Smooth manifolds lectures 16,17, & complex lecture 16

F4D1-Analysis-and-Geometry-on-Manifolds/Analysis_on_Manifolds_Notes.tex

@@ -44,6 +44,7 @@
 \maketitle
 \section*{Preliminaries}
 These notes roughly correspond to the course \textbf{F4D1 -- Analysis and Geometry on Manifolds} taught by Prof. Laurent C\^{o}t\'{e} at the Universit\"{a}t Bonn in the Winter 2024/25 semester. These notes are \LaTeX-ed after the fact with significant alteration and are subject to misinterpretation and mistranscription. Use with caution. Any errors are undoubtedly my own and any virtues that could be ascribed to these notes ought be attributed to the instructor and not the typist. 
+\newpage
 \tableofcontents
 \include{Lecture 1}
 \include{Lecture 2}
@@ -60,6 +61,8 @@ \section*{Preliminaries}
 \include{Lecture 13}
 \include{Lecture 14}
 \include{Lecture 15}
+\include{Lecture 16}
+\include{Lecture 17}
 \newpage
 \printbibliography
 \end{document}

F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 15.tex

@@ -74,6 +74,7 @@ \section{Lecture 15 -- 29th November 2024}\label{sec: lecture 15}
 \end{remark}
 Given that tangent spaces can be understood in terms of derivations, so too can vector fields. 
 \begin{lemma}\label{lem: smooth vector fields and functions}
+    Let $M$ be a smooth manifold.
     \begin{enumerate}[label=(\roman*)]
         \item If $X$ is a smooth tangent field on $M$ then $Xf$ is a smooth function. 
         \item If $X$ is a coarse vector field and $Xf$ is smooth for all $f\in C^{\infty}(M)$ then $X$ is smooth. 

F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 16.tex

@@ -0,0 +1,96 @@
+\section{Lecture 16 -- 3rd December 2024}\label{sec: lecture 16}
+We show that the formation of Lie brackets is natural in the following sense. 
+\begin{lemma}\label{lem: naturality of Lie brackets}
+    Let $M,N$ be smooth manifolds and $F:M\to N$ a smooth map. Let $X_{1},X_{2}\in\Xfrak(M),Y_{1},Y_{2}\in\Xfrak(N)$ with $X_{i}$ $F$-related to $Y_{i}$ for $i\in\{1,2\}$. Then $[X_{1},X_{2}]$ and $[Y_{1},Y_{2}]$ are $F$-related. 
+\end{lemma}
+\begin{proof}
+    Let $f\in C^{\infty}(N)$ and consider $X_{1}X_{2}(f\circ F)$ which by $F$-relatedness yields $X_{1}((Y_{2}f)\circ F)=(Y_{1}Y_{2}f)\circ F$ and similarly $X_{2}X_{1}(f\circ F)$ which by $F$-relatedness and an analogous calculation yields $X_{2}X_{1}(f\circ F)=(Y_{2}Y_{1}f)\circ F$. Thus in the Lie bracket we have 
+    \begin{align*}
+        [X_{1},X_{2}](f\circ F) &= (X_{1}X_{2}-X_{2}X_{1})(f\circ F) \\
+        &= (Y_{1}Y_{2}-Y_{2}Y_{1})(f)\circ F \\
+        &= [Y_{1},Y_{2}](f)\circ F
+    \end{align*}
+    showing $F$-relatedness, as desired. 
+\end{proof}
+We now consider coordinates on vector fields, in analogy to coordinates on the tangent bundle \Cref{def: coordinates on smooth manifold}. Recall that on a smooth manifold $M$ we have coordinates $\partial_{x_{i}}$ on the tangent bundle $TM$ which are the preimages of $i$th coordinate functions of a chart $(U,\phi)$ on $M$ under the canonical identification of the Euclidean space with its tangent bundle. 
+\begin{lemma}\label{lem: coordinate vector fields are smooth}
+    Let $M$ be a smooth manifold. The map $M\to TM$ by $p\mapsto (\partial_{x_{i}})_{p}$ defines a smooth vector field on an open set $U\subseteq M$. 
+\end{lemma}
+\begin{proof}
+    This is a coarse section by inspection, and smoothness follows from the smoothness of charts and projection maps. 
+\end{proof}
+\Cref{lem: coordinate vector fields are smooth} justifies the following definition. 
+\begin{definition}[Coordinate Vector Field]\label{def: coordinate vector field}
+    Let $M$ be a smooth manifold. The section $p\mapsto(\partial_{x_{i}})_{p}$ defines the coordinate vector field $\partial_{x_{i}}$. 
+\end{definition}
+Using this, we can define frames. 
+\begin{definition}[Local Frame]\label{def: local frame}
+    Let $M$ be a smooth $m$-manifold and $X^{1},\dots,X^{m}\in\Xfrak(M)$ be vector fields. $(X^{1},\dots,X^{m})$ is a local frame at $p$ if the tangent vectors $X^{1}_{p},\dots,X^{m}_{p}$ span $T_{p}M$. 
+\end{definition}
+\begin{definition}[Global Frame]\label{def: global frame}
+    Let $M$ be a smooth $m$-manifold and $X^{1},\dots,X^{m}\in\Xfrak(M)$ be vector fields. $(X^{1},\dots,X^{m})$ is a global frame if the tangent vectors $X^{1}_{p},\dots,X^{m}_{p}$ span $T_{p}M$ for all $p\in M$. 
+\end{definition}
+As expected, being a local frame is a local condition. 
+\begin{lemma}\label{lem: local frame is local}
+    Let $M$ be a smooth $m$-manifold and $X^{1},\dots,X^{m}$ a local frame at $p\in M$. Then there exists an open subset $U\subseteq M$ containing $p$ such that $(X^{1},\dots,X^{m})$ is a global frame on $U$. 
+\end{lemma}
+\begin{proof}
+    The $(X^{1},\dots,X^{m}_{p})$ considered as a matrix is of full rank, and by openness of the full rank condition in $p$ and \Cref{lem: subset of matrices of full rank is open} shows that there exists such $U$. 
+\end{proof}
+\begin{example}
+    The vector fields $\partial_{x_{i}}$ is a form a global frame, though not all vector fields arise as coordinate vector fields -- that is, admit a representation as a coordinate vector field for some chart. 
+\end{example}
+Now define integral curves. 
+\begin{definition}[Integral Curve]\label{def: integral curve}
+    Let $M$ be a smooth manifold and $X\in\Xfrak(M)$. An integral curve for $X$ is a curve $\gamma:(a,b)\to M$ with $(a,b)\subseteq\RR$ containing zero and such that $d\gamma_{t}(\partial_{t})=X_{\gamma(t)}$.  
+\end{definition}
+\begin{definition}[Starting Point]\label{def: starting point}
+    Let $M$ be a smooth manifold and $\gamma:(a,b)\to M$ an integral curve for $X\in\Xfrak(M)$. The starting point of $\gamma$ is $\gamma(0)\in M$. 
+\end{definition}
+To describe integral curves of \Cref{def: integral curve} more explicitly, we consider the diagram 
+$$% https://q.uiver.app/#q=WzAsNCxbMCwxLCIoYSxiKSJdLFsyLDEsIk0iXSxbMCwwLCJUXFxSUj1UKGEsYikiXSxbMiwwLCJUTSJdLFswLDEsIlxcZ2FtbWEiLDJdLFsyLDMsImRcXGdhbW1hIl0sWzMsMV0sWzIsMF0sWzAsMiwiXFxwYXJ0aWFsX3t0fSIsMCx7ImN1cnZlIjotMX1dLFswLDMsIlxcZG90e1xcZ2FtbWF9Il1d
+\begin{tikzcd}
+	{T\RR=T(a,b)} && TM \\
+	{(a,b)} && M.
+	\arrow["{d\gamma}", from=1-1, to=1-3]
+	\arrow[from=1-1, to=2-1]
+	\arrow[from=1-3, to=2-3]
+	\arrow["{\partial_{t}}", curve={height=-6pt}, from=2-1, to=1-1]
+	\arrow["{\dot{\gamma}}", from=2-1, to=1-3]
+	\arrow["\gamma"', from=2-1, to=2-3]
+\end{tikzcd}$$
+For a fixed vector field $X\in\Xfrak(M)$, an integral curve is a curve in $M$ that ``flows along'' the vector field in the sense that for each time $t$, the derivative of the curve $\dot{\gamma}(t)$ at a time $t$ is precisely the vector $X_{\gamma(t)}$ -- the element of $T_{\gamma(t)}M$ corresponding to $X_{\gamma(t)}$. 
+
+\begin{example}
+    Let $M=\RR^{2}$ and consider the vector field $\partial_{x}$ which associates to each point $(x,y)\in\RR^{2}$ the vector $(1,0)\in\RR^{2}=T_{p}\RR^{2}$ for all $p\in\RR^{2}$. The integral curves are curves of the form $t\mapsto p+t(1,0)$ with starting point $p$.  
+\end{example}
+\begin{example}
+    Let $M=\RR^{2}$ and consider the vector field $x\partial_{y}-y\partial_{x}$ which associates to each point $(x,y)$ the vector $(-y, x)$. Suppose that $\gamma:\RR\to\RR^{2}$ is an integral curve. Given as a component function, $\gamma$ necessarily satisfies $\dot{\gamma}^{1}(t)=-\gamma^{2}(t)$ and $\dot{\gamma}^{2}(t)=\gamma^{1}(t)$. For $p=(a,b)$ this system of ordinary differential equations is satisfied by the path $t\mapsto(a\cos(t)-b\sin(t), a\sin(t)+b\cos(t))$. 
+\end{example}
+More generally, the existence of integral curves is given by the solution to a system of ordinary differential equations. 
+\begin{theorem}[Existence-Uniqueness Theorem for Integral Curves]\label{thm: existence-uniqueness theorem integral curves}
+    Let $M$ be a smooth $m$-manifold and $X\in\Xfrak(M)$. For $p\in M$ there is an open interval $J\subseteq\RR$ containing 0 and an integral curve $\gamma:J\to M$ with starting point $p$. Moreover, $\gamma$ is unique. 
+\end{theorem}
+\begin{proof}
+    Let $M\subseteq\RR^{n}$ be open and we solve the system of differential equations 
+    $$\begin{cases}
+        \dot{\gamma}^{1}(t)=X^{1}_{\gamma(t)} \\
+        \vdots \\
+        \dot{\gamma}^{m}(t)=X^{m}_{\gamma(t)}
+    \end{cases}$$
+    which exists and is unique by the Picard-Lindel\"{o}f existence-uniqueness theorem for solutions to a system of differential equations. 
+\end{proof}
+The construction of integral curves is preserved under $F$-related vector fields. 
+\begin{lemma}\label{lem: integral curves preserved by F-related fields}
+    Let $F:M\to N$ be a morphism of smooth manifolds, $X\in\Xfrak(M), Y\in\Xfrak(N)$ with $X,Y$ $F$-related. Then $F$ takes integral curves of $X$ to integral curves of $Y$. 
+\end{lemma}
+\begin{proof}
+    Let $\gamma:(a,b)\to M$ be an integral curve. We compute $(F\circ \gamma)'(t)=dF_{\gamma(t)}\dot{\gamma}(t)$ which by $F$-relatedness is $Y_{F\circ\gamma(t)}$ as desired. 
+\end{proof}
+We conclude with the following definition. 
+\begin{definition}[Complete Vector Field]\label{def: complete integral curve}
+    Let $M$ be a smooth manifold and $X\in\Xfrak(M)$ a vector field. $X$ is a complete vector field if for all $p\in M$ the maximal integral curve at $p$ is defined on all of $\RR$. 
+\end{definition}
+\begin{example}
+    The vector field $\partial_{x}$ on $M=\RR\setminus\{0\}$ is an incomplete vector field. For $p=-1$ in $\RR$, teh integral curve starting at $p$ is the map $t\mapsto-1+t$ which is only defined on $(-\infty,1)$. 
+\end{example}

F4D1-Analysis-and-Geometry-on-Manifolds/Lecture 17.tex

@@ -0,0 +1,62 @@
+\section{Lecture 17 -- 7th December 2024}\label{sec: lecture 17}
+We begin with a discussion of flows. 
+\begin{definition}[Flow Domain]\label{def: flow domain}
+    Let $M$ be a smooth manifold. A flow domain is an open set $\Dcal\subseteq\RR\times M$ such that for all $p\in M$
+    $$\Dcal^{(p)}=\{t\in\RR:(t,p)\in\Dcal\}\subseteq\RR$$
+    is an open interval containing $0$. 
+\end{definition}
+More explicitly, $\Dcal$ admits two projection maps 
+$$% https://q.uiver.app/#q=WzAsMyxbMSwwLCJcXERjYWwiXSxbMCwxLCJcXFJSIl0sWzIsMSwiTSJdLFswLDFdLFswLDJdXQ==
+\begin{tikzcd}
+	& \Dcal \\
+	\RR && M
+	\arrow[from=1-2, to=2-1]
+	\arrow[from=1-2, to=2-3]
+\end{tikzcd}$$
+where the condition states that the fiber of the projection map $\Dcal\to M$ over $p\in M$ is an open interval of $\RR$ containing 0. 
+\begin{definition}[Flow]\label{def: flow}
+    Let $M$ be a smooth manifold and $\Dcal$ a flow domain. A flow is a map $\Psi:\Dcal\to M$ such that $\Psi(0,p)=p$ and $\Psi(t,\Psi(s,p))=\Psi(s+t,p)$ when $s,t,s+t\in\Dcal^{(p)}$. 
+\end{definition}
+\begin{remark}
+    Note that for fixed $t$, $\Psi(t,-)=\Psi_{t}:M\to M$ is a diffeomorphism as it is smooth with smooth inverse $\Psi(-t,-):M\to M$ and for fixed $p\in M$, $\psi(-,p)=\Psi^{(p)}:\Dcal^{(p)}\to M$ is a path in $M$ as $\Dcal^{(p)}$ is an interval in $\RR$ containing 0. 
+\end{remark}
+We can recover smooth vector fields from flows. 
+\begin{lemma}\label{lem: smooth vector field of a flow}
+    Let $M$ be a smooth manifold, $\Dcal$ a flow domain, and $\Psi:\Dcal\to M$ a flow. The map $p\mapsto \frac{d}{dt}|_{t=0}\Psi^{(p)}(t)$ defines a smooth vector field $X^{\Psi}$ on $M$ such that $\Psi^{(p)}:\Dcal^{(p)}\to M$ are integral curves of $X^{\Psi}$ starting at $p$.  
+\end{lemma}
+\begin{proof}
+    By \Cref{lem: smooth vector fields and functions} suffices to show that for all smooth functions $f\in C^{\infty}(M)$ that $X^{\Psi}f$ is smooth. We then compute for $p\in M$ that $X^{\Psi}f(p)=X^{\Psi}_{p}f=\frac{d}{dt}|_{t=0}(f\circ\Psi^{(p)}(t))=\partial_{t}(f\circ\Psi)(0,p)$ which is a smooth vector field by inspection. 
+
+    To show that $\Psi^{(p)}$ are the integral curves of $X^{\Psi}$ we fix some $t_{0}\in\Dcal^{(p)}, f\in C^{\infty}(M)$ and $q=\Psi(t_{0},p)$. We then compute 
+    \begin{align*}
+        X^{\Psi}_{q}f &= \Psi^{(q)}(0)f \\
+        &= \frac{d}{dt}|_{t=0}f(\Psi^{(q)}(t)) \\
+        &= \frac{d}{dt}|_{t=0}f(\Psi(t,\Psi(t_{0},p))) \\
+        &= \frac{d}{dt}|_{t=0}f(\Psi(t+t_{0},p)) \\
+        &= \frac{d}{dt}|_{t=0}f(\Psi^{(p)}(t+t_{0})) \\
+        &= \dot{\Psi}^{(p)}(t_{0})f
+    \end{align*}
+    as desired. 
+\end{proof}
+We now show the main result on flows. 
+\begin{theorem}[Flow]\label{thm: flow}
+    Let $M$ be a smooth manifold and $X\in\Xfrak(M)$ a smooth vector field. There exists a unique flow domain $\Dcal$ and a unqiue flow $\Psi:\Dcal\to M$ such that $X^{\Psi}=X$ and $\Psi^{(p)}:\Dcal^{(p)}\to M$ are the maximal integral curves of $X$. 
+\end{theorem}
+\begin{proof}
+    To define $\Psi$, we take $(t,p)\in\Dcal$ to $\gamma^{(p)}$ the unique integral curve starting at $p$, the existence of which is given by \Cref{thm: existence-uniqueness theorem integral curves}. This satisfies the hypotheses of \Cref{lem: smooth vector field of a flow} which gives the desired construction. 
+\end{proof}
+
+We now turn to a discussion of the Lie derivative, which seeks to generalize derivatives of functions on an arbitrary manifold. 
+\begin{definition}[Lie Derivative]\label{def: Lie derivative}
+    Let $M$ be a smooth manifold and $X,Y\in\Xfrak(M)$ smooth vector fields. The Lie derivative of $Y$ with respect to $X$ at a point $p\in M$ is given by 
+    $$\lim_{t\to0}(\Lcal_{X}Y)_{p}=\frac{d}{dt}|_{t=0}\left(\frac{Y_{p+tX_{p}}-Y_{p}}{t}\right).$$
+\end{definition}
+This in fact defines a smooth vector field. 
+\begin{lemma}\label{lem: Lie derivative is a smooth vector field}
+    Let $M$ be a smooth manifold and $X,Y\in\Xfrak(M)$ smooth vector fields. The Lie derivative $\Lcal_{X}Y$ of $Y$ with respect to $X$ is a smooth vector field. 
+\end{lemma}
+\begin{proof}
+    Let $\theta$ be the flow for $X$ and for $p\in M$, let $(U,\phi)$ be a smooth chart of $M$ containing $p$. Let $J\subseteq\RR$ be an open interval containing 0 and $U_{0}\subseteq U$ containing $p$ such that $\theta(J_{0}\times U_{0})\subseteq U$. For $(t,q)\in J_{0}\times U_{0}$, the Jacobian matrix of the differential is given by 
+    $$(d\Psi_{-t})_{\Psi(t,x)}=(\partial_{x_{i}}\Psi^{j}(-t,x))_{1\leq i,j\leq n}.$$
+    As such, we have $d\Psi_{-t}Y_{\Psi(t,x)}=(\partial_{x_{i}}\Psi^{j}(-t,x))_{1\leq i,j\leq n}$ giving the claim. 
+\end{proof}

V5A2-Habiro-Rings/Habiro_Rings_Notes.tex

@@ -61,7 +61,8 @@ \section*{Preliminaries}
 \include{Lecture 7}
 \include{Lecture 8}
 \include{Lecture 8 Computations}
-\newpage
+
+
 \printbibliography
 
 \end{document}

V5B1-Advanced-Topics-in-Complex-Analysis/Advanced_Complex_Analysis_Notes.tex

@@ -45,6 +45,7 @@
 \maketitle
 \section*{Preliminaries}
 These notes roughly correspond to the course \textbf{V5B1 - Advanced Topics in Complex Analysis} taught by Prof. Ingo Lieb at the Universit\"{a}t Bonn in the Winter 2024/25 semester. These notes are \LaTeX-ed after the fact with significant alteration and are subject to misinterpretation and mistranscription. Use with caution. Any errors are undoubtedly my own and any virtues that could be ascribed to these notes ought be attributed to the instructor and not the typist. 
+\newpage
 \tableofcontents
 \include{Lecture 1}
 \include{Lecture 2}
@@ -61,6 +62,7 @@ \section*{Preliminaries}
 \include{Lecture 13}
 \include{Lecture 14}
 \include{Lecture 15}
+\include{Lecture 16}
 \appendix
 \include{Appendix}
 \newpage

V5B1-Advanced-Topics-in-Complex-Analysis/Lecture 13.tex

@@ -24,10 +24,10 @@ \section{Lecture 13 -- 21st November 2024}\label{sec: lecture 13}
 \end{definition}
 We can provide a sufficient criterion for convergence. 
 \begin{proposition}\label{prop: sufficient conditions for convergence}
-    Let $f(z)=\sum_{\alpha\in\ZZ_{\geq0}^{n}}c_{\alpha}z^{\alpha}$ be a formal complex power series and $\tau\in\CC^{n}\setminus\{0\}$ such that $|c_{\alpha}\tau^{\alpha}|$ is bounded. Then $f(z)$ is uniformly convergent for all $z\in \overline{B_{|\tau|}(0)}^{\circ}$.\marginpar{Proposition 3.1} 
+    Let $f(z)=\sum_{\alpha\in\ZZ_{\geq0}^{n}}c_{\alpha}z^{\alpha}$ be a formal complex power series and $\tau\in\CC^{n}\setminus\{0\}$ such that $|c_{\alpha}\tau^{\alpha}|$ is bounded. Then $f(z)$ is uniformly convergent for all $z\in B_{|\tau|}(0)$.\marginpar{Proposition 3.1} 
 \end{proposition}
 \begin{proof}
-    For any $z\in \overline{B_{|\tau|}(0)}^{\circ}$, we have $|z|=\lambda|\tau|$ for some $0<\lambda<1$. Thus by boundedness of $|c_{\alpha}z^{\alpha}|\leq|c_{\alpha}\tau^{\alpha}|\cdot\lambda^{|\alpha|}$ and the claim follows from $\sum_{\alpha\in\ZZ_{\geq0}^{n}}\lambda^{|\alpha|}$ being convergent. 
+    For any $z\in B_{|\tau|}(0)$, we have $|z|=\lambda|\tau|$ for some $0<\lambda<1$. Thus by boundedness of $|c_{\alpha}z^{\alpha}|\leq|c_{\alpha}\tau^{\alpha}|\cdot\lambda^{|\alpha|}$ and the claim follows from $\sum_{\alpha\in\ZZ_{\geq0}^{n}}\lambda^{|\alpha|}$ being convergent. 
 \end{proof}
 More generally, we can write holomorphic functions as a convergent power series in the following way. 
 \begin{theorem}\label{thm: power series expansion of holomorphic functions}

V5B1-Advanced-Topics-in-Complex-Analysis/Lecture 15.tex

@@ -19,11 +19,12 @@ \section{Lecture 15 -- 28th November 2024}\label{sec: lecture 15}
 \begin{proposition}\label{prop: relatively prime germs}
     Let $U\subseteq\CC^{n}$ open containing the origin and $f,g\in\Ocal(U)$. If $f_{0},g_{0}$ are relatively prime in $\Ocal_{0}$ then there is a neighborhood $V$ of the origin such that $f_{z},g_{z}$ are relatively prime for all $z\in V$. 
 \end{proposition}
+In turn:
 \begin{theorem}\label{thm: convergent power series is UFD}
     The ring of convergent power series $\CC\{z_{1},\dots,z_{n}\}$ is a unique factorization domain. 
 \end{theorem}
 We now discuss meromorphic functions. 
 \begin{definition}[Sheaf of Meromorphic Functions]\label{def: sheaf of meromorphic functions}
-    The sheaf of meromorphic functions $\Kcal$ on $\CC^{n}$ is given by $\coprod_{z\in\CC^{n}}\Kcal_{z}$ where $\Kcal_{z}$ is the germs of meromorphic functions at $z$. 
+    The sheaf of meromorphic functions $\Kcal$ on $\CC^{n}$ is given by $\coprod_{z\in\CC^{n}}\Kcal_{z}$ where $\Kcal_{z}$ is the ring of germs of meromorphic functions at $z$. 
 \end{definition}
 Sections of $\Kcal$ give rise to meromorphic functions in the same way as \Cref{def: continuous section}.