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\newcommand{\fpause}{\pause\vfill}

\title{All kinds of big: Hadwiger's theorem}
\author[Staecker]{P. Christopher Staecker}
\institute[Fairfield U.]{Fairfield University, Fairfield CT}
\date[]{July 16, 2013}

\begin{document}

\frame{\titlepage}

\frame{
Goal: \pause Describe every possible type of notion of ``bigness'' for subsets in space.
\fpause
{\bf Hadwiger's Theorem}: 

If $v$ is a measure of bigness for sets in $\R^n$, then $v$ must have the form \dots
}

\frame{
\[ \includegraphics[width=300px]{jokes}\pause \] 
\[\text{A graph of jokes per slide.}\]
}

\frame{
\[\text{\it every possible type of measure of ``bigness'' for subsets in space} \]
\fpause
``measure of bigness for subsets in space'' means a function $v$ which assigns a real number ``size'' to a subset of $\R^n$
\fpause
Such a function should obey three properties:\pause
\begin{itemize}
\item \emph{Rigid-motion invariant} The size never changes if you translate or rotate the set\pause
\item \emph{Continuity} The size changes a little bit if we change the set a little bit\pause
\item \emph{Valuation} $v(\emptyset) = 0$ and 
\[ v(A \cup B) = v(A) + v(B) - v(A\cap B). \]
\pause
``inclusion-exclusion''
\end{itemize}
}

\frame{
\[ v(A \cup B) = v(A) + v(B) - v(A\cap B) \]
\pause
\[ 
\only<2>{\includegraphics[width=200px, height=100px]{bigshape1}}
\only<3>{\includegraphics[width=200px, height=100px]{bigshapeA}}
\only<4>{\includegraphics[width=200px, height=100px]{bigshapeB}}
\only<5->{\includegraphics[width=200px, height=100px]{bigshapeAB}}
 \]
\pause Split it into subsets $A$ and $B$.
\fpause\pause\pause
Then this says:
\[ v \left( \vcenter{\hbox{\includegraphics[width=80px]{AcupB}}} \right) = 
v \left(\vcenter{\hbox{\includegraphics[width=50px, ]{A}}} \right) +
v \left(\vcenter{\hbox{\includegraphics[width=50px, ]{B}}} \right) -
v \left(\vcenter{\hbox{\includegraphics[width=30px, ]{AcapB}}} \right)
 \]
}

\frame{
So by ``measure of bigness'' we technically mean:
\fpause
A continuous\pause\ invariant\pause\ valuation\pause\ defined on subsets of $\R^n$
\fpause
Big words, but this is the bare minimum of what ``bigness'' could mean.
\fpause
The area is one such function, but there are many others.
}

\frame{
Technical interlude:
\fpause
Actually we need to be a bit careful about what kinds of subsets are allowed.\pause\ Crazy subsets will mess up the theory.
\fpause
If \emph{any} subsets are allowed, we'll have the ``Banach-Tarski Paradox''\pause: a set whose volume is 1 can be chopped up into crazy subsets and reassembled so that the volume is 2.
\fpause
Moral: volume and area of ``pathological'' sets don't add up the way we expect. 
}

\frame{
To disallow this kind of pathological behavior, we will require our subsets to be closed and ``polyconvex''.
\fpause
A set is convex when the straight line connecting any two points in the set lies entirely in the set.
\fpause
Polyconvex means any finite union of convex sets.
\fpause
Any polygonal-type shape is polyconvex, \pause and any ``ordinary'' shape you can think of is arbitrarily close to a polyconvex set.
\fpause
Our continuity assumption is actually ``continuity on convex sets''
}

\frame{
Some examples of continuous invariant valuations: 
\fpause
In $\R^2$, the area.
\[ v \left( \vcenter{\hbox{\includegraphics[width=80px]{AcupB}}} \right) = 
v \left(\vcenter{\hbox{\includegraphics[width=50px, ]{A}}} \right) +
v \left(\vcenter{\hbox{\includegraphics[width=50px, ]{B}}} \right) -
v \left(\vcenter{\hbox{\includegraphics[width=30px, ]{AcapB}}} \right)
 \]
\pause
Also the perimeter!
}

\frame{
In $\R^3$:
\[ \includegraphics[width=150px]{spheres} \]
\pause we have:
\begin{itemize} 
\item the surface area\pause: $3 \cdot 4\pi = 12\pi$\pause
\item the ``perimeter''\pause: 3\pause
\item the volume\pause: $3 \cdot (\frac43 \pi) = 4\pi$
\end{itemize}
}

\frame{
\begin{itemize} 
\item the volume: \pause 3 dimensional size\pause
\item the surface area: \pause size of the 2 dimensional ``edge''\pause
\item the ``perimeter'': \pause size of the 1 dimensional ``edge'' (if any)
\end{itemize}
\fpause
These are the \emph{intrinsic volumes} of dimension 3, 2, 1.
\fpause
For higher dimensional spaces, there are higher dimensional intrinsic volumes.
}

\frame{
\[ \includegraphics[width=300px]{jokesbefore} \]
}

\frame{
The intrinsic volumes in each dimension are continuous invariant valuations.
\fpause
Are there any others?
\fpause
Yes there are.
}

\frame{
Our goal is to describe all possible continuous invariant valuations.
\fpause
``All kinds of big''
}

\newcommand{\pp}[2]{%
\frame{
\[ \includegraphics[#2]{#1} \]
}%
}

\pp{bigposter}{width=100px}
\pp{mrbigsex}{height=150px}
\pp{mrbigband}{height=150px}
\pp{mrbigbond}{height=150px}
\pp{biggie.jpg}{height=200px}

\frame{
\[\includegraphics[width=300px]{jokesafter} \]
}

\frame{
Besides the intrinsic volumes, are there any other continuous invariant valuations?
\fpause
Stupid answer: \pause ``2 times the area'' \pause\ (it's not the same as the area!)
\fpause
Actually any continuous invariant valuation can be multiplied by a constant and the result is another continuous invariant valuation.
\fpause
Really stupid answer: \pause zero
}

\frame{
You could also do ``perimeter plus area''
\fpause
Any sum of two continuous invariant valuations is a continuous invariant valuation.
\fpause
So the set of continuous invariant valuations is a\pause\ vectorspace.
\fpause
So there are infinitely many of them, but we can still try to find a basis for the space.
}

\frame{
Other than the intrinsic volumes, are there any other \emph{really different} continuous invariant valuations?
\fpause
There are!
}

\frame{
Define a valuation $\chi$ like so:\pause\ If $A$ is convex, then $\chi(A)= 1$. \pause\ Otherwise, compute $\chi$ in terms of smaller convex sets using the valuation property.
\fpause
So for this:
\[ \includegraphics[width=150px]{chi1} \]
we have $\chi(X) = 1$.
}

\frame{
What about this:
\[ 
\only<1>{\includegraphics[width=150px]{chi2} }
\only<2->{\includegraphics[width=150px]{chi3}}
\]
Not convex, so break it up.\pause\pause
\begin{align*} 
\onslide<3->{\chi\left( \vcenter{\hbox{\includegraphics[width=40px]{chi2}}} \right)  &= 
\chi\left( \vcenter{\hbox{\includegraphics[width=15px]{sector}}} \right)  +
\chi\left( \vcenter{\hbox{\includegraphics[width=15px, angle=90]{sector}}} \right)  + 
\chi\left( \vcenter{\hbox{\includegraphics[width=15px, angle=180]{sector}}} \right)  + 
\chi\left( \vcenter{\hbox{\includegraphics[width=15px, angle=270]{sector}}} \right)  \\
&\qquad- \chi(\,\pmb{\diagup}\,) - \chi(\,\pmb{\diagdown}\,) - \chi(\,\pmb{\diagup}\,) - \chi(\,\pmb{\diagdown}\,)} 
\\ 
\onslide<4->{&= 1 + 1 + 1 + 1 - 1 - 1 - 1 - 1 = 0}
\end{align*}
}

\frame{
We can do this computation in a more systematic way:
\[ \includegraphics{triangulation} \]
\pause
Make a \emph{triangulation}.\pause\ Decompose the space as faces, edges, vertices.
\fpause
Then $\chi$ is:
\[ (\#\text{faces}) \pause - (\#\text{edges}) \pause + (\#\text{vertices}) \]
}

\frame{
This is the Euler characteristic!
\fpause
So the continuous invariant valuations in $\R^n$ include:\fpause

\begin{itemize}
\item The intrinsic volumes of dimensions $1, \dots, n$\pause
\item The Euler characteristic\pause\ (``the intrinsic volume of dimension 0'')\pause
\item any linear combination of these
\end{itemize}
\fpause
Any more?
\fpause
No!
}

\frame{

{\bf Hadwiger's Theorem} (1957) The intrinsic volumes of dimension $0,\dots, n$ are a \emph{basis} for the vectorspace of continuous invariant valuations on $\R^n$.
\fpause
So any measure of bigness is some (unique) combination of intrinsic volumes and Euler characteristic.
\fpause
In $\R^3$, this means that any measure of bigness has the specific form:
\[ v(X) = c_0\chi(X) + c_1 P(X) + c_2 A(X) + c_3 V(X) \]
where $\chi$ is the Euler characteristic, $v_1$ is the perimeter, $v_2$ is the surface area, $v_3$ is the volume, and $c_i$ are constants.
}

\frame{
This is actually a beautiful theorem.
\fpause
The valuation property seems very general. \pause\ Many many functions ought to obey this.
\fpause
But it turns out that the valuation property is very restrictive.
\fpause
The classification is much simpler than it should be.
}

\frame{
Topologically this is very interesting:
\fpause
The Euler characteristic is the only topologically invariant valuation.
\fpause
Much of topology is about assigning ``invariants'' to spaces based on their structure,\pause\ and the valuation property is a very natural kind of thing that we'd want to satisfy.
\fpause 
Similar properties exist for fundamental groups (Van Kampen's theorem) and homology groups (Mayer-Vietoris sequence)
\fpause
If your invariant is going to be a $\R$-valued valuation, it must be the Euler characteristic.
}

\frame{
What remains:
\begin{itemize}
\item Why it's true
\item Real-world applications
\end{itemize}
}

\section{Why it's true}
\frame{
Let's try to give an idea of why any continuous invariant valuation must be some intrinsic volume.
\fpause
(Some familiar big ideas coming)
\fpause
The intrinsic volumes break down nicely into dimensions.
\fpause
Let's just show that the only ``dimension 2'' valuation in $\R^2$ is the area.
\fpause
Specifically we'll show that the only (continuous invariant) valuation (which is zero on sets of dimension less than 2) is (a constant times) the area.
}

\frame{
Let $v$ be any continuous invariant valuation in $\R^2$ which is zero on sets of dimension less than 2
\fpause
We'll show that $v(X) = c\cdot A(X)$ where $A$ is the area.
\fpause
First consider the unit square $S$: \pause it has some value $v(S) = c$. 
\fpause
By invariance, any square of area 1 will have value $v(S) = c$.
}

\frame{
\[ \includegraphics[width=80px]{square} \]
\fpause
By the valuation property:
\begin{align*}
c = v \left( \vcenter{\hbox{\includegraphics[width=40px]{square}}} \right) &= 
v \left( \vcenter{\hbox{\includegraphics[width=40px]{halfsquare}}} \right) +
v \left( \vcenter{\hbox{\includegraphics[width=40px, angle=180]{halfsquare}}} \right) -
v \left( \vcenter{\hbox{\includegraphics[width=40px,angle=180]{squareline}}} \right)
\end{align*}
}

\newcommand{\sqpic}{\vcenter{\hbox{\includegraphics[width=30px]{square}}}}
\newcommand{\leftpic}{\vcenter{\hbox{\includegraphics[width=30px]{halfsquare}}}}
\newcommand{\rightpic}{\vcenter{\hbox{\includegraphics[width=30px,angle=180]{halfsquare}}}}
\newcommand{\linepic}{\vcenter{\hbox{\includegraphics[width=30px,angle=180]{squareline}}}}


\frame{
%$v(|) = 0$ since $v$ is zero on sets of dimension less than 2, so 
\begin{align*}
c &=  v \left( \leftpic \right) + v \left( \rightpic \right) -
v \left( \linepic \right) \\
\onslide<2->{&= v \left( \leftpic \right) + v \left( \rightpic \right) \\}
\onslide<3->{&= v \left( \leftpic \right) + v \left( \leftpic \right) \\}
\onslide<4->{&= 2 \cdot v \left( \leftpic \right) }
\end{align*}
\fpause
\onslide<5->{So $v\left( \leftpic \right) = \frac12 c$.}
}

\frame{
So $v$ on the unit square has value $c \cdot 1$
\fpause
$v$ on this rectangle with area 1/2 has value $c \cdot \frac12$. 
\fpause
By cutting up different ways, easy to show that $v$ on a rectangle with area $q \in \Q$ is $c \cdot q$.
\fpause
Already it's starting to look like $v$ is always just $c$ times the area, but we showed it only for rectangles.
}

\frame{
What if our shape isn't a rectangle?
\[ \includegraphics[width=100px]{rectshape} \]
\pause
Just break it up into rectangles! 
\fpause
For a shape like this, still $v$ must be $c$ times the actual area.
}

\frame{
What if the shape is curvy?
\fpause
\[ \includegraphics[height=100px]{parabola} \]
\fpause
Cover it with rectangles!
}

\frame{
COVER IT WITH RECTANGLES!
\[ \includegraphics[height=100px]{parabolarect} \]
\fpause
The area of the ``rectified'' region is close to the area of the curved region,
\fpause
and as the rectanglular approximations get smaller, the rectified area approaches the actual area.
\fpause
Is the same true for $v$?
}

\frame{
We already know $v$ is $c$ times the area for the rectified areas. 
\fpause
Will it also be true for the curvy area?
\fpause
It will because $v$ is continuous!
}

\frame{
{The whole idea at once}

Say $v$ has value $c$ on the unit square.
\fpause
The value on the square dictates exactly what the value must be on any rectangles, and this dictates the value on any curvy area.
\fpause
So any dimension 2 measurement which can be ``broken down'' additively must actually be the area (times a constant).
}

\section{Real-world applications}
\frame{{Real-world applications}
Any continuous invariant valuation is a combination of intrinsic volumes.
\fpause
Most things in nature are continuous and invariant.
\fpause
So if you encounter a valuation in nature, it must be a combination of intrinsic volumes.
\fpause
What I'm about to say is mostly true.
}

\frame{
Some examples are collected in: ``Additivity, Convexity, and Beyond: Applications of Minkowski Functionals in Statistical Physics'' by Mecke (2000) \pause 74 pages!
\fpause
One is ``curvature energy of a membrane''.
\fpause
Given a flexible flat membrane (zero or uniform thickness), how much energy is required to bend it?
\fpause
This will depend on what the membrane is made of, its temperature, etc.
\fpause
Let's ignore all that- assume constant temperature, etc.\pause\ We care only about the shape of it.
}

\frame{
What could the curvature energy depend on? (in terms of the shape)
\fpause
Obviously it might depend on the total area.\pause\ But how exactly?
\fpause
Probably something like
\[ E \propto A^{2.4} + A\log A - 8e^{\sqrt A} \]
\pause
Actually we have no idea.
}

\frame{
Beyond just the area, it probably depends somehow on the shape.
\fpause
Specifically: Is the curvature energy the same for these?
\[ \includegraphics[width=100px]{shapes1} \]
\fpause
They have the same area, but they're different shapes.
}

\frame{
How about these?
\[ \includegraphics[width=200px]{shapes2} \]
\fpause
How about these?
\[ \includegraphics[width=200px]{shapes3} \]
}

\frame{
So we expect the curvature energy to depend on the shape, probably in a very complicated way.
\fpause
The best imaginable goal would be a simple mathematical formula for $E$ in terms of some geometric information.\pause\ But this seems probably impossible.
\fpause
But it turns out the curvature energy is a valuation:
\[ 
E \left( \vcenter{\hbox{\includegraphics[width=70px]{AcupB}}} \right) = 
E \left(\vcenter{\hbox{\includegraphics[width=50px, ]{A}}} \right) +
E \left(\vcenter{\hbox{\includegraphics[width=50px, ]{B}}} \right) -
E \left(\vcenter{\hbox{\includegraphics[width=30px, ]{AcapB}}} \right)
\]
\fpause
It is obviously continuous and invariant.
}

\frame{
So by Hadwiger's theorem the curvature energy must have this form:
\[ E(X) = c_1 \chi(X) + c_2 P(X) + c_3 A(x) \]
where $\chi$ is the Euler characteristic, $P$ is the perimeter, and $A$ is the area.
\fpause
This is a very simple formula for $E$ obtained purely mathematically! (no experiments necessary)
\fpause
The only things we need to test experimentally are the constants.
}

\frame{
We can even answer conclusively each of the questions above.
\fpause
Remember the curvature energy depends \emph{only} on area, perimeter, and Euler characteristic.
\fpause
Is the curvature energy the same for these?
\[ \includegraphics[width=100px]{shapes1} \]
\fpause
No- different perimeters.
}

\frame{
Is the curvature energy the same for these?
\[ \includegraphics[width=200px]{shapes2} \]
\fpause
Yes- same areas, same perimeters, same Euler characteristic.
}

\frame{
Is the curvature energy the same for these?
\[ \includegraphics[width=200px]{shapes3} \]
\fpause
Probably not- same areas \& perimeters, but different Euler characteristic. \pause\ (top is 0, bottom is 1)
}

\frame{
So if you encounter a valuation in nature, Hadwiger's theorem gives you a formula for free.
\fpause
Other examples from ``Additivity, Convexity, and Beyond'' are
\begin{itemize}
\item Percolation in porous solids\pause
\item ``Hearing the shape of a drum''\pause
\end{itemize}
}

\frame{{How I came to this}
I'm interested in the Euler characteristic, and there is another theorem by Watts, which looks just like Hadwiger's theorem in dimension 0.
\begin{thm} (Hadwiger) The Euler characteristic $\chi$ is the unique function with:
\begin{itemize}
\item $\chi(A \cup B) =  \chi(A) +  \chi(B) - \chi(A\cap B)$
\item When $X$ is convex, $\chi(X) = 1$
\end{itemize}
\end{thm}
\fpause
\begin{thm}(Watts, 1962) The ``reduced Euler characteristic'' $\bar \chi = \chi - 1$ is the unique function with:
\pause
\begin{itemize}
\item When $A \subseteq B$, $\bar \chi(B) = \bar \chi(A) - \bar \chi(B/A)$\pause
\item $\bar \chi(S^0) = 1$
\end{itemize}
\end{thm}
}

\frame{
My research is in topological fixed point theory.
\fpause
A major tool is the \emph{Lefschetz number} $L(f)$ of a map from a space to itself.
\fpause
Always $L(\id) = \chi(X)$, so $L(f)$ is a generalization of the Euler characteristic.
\fpause
Think of $L(f)$ like an Euler characteristic for a function.
}

\frame{
From 1962 we have Watts's theorem about ``$\chi$ is the unique function satisfying\dots''
\fpause
We should try to prove the same thing about $L(f)$.
\fpause
In 2004, Arkowitz \& Brown proved that $\bar L(f)$ is the unique function satisfying \dots''
\fpause
Also in 2004, Furi, Pera, \& Spadini proved \emph{another} uniqueness theorem for $L(f)$.
\fpause
I did some stuff with this too.
}

\frame{
So when I saw Hadwiger's theorem, I knew immediately that it would give yet another theorem about $L(f)$.
\fpause
\begin{thm}
There is a unique function $\Lambda:N(X)\to \R$ satisfying:
\begin{itemize}
\item Let $A,B$ be subcomplexes of some common subdivision of $X$. Then $\Lambda(f,\emptyset)=0$, and 
\[ \Lambda(f,A\cup B) = \Lambda(f,A) + \Lambda(f,B) - \Lambda(f,A\cap B). \]
\item Let $f$ be a Hopf simplicial map and $x$ be a simplex. If $x$ is not a maximal simplex we have $\Lambda(f,x) = 0$, and if $x$ is a maximal simplex we have 
\[ \Lambda(f,x) = (-1)^{\dim X} c(f, x). \]
\item $\Lambda(f,A)$ depends continuously on $f$. 
\end{itemize}
\end{thm}
}

\frame{
Why hadn't anybody else done this?
\fpause
People in fixed point theory don't know about Hadwiger's theorem.
\fpause
This is called: \pause\ ``low-hanging fruit''
\fpause
Currently looking at higher dimensions.
}

\begin{comment}

\frame{{Hearing the shape of a drum}
A drum makes a complex sound including a series of \emph{overtones}. 
\fpause
The exact overtones produced depend on the size and shape of the drum.
\fpause
M. Kac (1966) asked: By analyzing the sequence of overtones, can we determine the shape of the drum?
\fpause
Kac didn't answer the question, but he showed that the spectrum of overtones can be used to infer:\pause
\begin{itemize}
\item the area\pause
\item the perimeter\pause
\item the Euler characteristic
\end{itemize}
}

\frame{
So Kac showed that you \emph{can} hear the intrinsic volumes. 
\fpause
Can you hear everything about the shape?
\fpause
Milnor immediately gave an example of two 16-dimensional spaces with different shapes but the same spectra. \pause (So ``no'')
\fpause
Not really what Kac was looking for.
}

\frame{
Eventually in 1992, Gordon, Webb, and Wolpert gave this example:\pause
\[ \includegraphics[width=200px]{isospectral} \]
\pause
GWW show these shapes are ``isospectral''\pause, so the answer to Kac's question even in dimension 2 is ``no''.
\fpause
Note they have the same intrinsic volumes.
}

\frame{
Buser, Conway, Doyle, and Semmler ``Some planar isospectral domains'', 1994:\pause
\[\includegraphics[width=150px]{gallery} \]
}

\frame{
So: resonant frequencies of membranes have a lot to do with intrinsic volumes
\fpause
Hadwiger's theorem can say a lot about many questions related to this.
}
\end{comment}

\frame{
That's all!
}



\end{document}