
> The section belows are from the lecture notes [Holomorphic method in analysis and mathematical physics](https://arxiv.org/pdf/quant-ph/9912054)
## Complex function spaces
### Holomorphic spaces
Let $U$ be a non-empty open set in $\mathbb{C}^d$. Let $\mathcal{H}(U)$ be the space of holomorphic (or analytic) functions on $U$.
Let $f\in \mathcal{H}(U)$, note that by definition of holomorphic on several complex variables, $f$ is continuous and holomorphic in each variable with the other variables fixed.
Let $\alpha$ be a continuous, strictly positive function on $U$.
$$
\mathcal{H}L^2(U,\alpha)=\left\{F\in \mathcal{H}(U): \int_U |F(z)|^2 \alpha(z) d\mu(z)<\infty\right\},
$$
where $\mu$ is the Lebesgue measure on $\mathbb{C}^d=\mathbb{R}^{2d}$.
#### Theorem of holomorphic spaces
1. For all $z\in U$, there exists a constant $c_z$ such that
$$
|F(z)|^2\le c_z \|F\|^2_{L^2(U,\alpha)}
$$
for all $F\in \mathcal{H}L^2(U,\alpha)$.
2. $\mathcal{H}L^2(U,\alpha)$ is a closed subspace of $L^2(U,\alpha)$, and therefore a Hilbert space.
<details>
<summary>Proof</summary>
First we check part 1.
Let $z=(z_1,z_2,\cdots,z_d)\in U, z_k\in \mathbb{C}$. Let $P_s(z)$ be the "polydisk"of radius $s$ centered at $z$ defined as
$$
P_s(z)=\{v\in \mathbb{C}^d: |v_k-z_k|<s, k=1,2,\cdots,d\}
$$
If $z\in U$, we cha choose $s$ small enough such that $\overline{P_s(z)}\subset U$ so that we can claim that $F(z)=(\pi s^2)^{-d}\int_{P_s(z)}F(v)d\mu(v)$ is well-defined.
If $d=1$. Then by Taylor series at $v=z$, since $F$ is analytic in $U$ we have
$$
F(v)=F(z)+\sum_{k=1}^{\infty}a_n(v-z)^n
$$
Since the series converges uniformly to $F$ on the compact set $\overline{P_s(z)}$, we can interchange the integral and the sum.
Using polar coordinates with origin at $z$, $(v-z)^n=r^n e^{in\theta}$ where $r=|v-z|, \theta=\arg(v-z)$.
For $n\geq 1$, the integral over $P_s(z)$ (open disk) is zero (by Cauchy's theorem).
So,
$$
\begin{aligned}
F(z)&=(\pi s^2)^{-1}\int_{P_s(z)}F(z)+\sum_{k=1}^{\infty}a_n(v-z)^n d\mu(v)\\
&=(\pi s^2)^{-1}F(z)+(\pi s^2)^{-1}\sum_{k=1}^{\infty}a_n\int_{P_s(z)}r^n e^{in\theta} d\mu(v)\\
&=(\pi s^2)^{-1}F(z)
\end{aligned}
$$
For $d>1$, we can use the same argument to show that
Let $\mathbb{I}_{P_s(z)}(v)=\begin{cases}1 & v\in P_s(z) \\0 & v\notin P_s(z)\end{cases}$ be the indicator function of $P_s(z)$.
$$
\begin{aligned}
F(z)&=(\pi s^2)^{-d}\int_{U}\mathbb{I}_{P_s(z)}(v)\frac{1}{\alpha(v)}F(v)\alpha(v) d\mu(v)\\
&=(\pi s^2)^{-d}\langle \mathbb{I}_{P_s(z)}\frac{1}{\alpha},F\rangle_{L^2(U,\alpha)}
\end{aligned}
$$
By definition of inner product.
So $\|F(z)\|^2\leq (\pi s^2)^{-2d}\|\mathbb{I}_{P_s(z)}\frac{1}{\alpha}\|^2_{L^2(U,\alpha)} \|F\|^2_{L^2(U,\alpha)}$.
All the terms are bounded and finite.
For part 2, we need to show that $\forall z\in U$, we can find a neighborhood $V$ of $z$ and a constant $d_z$ such that
$$
|F(z)|^2\leq d_z \|F\|^2_{L^2(U,\alpha)}
$$
Suppose we have a sequence $F_n\in \mathcal{H}L^2(U,\alpha)$ such that $F_n\to F$, $F\in L^2(U,\alpha)$.
Then $F_n$ is a cauchy sequence in $L^2(U,\alpha)$. So,
$$
\sup_{v\in V}|F_n(v)-F_m(v)|\leq \sqrt{d_z}\|F_n-F_m\|_{L^2(U,\alpha)}\to 0\text{ as }n,m\to \infty
$$
So the sequence $F_m$ converges locally uniformly to some limit function which must be $F$ ($\mathbb{C}^d$ is Hausdorff, unique limit point).
Locally uniform limit of holomorphic functions is holomorphic. (Use Morera's Theorem to show that the limit is still holomorphic in each variable.) So the limit function $F$ is actually in $\mathcal{H}L^2(U,\alpha)$, which shows that $\mathcal{H}L^2(U,\alpha)$ is closed.
which shows that $\mathcal{H}L^2(U,\alpha)$ is closed.
</details>
> [!TIP]
>
> [1.] states that point-wise evaluation of $F$ on $U$ is continuous. That is, for each $z\in U$, the map $\varphi: \mathcal{H}L^2(U,\alpha)\to \mathbb{C}$ that takes $F\in \mathcal{H}L^2(U,\alpha)$ to $F(z)$ is a continuous linear functional on $\mathcal{H}L^2(U,\alpha)$. This is false for ordinary non-holomorphic functions, e.g. $L^2$ spaces.
#### Reproducing kernel
Let $\mathcal{H}L^2(U,\alpha)$ be a holomorphic space. The reproducing kernel of $\mathcal{H}L^2(U,\alpha)$ is a function $K:U\times U\to \mathbb{C}$, $K(z,w),z,w\in U$ with the following properties:
1. $K(z,w)$ is holomorphic in $z$ and anti-holomorphic in $w$.
$$
K(w,z)=\overline{K(z,w)}
$$
2. For each fixed $z\in U$, $K(z,w)$ is a square integrable $d\alpha(w)$. For all $F\in \mathcal{H}L^2(U,\alpha)$,
$$
F(z)=\int_U K(z,w)F(w) \alpha(w) dw
$$
3. If $F\in L^2(U,\alpha)$, let $PF$ denote the orthogonal projection of $F$ onto closed subspace $\mathcal{H}L^2(U,\alpha)$. Then
$$
PF(z)=\int_U K(z,w)F(w) \alpha(w) dw
$$
4. For all $z,u\in U$,
$$
\int_U K(z,w)K(w,u) \alpha(w) dw=K(z,u)
$$
5. For all $z\in U$,
$$
|F(z)|^2\leq K(z,z) \|F\|^2_{L^2(U,\alpha)}
$$
<details>
<summary>Proof</summary>
For part 1, By [Riesz Theorem](../../Math429/Math429_L27#theorem-642-riesz-representation-theorem), the linear functional evaluation at $z\in U$ on $\mathcal{H}L^2(U,\alpha)$ can be represented uniquely as inner product with some $\phi_z\in \mathcal{H}L^2(U,\alpha)$.
$$
F(z)=\langle F,\phi_z\rangle_{L^2(U,\alpha)}=\int_U F(w)\overline{\phi_z(w)} \alpha(w) dw
$$
And assume part 2 is true, then we have
$K(z,w)=\overline{\phi_z(w)}$
So part 1 is true.
For part 2, we can use the same argument
$$
\phi_z(w)=\langle \phi_z,\phi_w\rangle_{L^2(U,\alpha)}=\overline{\langle \phi_w,\phi_z\rangle_{L^2(U,\alpha)}}=\overline{\phi_w(z)}
$$
... continue if needed.
</details>
#### Construction of reproducing kernel
Let $\{e_j\}$ be any orthonormal basis of $\mathcal{H}L^2(U,\alpha)$. Then for all $z,w\in U$,
$$
\sum_{j=1}^{\infty} |e_j(z)\overline{e_j(w)}|<\infty
$$
and
$$
K(z,w)=\sum_{j=1}^{\infty} e_j(z)\overline{e_j(w)}
$$
### Bargmann space
The Bargmann spaces are the holomorphic spaces
$$
\mathcal{H}L^2(\mathbb{C}^d,\mu_t)
$$
where
$$
\mu_t(z)=(\pi t)^{-d}\exp(-|z|^2/t)
$$
> For this research, we can tentatively set $t=1$ and $d=2$ for simplicity so that you can continue to read the next section.
#### Reproducing kernel for Bargmann space
For all $d\geq 1$, the reproducing kernel of the space $\mathcal{H}L^2(\mathbb{C}^d,\mu_t)$ is given by
$$
K(z,w)=\exp(z\cdot \overline{w}/t)
$$
where $z\cdot \overline{w}=\sum_{k=1}^d z_k\overline{w_k}$.
This gives the pointwise bounds
$$
|F(z)|^2\leq \exp(\|z\|^2/t) \|F\|^2_{L^2(\mathbb{C}^d,\mu_t)}
$$
For all $F\in \mathcal{H}L^2(\mathbb{C}^d,\mu_t)$, and $z\in \mathbb{C}^d$.
> Proofs are intentionally skipped, you can refer to the lecture notes for details.
#### Lie bracket of vector fields
Let $X,Y$ be two vector fields on a smooth manifold $M$. The Lie bracket of $X$ and $Y$ is an operator $[X,Y]:C^\infty(M)\to C^\infty(M)$ defined by
$$
[X,Y](f)=X(Y(f))-Y(X(f))
$$
This operator is a vector field.
## Complex Manifolds ## Complex Manifolds
@@ -312,6 +44,40 @@ A **holomorphic line bundle** is a holomorphic vector bundle with rank 1.
> Intuitively, a holomorphic line bundle is a complex vector bundle with a complex structure on each fiber. > Intuitively, a holomorphic line bundle is a complex vector bundle with a complex structure on each fiber.
### Simplicial, Sheafs, Cohomology and homology
What is homology and cohomology?
> This section is based on extension for conversation with Professor Feres on [11/05/2025].
#### Definition of meromorphic function
Let $Y$ be an open subset of $X$. A function $f$ is called meromorphic function on $Y$, if there exists a non-empty open subset $Y'\subset Y$ such that
1. $f:Y'\to \mathbb{C}$ is a holomorphic function.
2. $A=Y\setminus Y'$ is a set of isolated points (called the set of poles)
3. $\lim_{x\to p}|f(x)|=+\infty$ for all $p\in A$
> Basically, a local holomorphic function on $Y$.
#### De Rham Theorem
This is analogous to the Stoke's Theorem on chains, $\int_c d\omega=\int_{\partial c} \omega$.
$$
H_k(X)\cong H^k(X)
$$
Where $H_k(X)$ is the $k$-th homology of $X$, and $H^k(X)$ is the $k$-th cohomology of $X$.
#### Simplicial Cohomology
Riemann surfaces admit triangulations. The triangle are 2 simplices. The edges are 1 simplices. the vertices are 0 simplices.
Our goal is to build global description of Riemann surfaces using local description on each triangulation.
#### Singular Cohomology
### Riemann-Roch Theorem (Theorem 9.64) ### Riemann-Roch Theorem (Theorem 9.64)
Suppose $M$ is a connected compact Riemann surface of genus $g$, and $L\to M$ is a holomorphic line bundle. Then Suppose $M$ is a connected compact Riemann surface of genus $g$, and $L\to M$ is a holomorphic line bundle. Then

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@@ -1,3 +1,4 @@
export default { export default {
index: "Math 401, Fall 2025: Overview of thesis", index: "Math 401, Fall 2025: Overview of thesis",
} }