add a graphicspath to everything that had graphics, except Real Analy…

…sis because it only had one graphic

Algebraic Topology/algebraic-topology.tex

@@ -27,6 +27,8 @@
   \end{dateenv}
 }
 
+\graphicspath{{./images/}}
+
 \numberwithin{thm}{section}
 
 \everymath{\displaystyle}
@@ -259,7 +261,7 @@ \section{Homotopy}
 \begin{example}
   The isomorphism above is not canonical, because we could use path $p$ or $q$ to create the isomorphism, but $p$ and $q$ are not homotopic, as shown below.
 
-  \includegraphics[scale=0.2]{images/isomorphism-not-canonical}
+  \includegraphics[scale=0.2]{isomorphism-not-canonical}
 
   Note that $\fund X$ is only well defined up to inner automorphism.  ((What does this mean?))
 \end{example}
@@ -282,7 +284,7 @@ \section{Homotopy}
   Given a fundamental group $\fund(X, x)$, then we can restrict $f_*$
   to the functor $$f_* \from \fund(X, x) \to \fund(Y, f(x)).$$ which is a group homomorphism.
 
-  \includegraphics[scale=0.17]{images/fundamental-functor}
+  \includegraphics[scale=0.17]{fundamental-functor}
 \end{prop}
 \begin{prop}
   If $X,Y,Z$ are metric spaces and $f \from X \to Y$, $g \from Y \to Z$ are
@@ -615,12 +617,12 @@ \section{Classification of covering spaces (Hatcher 1.3)}
   Let $K \subset \R^3$ be a knot, that is, a smoothly embedded $S^1$.
   In this example, we use $K =$ ``trefoil'', below
 
-  \includegraphics[scale=0.15]{images/trefoil.jpg}
+  \includegraphics[scale=0.15]{trefoil.jpg}
 
   What is $\fund(\R^3\minus K)$?  Draw two sets of open intervals,
   $A'$ in red and $B'$ in blue, as shown.
 
-  \includegraphics[scale=0.2]{images/trefoil-with-red-and-blue.jpg}
+  \includegraphics[scale=0.2]{trefoil-with-red-and-blue.jpg}
 
   What looks like the ``ends'' of the ``line segments'' in $A'$, for
   example, are in fact the height of the lines going down to negative
@@ -648,12 +650,12 @@ \section{Classification of covering spaces (Hatcher 1.3)}
 
   ((do we invoke Van Kampen's theorem here?))
 
-  \includegraphics[scale=0.2]{images/mountains.jpg}
+  \includegraphics[scale=0.2]{mountains.jpg}
 
   By sliding $a$ under the bridge, we see that $a$ and $b$ are
   conjugate via $c$!
 
-  \includegraphics[scale=0.2]{images/mountains-with-explanation.jpg}
+  \includegraphics[scale=0.2]{mountains-with-explanation.jpg}
 
   Finally, to compute the fundamental group of $K^C$, we pick an
   arbitrary basepoint $x$ and put loops around the knot as follows.
@@ -668,7 +670,7 @@ \section{Classification of covering spaces (Hatcher 1.3)}
   is $${F_m}/{(a=c*b*c^{-1}, b=a*c*a^{-1}, c=b*a*b^{-1})}$$, as
   illustrated below.
 
-  \includegraphics[scale=0.2]{images/trefoil-fully-described.jpg}
+  \includegraphics[scale=0.2]{trefoil-fully-described.jpg}
 \end{example}
 \begin{thm}[Wirtinger]
   For any knot $K$, $\fund(\R^3\minus K)$ is generated by loops
@@ -770,7 +772,7 @@ \section{CW complexes and the fundamental group}
   If $X$ is a space and $A$ is a subspace, then the \de{quotient space} $X/A$ is the partition of $X$ resulting from the equivalence relation $$(x \sim x') \quad \text{iff} \quad (x = x' \quad \text{or} \quad x, x' \in A).$$
 \end{defn}
 \begin{example}
-  \includegraphics[scale=0.2]{images/quotient-space}
+  \includegraphics[scale=0.2]{quotient-space}
 \end{example}
 \begin{thm}
   If $X$ is a CW complex and $A$ a contractible subcomplex, $\fund(X) = \fund(X/A)$.
@@ -808,7 +810,7 @@ \subsection{How to glue one disk onto $X_1$}
   So it's just like a group presentation!
 \end{proof}
 \begin{example}
-  \includegraphics[scale=0.4]{images/cw-complex-group-presentation}
+  \includegraphics[scale=0.4]{cw-complex-group-presentation}
 \end{example}
 \begin{cor}
   Every finitely presented group is the fundamental group of a finite CW complex.
@@ -890,7 +892,7 @@ \subsection{How to glue one disk onto $X_1$}
   Given $X$ is connected and SLSC, and $p_1$ and $p_2$ are covering maps, then if a map from $X_1$ to $X_2$ exists, it must be unique and is $\fund$ - equivariant.
 \end{thm}
 
-\includegraphics[scale=0.4]{images/fund-equivariant}
+\includegraphics[scale=0.4]{fund-equivariant}
 
 \begin{defn}
   A \de{$\fund$-set} is a set that $\fund$ acts on.
@@ -921,7 +923,7 @@ \subsection{How to glue one disk onto $X_1$}
   $gHg^{-1} \isom \fund(X^U/H, (q_0, gH) ) \to \fund(X_1, x_0)$
 \end{proof}
 
-\includegraphics[scale=0.3]{images/universal-cover-f2}
+\includegraphics[scale=0.3]{universal-cover-f2}
 
 \begin{example}
   We will use $H =  \langle a \rangle $ in our image above.
@@ -930,12 +932,12 @@ \subsection{How to glue one disk onto $X_1$}
 
   Now look at the image of $\fund(X^U/ \langle a \rangle )$.
 
-  \includegraphics[scale=0.2]{images/loop-tree-cover}
+  \includegraphics[scale=0.2]{loop-tree-cover}
 
   Note that this is homotopic to a loop!  Woohoo!
 \end{example}
 \begin{example}
-  \includegraphics[scale=0.26]{images/loopy}
+  \includegraphics[scale=0.26]{loopy}
 
   $1 \mapsto a$
 
@@ -1078,13 +1080,13 @@ \subsection{Covering spaces are analagous to Galois theory}
 
 So if $g_1, g_2 \in G$, then $g_1$, $g_2$, and $g_1g_2$ are all generators!  They each get a loop!.  But then, of course, we will need to glue faces to build the group.
 
-\includegraphics[scale=0.3]{images/group-as-fund-group}
+\includegraphics[scale=0.3]{group-as-fund-group}
 
 So for example, $g_1.g_2 = g_1g_2$, where the LHS is the directed edge concatenation, and the RHS is its very own directed edge.  The picture for this is the triangle.
 
 Similarly, we can create higher dimensional simplices.  For example $g_1g_2g_3 = g_1.(g_2g_3)$ in the tetrahedron simplex:
 
-\includegraphics[scale=0.4]{images/tetrahedron}
+\includegraphics[scale=0.4]{tetrahedron}
 
 I have an $n$-cell for every $n$-tuple
 
@@ -1140,9 +1142,9 @@ \section*{Simplicial homology}
   convenient than simplicial complexes.  The following are examples of
   $\Delta$-complexes that are not simplicial complexes.
   \begin{itemize}
-  \item \includegraphics[scale=0.22]{images/not-delta-complex1}
-  \item \includegraphics[scale=0.2]{images/not-delta-complex2}
-  \item \includegraphics[scale=0.2]{images/not-delta-complex3}
+  \item \includegraphics[scale=0.22]{not-delta-complex1}
+  \item \includegraphics[scale=0.2]{not-delta-complex2}
+  \item \includegraphics[scale=0.2]{not-delta-complex3}
   \end{itemize}
   $\Delta$-complexes allow you to glue vertices together, but
   simplicial complexes do not. ((John Harnois))
@@ -1194,7 +1196,7 @@ \section*{Simplicial homology}
 \begin{example}[Hatcher, p.105]
   \quad
 
-  \includegraphics[scale=0.5]{images/boundary-of-oriented-simplex}
+  \includegraphics[scale=0.5]{boundary-of-oriented-simplex}
 \end{example}
 
 Claim: $\d^2 = 0$.
@@ -1205,7 +1207,7 @@ \section*{Simplicial homology}
 \end{defn}
 
 \begin{example}
-  \includegraphics[scale=0.2]{images/klein-bottle-CW}
+  \includegraphics[scale=0.2]{klein-bottle-CW}
 
   Consider $K =$ the klein bottle above.  The chain complex is
 
@@ -1460,7 +1462,7 @@ \subsection{Reduced homology $\rhoml(X)$}
   We say that two cycles $x, y \in C_n$ are \de{homologous} if their difference is a boundary.
 \end{defn}
 \begin{example}
-  \includegraphics[scale=0.4]{images/homologous}
+  \includegraphics[scale=0.4]{homologous}
 
   In $C_1(X,A)$, consider $x$ and $y$ are the same in $X \minus A$, but different in $A$.  Then $x ~ a+b = a  = a+c ~ y$.
 \end{example}
@@ -1560,12 +1562,12 @@ \subsection{Reduced homology $\rhoml(X)$}
 \begin{example}
   What is $T(\text{interval})$?
 
-  \includegraphics[scale=0.3]{images/table-example-2d}
+  \includegraphics[scale=0.3]{table-example-2d}
 \end{example}
 \begin{example}
   What is $T(\text{triangle})$?
 
-  \includegraphics[scale=0.3]{images/table-example-3d}
+  \includegraphics[scale=0.3]{table-example-3d}
 \end{example}
 \begin{thm}
   $\d T + T \d = 1 - S$, where $1$ is the identity function.
@@ -1619,7 +1621,7 @@ \subsection{Reduced homology $\rhoml(X)$}
   $(X, A)$ is a \de{good pair} if $A$ is a closed subspace of $X$ and there exists an open $V \supset A$ s.t. $A \into V$ is a deformation retract.
 \end{defn}
 \begin{example}
-  \includegraphics[scale=0.18]{images/good-pair-1} \includegraphics[scale=0.16]{images/good-pair-2}
+  \includegraphics[scale=0.18]{good-pair-1} \includegraphics[scale=0.16]{good-pair-2}
 \end{example}
 \begin{thm}
   If $A \subset X$ is a good pair, then $$\homl(X, A) \isom \rhoml(X/A) \isom \homl(X/A, A/A).$$
@@ -2118,7 +2120,7 @@ \subsection{Mayer-Vietoris sequence}
 
 Let \(i \from A \intersect B \to A\), \(i' \from A \intersect B \to
 B\), \(j \from A \to X\), and \(j' \from B \to X\) be the standard
-inclusions. Then, we have the following commutative diagram. (We need that $i_* j_* = i'_* j'_*$, so we make one of the i's negative.) 
+inclusions. Then, we have the following commutative diagram. (We need that $i_* j_* = i'_* j'_*$, so we make one of the i's negative.)
 
 $$\begin{tikzcd}
   &\, & \homl(A) \arrow[rd, "j_*"] \\
@@ -2142,7 +2144,7 @@ \subsection{Mayer-Vietoris sequence}
   we get the sequence \(0 \to \homl(X) \to \homl[n-1](A \intersect B)
   \to 0 \oplus 0 \isom 0\). However, since \(\homl[n-1](A \intersect
   B) \isom \homl[n-1](S^{n-1}) \isom \Z\), we get that \(\homl(X)
-  \isom \Z\). 
+  \isom \Z\).
 \end{example}
 \begin{proof}
   Consider
@@ -2165,7 +2167,7 @@ \subsection{Mayer-Vietoris sequence}
 \begin{example}
   Consider the Klein bottle $K^2$ with the CW-complex structure:
 
-  \includegraphics[scale=0.15]{images/klein-cw-mayer-vietoris}
+  \includegraphics[scale=0.15]{klein-cw-mayer-vietoris}
 
   where $A$ and $B$ are both mobius bands, and so denoted by $M$.
 
@@ -2190,7 +2192,7 @@ \subsection{Mayer-Vietoris sequence}
 \begin{example}
   Consider the torus $T^2$ with the structure:
 
-  \includegraphics[scale=0.15]{images/torus-cw-mayer-vietoris}
+  \includegraphics[scale=0.15]{torus-cw-mayer-vietoris}
 
   where $A$ and $B$ are both annuli ($S^1 \x I$), and so denoted by $M$.
 
@@ -2356,7 +2358,7 @@ \subsection{Mayer-Vietoris sequence}
 
 $k = 1, n = 3$.  No!  There exist knots!  For example,
 
-\includegraphics[scale=0.1]{images/trefoil}
+\includegraphics[scale=0.1]{trefoil}
 
 $k = n-2$.  Higher dimensional knot theory.
 
@@ -2367,7 +2369,7 @@ \subsection{Mayer-Vietoris sequence}
 
   The Alexander horned sphere is obtained as follows. Take a torus, cut it, and insert two handles at the ends that interlock.  Then cut each handle and perform the same trick.  Repeat this process forever.
 
-  \includegraphics[scale=0.05]{images/Alexander_horned_sphere}
+  \includegraphics[scale=0.05]{Alexander_horned_sphere}
 
   Since the handles split up forever, they never actually connect back to themselves.  So this surface actually has genus 0 and is homeomorphic to $S^2$.  Wowzers!
 \end{example}

Complex Analysis/complex-analysis-notes.tex

@@ -1,6 +1,7 @@
 \documentclass[11pt,leqno,oneside]{amsart}
 
 \usepackage{../notes}
+\graphicspath{{./images/}}
 \numberwithin{thm}{section}
 
 \newcommand{\Arg}{\operatorname{Arg}}
@@ -385,13 +386,13 @@ \section{(9/6/2016) Lecture 5: Landau Notation and Cauchy-Riemann Equations}
 
     \begin{figure}[h]
         \centering
-        \includegraphics[scale=0.2]{images/2i-to-one-third.png}
+        \includegraphics[scale=0.2]{2i-to-one-third.png}
         \caption{$2i^{\frac{1}{3}}$}
         \label{fig:2i13}
     \end{figure}
     \begin{figure}[h]
         \centering
-        \includegraphics[scale=0.2]{images/2i-to-1-over-pi.png}
+        \includegraphics[scale=0.2]{2i-to-1-over-pi.png}
         \caption{$2i^{\frac{1}{\pi}}$ first $k = 0,1, \ldots, 100$}
         \label{fig:2i1pi}
     \end{figure}
@@ -1252,7 +1253,7 @@ \section{(10/6/2016) Lecture 13}
   on overlaps. If such an extension exists, it is unique. Here is a
   picture illustrating this on the function $\log(z)$. The graphic is
   from Wikipedia and created by Yamashita Makoto. \\
-  \includegraphics{images/Imaginary_log_analytic_continuation.png}
+  \includegraphics{Imaginary_log_analytic_continuation.png}
   In general, a Riemann surface is equivalent to the set of analytic
   continuations modulo the equivalence relation that two analytic
   continuations yield the same function at the endpoint.
@@ -1710,7 +1711,7 @@ \section{(10/6/2016) Lecture 13}
   \subsection*{Integrands with branch points.}
   Integrands involving $\log x$ or $x^a$ have branch points and one
   way to deal with these is to integrate around a ``keyhole contour.'' \\
-  \includegraphics[scale=0.5]{images/keyhole_contour.png} \\
+  \includegraphics[scale=0.5]{keyhole_contour.png} \\
   \begin{example}
     Consider $\int_0^\infty \frac{x^{-a}}{1+x}dx, 0 < a < 1$. We will
     compute $\oint \frac{z^{-a}}{1+z}dz$ where the branch