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\handout{COMS 6998: Advanced Complexity}{Spring 2017}
{Lecturer: Rocco Servedio}
{Scribes: Aaaa B. Dddd and Eeee F. Gggg}{Lecture 1: January 19, 2017}

\thispagestyle{plain}


\section{Introduction}

This is a template file for scribe notes for COMS 6998. This class is about Boolean functions $f:\zo^n\rightarrow \zo$.  
Here is some filler text.  

Check out the style files 6998.sty and 6998-thm.sty (in the same directory where you found this file); those files contain macros, etc that can make your notes easier to write and more attractive to read.

Here's an example of an embedded link (in this case, a link to a free online version of Wegener's ``blue book''):
\href{https://eccc.weizmann.ac.il/static/books/The_Complexity_of_Boolean_Functions/}{{\it The Complexity of Boolean Functions}}.


\section{Some hints about organization}

It's a good idea to organize your scribe notes into sections according to logical break-points in the material.  Similarly, it's a good idea to organize technical material into lemmas, claims, theorems, et cetera.  LaTeX is great at automatically handling references to material in your text.  Here's a simple example:

\begin{theorem} \label{thm:dummy}
This is a dummy theorem.
\end{theorem}

\noindent By labeling the theorem in the LaTeX source it's easy to refer to it later in an automatic way 
 (for example, this parenthetical references Theorem~\ref{thm:dummy} --- see the LaTeX source).




\section{Some hints about exposition}

You should strive for clarity in your writeup of scribe notes --- don't be too concise, but also avoid unnecessary over-explanation.  A good imaginary reader to keep in mind is a friend who's smart and mathematically mature, but doesn't have any special expertise or knowledge of this area.

\section{Some hints about presentation}

Pictures/diagrams/figures are extremely helpful!  It's a good idea to include as many of these as you need.  You can draw them in LaTeX itself or import files, whatever you prefer --- the key thing is that your figures are legible and informative.  See Appendix~\ref{ap:figure} for an example of a figure. (It's taken out of context so don't worry about the content -- but hopefully you are convinced that in whatever context it came from, the figure helps explain what is going on!)

Everybody likes to read good-looking mathematics; or rather, nobody likes to read math that's ugly or hard to parse.
(Subjectivity alert) Here is an example of some reasonably good-looking math:

\[
(16n)^s = 2^{\log(16n) \cdot 2^n/2\log(n)}=2^{ \frac{4+\log(n)}{2\log(n)} \cdot 2^n}
<2^{0.51\cdot 2^n }\ll \left(1 - {\frac 1 {2^n}}\right) \cdot 2^{2^n}.
\]


Finally, a typical set of scribe notes should provide references to relevant papers, books, etc. as appropriate.  This sentence contains several references to various papers \cite{Shannon-1949,Subbotovskaya-1961,Hastad-1986}.


\bibliography{the}
\bibliographystyle{alpha}

\appendix

\section{An example figure} \label{ap:figure}


\begin{figure}[p!]
\begin{center}
\includegraphics[width=16.5cm]{levels-new-3}
\end{center}
\caption{The figure illustrates a typical $\tau \in \{\bullet,\circ,\ast\}^{A_{k}}$.   For $a \in A_{k-1}$,  $\tau_a$ is a block of length $w_{k-1}$,  i.e.\ a string in $\{\bullet,\circ,\ast\}^{w_{k-1}}$. We may think of the block $\tau_a$ as being located at level $k$.  By Condition (1) of Definition~72, for every $a \in A_{k-1}$ we have that $|\tau_a^{-1}(\ast)|$, the number of $\ast$'€™s in $\tau_a$, is roughly $qw = \tilde{\Theta}(\sqrt{w})$.
The lift $\hat{\tau}$ of $\tau$ is a string in $\{\bullet,\circ,\ast\}^{A_{k-1}}$, and for $\alpha \in A_{k-2}$, $\hat{\tau}_\alpha$ is a block of length $w_{k-2}$. We may think of the block $\hat{\tau}_\alpha$ as being located at level $k-1$.  As stipulated by equation (86), for every $\alpha \in A_{k-2}$, the string $\hat{\tau}_\alpha$ belongs to $\{\ast,\circ\}^{w_{k-2}}$.  By Condition (2) of Definition~72, for every $\alpha \in A_{k-2}$, we have that $|(\hat{\tau}_\alpha)^{-1}(\ast)|$, the number of $\ast$'€™s in $\hat{\tau}_\alpha$, is at least $w_{k-2} - w^{4/5} = w_{k-2}(1-o(1))$.  Finally, we observe that equation (55) and Condition~(2) of Definition~72 imply that $\hat{\hat\tau}_\alpha = \ast$ for every $\alpha \in A_{k-2}.$
}
\label{figure:the-figure}
\end{figure}



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