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\input{nielsenmacros}

\title[Circle bouquets]{Fixed point and coincidence theory on circle bouquets}
\subtitle{in ten minutes}
\author[Staecker]{Chris Staecker}
\date[MEB]{MEB school meeting, March 25 2009}

\begin{document}

\frame{\titlepage}

\frame{
  \frametitle{Fixed Points}
  A fixed point is some point $x$ with $f(x) = x$. We can find them on a
  graph like so:
  \begin{center}
  \only<1>{\includegraphics[scale=.5]{fpgraph}}
  \only<2>{\includegraphics[scale=.5]{fpgraphdiag}}
  \only<3>{\includegraphics[scale=.5]{fpgraphcircled}}
\end{center}
}

\frame{
  \frametitle{Doing it on a circle}
  Let's try the same thing, only instead of an ordinary function with
  numbers, we want $f$ to be a function which takes points on a circle
  and gives back other points on a circle.
  \pause

  We could visualize it like so:
  
  \begin{center}
\includegraphics[scale=.3]{a5arrow.jpg}
\end{center}
}

\frame{
  \frametitle{Doing it on a circle}
  Really we only need to draw the first picture.

\begin{center}
\includegraphics[scale=.25]{a5alone.jpg}
\end{center}

  This function we write algebraically as
\[ f: a \mapsto a^5. \]
}

\frame{
  \frametitle{The fixed points}
  To find the fixed points, we compare the images to the circle
  itself, and find the ``intersections'', i.e.\ the places where the
  color matches up:

\begin{center}
\only<1>{\includegraphics[scale=.25]{a5diag.jpg}}
\only<2,3,4>{\includegraphics[scale=.25]{a5circled.jpg}}
\end{center}

\pause\pause
  Not too hard to see that the number of fixed points of $f(a)= a^n$
  is \pause $n-1$.
}

\frame{
  \frametitle{A circle bouquet}
  Instead of a circle, now lets do it on a set of circles sharing a
  common base point. 

\pause

  This we could visualize like:
\begin{center}
\includegraphics[scale=.25]{bigmaparrow.jpg}
\end{center}
}

\frame{
  \frametitle{A circle bouquet}
  Again, we only need to draw one of these. 

\begin{center}
\includegraphics[scale=.25]{bigmapalone.jpg}
\end{center}

This one is written algebraically as
\[ f: \begin{array}{rcl} a & \mapsto & baba^2 \\
                         b & \mapsto & a^2 b^{-1}ab^3 \end{array} \]
}

\frame{
  \frametitle{The fixed points}
  Again, to get the fixed points, we compare the ``graph'' to the
  plain circles $a$ and $b$ and find the spots where the color
  matches:

\begin{center}
\only<1>{\includegraphics[scale=.25]{bigmapdiag.jpg}}
\only<2>{\includegraphics[scale=.25]{bigmapcircled.jpg}}
\end{center}
}

\frame{
  \frametitle{Fadell \& Husseini}
Work in 1981 by Fadell \& Husseini: The fixed points of a function $f$
on a bouquet of circles obey the formula:
\[ \RT(f) = -1 + \sum_{x \in A} \frac{\pd}{\pd x} f(x). \]
Where $\pd$ is the ``Fox derivative'' (similar to the partial
derivative, but purely algebraic), and $A$ is the set of circles in
the bouquet.
}

\frame{
\frametitle{The example}
Let's apply the Fadell \& Husseini formula to our example from before:
\[ f: \begin{array}{rcl} a & \mapsto & baba^2 \\
                         b & \mapsto & a^2 b^{-1}ab^3 \end{array} \]

\begin{eqnarray*} 
\RT(f) &=& -1 + \sum_{x=a,b} \frac{\pd}{\pd x} f(x) \\
\pause &=& -1 + \left( \frac{\pd}{\pd a}(baba^2) + \frac{\pd}{\pd
  b}(a^2b^{-1}ab^3) \right) \\
\pause &=& -1 + b + bab + baba - a^2b^{-1} + a^2b^{-1}a + a^2b^{-1}ab
+ a^2b^{-1}ab^2 \\
\pause &=& 1 + b + bab - a^2b^{-1} + a^2b^{-1}a + a^2b^{-1}ab \quad \text{(reduced)}
\end{eqnarray*}
}

\frame{
\frametitle{Example}
Compare this to the picture:
\[ \RT(f) = 1 + b + bab - a^2b^{-1} + a^2b^{-1}ab + a^2b^{-1}a \]

\begin{center}
\includegraphics[scale=.25]{bigmapcircled.jpg}
\end{center}

Each term in the sum represents a fixed point, the signs tell you if
it was ``forward'' or ``backward'', and the algebraic words themselves
give very important topological information.
}

\frame{
\frametitle{Photo interlude}
I got interested in this at a conference in B\c{e}dlewo, Poland.

\begin{center}
\includegraphics[scale=.25]{bedlewocastle.jpg}
\end{center}
}

\frame{
\frametitle{Photo interlude}
Big birds, small church.

\begin{center}
\includegraphics[scale=.25]{bedlewostorks.jpg}
\end{center}
}

\frame{
\frametitle{Coincidence theory}
We want to do all this for \emph{coincidence points} of two functions:
\[ f(x) = g(x). \]

\begin{center}
\only<1>{\includegraphics[scale=.5]{coingraphs.jpg}}
\only<2>{\includegraphics[scale=.5]{coingraphscircled.jpg}}
\end{center}
}

\frame{
\frametitle{Circle bouquets in coincidence theory}
This time, we compare $f$ to $g$ directly and look for the
matchups. This example is for 
\[ f: \begin{array}{rcl} a & \mapsto & baba^2 \\
                         b & \mapsto & a^2 b^{-1}ab^3 \end{array}
\quad
g: \begin{array}{rcl} a & \mapsto & ba^{-1}b^{-2} \\
                      b & \mapsto & ab^2a \end{array} \]

\begin{center}
\only<1>{\includegraphics[scale=.25]{bigcoin.jpg}}
\only<2>{\includegraphics[scale=.25]{bigcoincircled.jpg}}
\end{center}
}

\frame{ 
\frametitle{A formula?}
My paper ``A formula for the coincidence Reidemeister trace of
selfmaps on bouquets of circles'' (\emph{Topological
  Methods in Nonlinear Analysis} {\bf 33} 2009) works towards an analogue of the Fadell \&
Husseini formula.
\pause

\vspace{.5in}
The paper gets it wrong though-- still much work to be done.
}

\frame{
\frametitle{Our example}
For our example above:
\begin{center}
\includegraphics[scale=.25]{bigcoincircled.jpg}
\end{center}
We can compute:
\begin{align*}
\RT(f,g) = \pm 1 \pm b^{-1}ab^{-1} \pm b^{-2}ab^{-1}a \pm a \pm  b
\end{align*}

We still don't know what the signs are-- some of them may even be zero!
\vfill
}

\frame{
\frametitle{Still working}
Still working\dots
\vspace{.5in}
It will take a while.
}

\end{document}
