# Friday, November 17

:::{.remark}
Today: scattering diagrams.
Fix the data of $N, M$, and a skew-symmetric pairing.
Note that the data of seed $(N, \ts{\wait, \wait})$ with a choice of basis $\ts{e_i}$ yields such data.
Define $N^+ \da \ts{a_ie_i \st a_i\geq 0, \sum a_i > 0 }$ -- imagine the first quadrant with the origin deleted.
A **wall** in $M_\RR$ is a pair $(d, f_d)$ where $d$ is a codimension 1 rational polyhedral cone in $M_\RR$ contained in $n^\perp$ for some $n\in N^+$, and $f_d = 1 + \sum_{k\geq 1} c_k z^{k p^*(n)}$ where $p^*$ is the map $n\mapsto \ts{n, \wait}$.
This can generally be a wall or a ray.
We say a wall is **incoming** if $p^*(n) \in d$ and is outgoing otherwise.
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:::{.remark}
The data of a fan with integral affine singularities satisfying a certain balancing condition is called a *scattering diagram*.
The rays are additionally decorated with arrows where rays with singularities point toward the origin.
Given a SCRPC $\sigma \subseteq M_\RR$ with $p^*(N^+) \subseteq \sigma$, set $P \da \sigma \intersect M$ as a monoid.
Then let $\CC[P]$ be the free algebra, and $m\da \gens{z^p \st p\in P\smz}_{\CC}$ as a maximal ideal.
Let $\widehat{\CC[P]}$ be the $m\dash$adic completion; this is where the function $f_d$ will live.
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:::{.definition title="Scattering diagrams"}
A **scattering diagram** $D$ is a union of walls $(d, f_d)$ such that $D_k = \ts{(d, f_d) \in D\st f_d \not\equiv 1 \mod m^{k+1}}$ is finite for all $k$.
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