Advanced handouts

Add missing file
Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Lattices/parts/0 intro.tex

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+\definition{}
+The \textit{integer lattice} $\mathbb{Z}^n \subset \mathbb{R}^n$ is the set of points with integer coordinates.
+
+\problem{}
+Draw $\mathbb{Z}^2$.
+
+\vfill
+
+
+\definition{}
+We say a set of vectors $\{v_1, v_2, ..., v_k\}$ \textit{generates} $\mathbb{Z}^n$ if every lattice point can be written uniquely as
+$$
+	a_1v_1 + a_2v_2 + ... + a_kv_k
+$$
+for integer coefficients $a_i$. \par
+It is fairly easy to show that $k$ must be at least $n$.
+
+\problem{}
+Which of the following generate $\mathbb{Z}^2$?
+\begin{itemize}
+	\item $\{ (1,2), (2,1) \}$
+	\item $\{ (1,0), (0,2) \}$
+	\item $\{ (1,1), (1,0), (0,1) \}$
+\end{itemize}
+
+\begin{solution}
+	Only the last.
+\end{solution}
+
+\vfill
+
+\problem{}
+Find a set of two vectors that generates $\mathbb{Z}^2$. \\
+Don't say $\{ (0, 1), (1, 0) \}$, that's too easy.
+
+\vfill
+
+\problem{}
+Find a set of vectors that generates $\mathbb{Z}^n$.
+
+
+
+\vfill
+\pagebreak
+
+\definition{}
+A \textit{fundamental region} of a lattice is the parallelepiped spanned by a generating set. The exact shape of this region depends on the generating set we use.
+
+\problem{}
+Draw two fundamental regions of $\mathbb{Z}^2$ using two different generating sets. Verify that their volumes are the same.
+
+\vfill
+\pagebreak