reveal.js

<h1 class="fragment"> I AM COUNTING THINGS</h1>

<u class="fragment fade-right" style="display:block;">What am I counting?</u>

<span class = "fragment fade-up" style="display: block;">Rigid indecomposable modules</span>

<u class = "fragment fade-right" style="display:block;">Why am I counting them?</u>

<span class = "fragment fade-up" style="display:block;"> They correspond to __cluster variables__ </span>

<u class = "fragment fade-right" style="display:block;">How am I counting them?</u>

<span class="fragment fade-up" style="display: block;"> _Carefully_ </span>

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<span class="fragment fade-up" style="display:block;">The homogeneous coordinate ring $\mathbb{C}[\mathrm{Gr}(k,n)]$ has a **cluster structure**.</span>

<span class="fragment fade-up" style="display:block;">This structure can be **categorified** by the module category $\mathrm{CM}C$ introduced by Jensen, King, and Su.</span>

<span class="fragment fade-up" style="display:block;"> Cluster character $\Phi \colon \mathrm{CM}C \to \mathbb{C}[\mathrm{Gr}(k,n)]$</span>

<span class="fragment fade-up" style="display:block;" >Rigid indecomposables $\longleftrightarrow$ Cluster variables</span>

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<div style="display: flex; justify-content: center;">
<table>
    <thead>
    <tr>
      <th class="fragment fade-right" data-fragment-index="0" style="border-right:2px solid white;">$k,n$</th>
      <th class=" fragment fade-down" data-fragment-index="1">$1,n$</th>
      <th class=" fragment fade-down" data-fragment-index="1">$2,n$</th>
      <th class=" fragment fade-down" data-fragment-index="1">$3,6$</th>
      <th class=" fragment fade-down" data-fragment-index="1">$3,7$</th>
      <th class=" fragment fade-down" data-fragment-index="1">$3,8$</th>
      <th class="fragment fade-down" data-fragment-index="3" style="color:#50FA7B;">$3,9$</th>
      <th class="fragment fade-down" data-fragment-index="3" style="color:#50FA7B;">$4,8$</th>
      <th class="fragment fade-left" data-fragment-index="2" style="color:#FF5555;">$4+,9+$</th>
    </tr>
    </thead>
    <tbody>
    <tr>
      <td class="fragment fade-right" data-fragment-index="0" style="font-weight:bold;border-right:2px solid white">#CV</td>
      <td class=" fragment fade-down" data-fragment-index="1">$n$</td>
      <td class=" fragment fade-down" data-fragment-index="1">$\frac{n(n-1)}{2}$</td>
      <td class=" fragment fade-down" data-fragment-index="1">22</td>
      <td class=" fragment fade-down" data-fragment-index="1">49</td>
      <td class=" fragment fade-down" data-fragment-index="1">136</td>
      <td class="fragment fade-down" data-fragment-index="3" style="color:#50FA7B;">$\infty$</td>
      <td class="fragment fade-down" data-fragment-index="3" style="color:#50FA7B;">$\infty$</td>
      <td class="fragment fade-left" data-fragment-index="2" style="color:#FF5555;">$\infty$</td>
    </tr>
    </tbody>
</table>
</div>

<br>

<span style="display:block;color:#50FA7B;" class="fragment fade-down" data-fragment-index="4">How many cluster variables have Plücker degree $r$?</span>

<span class="fragment fade-down" data-fragment-index="5" style="color:#50FA7B">How many rigid indecomposable $\mathrm{CM}C$ modules have rank $r$?</span>

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<div style="display: flex; justify-content: center;">
<div class="fragment fade-right">
  <p>$\mathrm{CM}C$ module for $3,9$</p>
  <img height="auto" src="./39-rank-7.png"></img>
</div>
<div style="" class="fragment fade-left">
  <p>Mysterious quiver with relations</p>
  <img height="auto" src="./subspace-quiver.png"></img>
</div>

</div>
</div>

---

### <span class="fragment fade-down" style="display:block;" data-fragment-index="1">Tubular algebra of type 2,3,6 </span>

<div>
  <span style="display:block;" class="fragment fade-down" data-fragment-index="1" ><img src="./tubular-ar-quiver.png"></img></span>
  </div>
  <div style="display:flex; justify-content: center; align-items: center;">
      <img class="fragment fade-right" width="15%"  src="./tube.png"></img>
      <div style="font-size:35px;display:block;width:50%;">
       <p class="fragment fade-left">'High enough' in an exceptional tube $\implies$ <span style="color:#FFB86C;">Rigid</span> </p>
       <p class="fragment fade-left">Count subspace configurations of <span style="color:#FFB86C;">ambient dimension $r$</span> that are high enough in their tubes </p>
      </div>

</div>

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$$
\begin{align*} N_6(r) &= 3\phi_6(2r-1) + 6\phi_6(2r) + 3\phi_6(2r+1)\\\\
N_3(r) &= 3\phi_3(r) \\\\
N_2(r) &= \begin{dcases} 0 & r \equiv 0 \mod 3\\\ 
\phi_2\left(\frac{2r-2}{3}\right) + \phi_2\left(\frac{2r+1}{3}\right) & r \equiv 1 \mod 3\\\\
\phi_2\left(\frac{2r-1}{3}\right) + \phi_2\left(\frac{2r+2}{3}\right) & r \equiv 2 \mod 3
\end{dcases}
\end{align*}
$$

<span class="fragment fade-up" style="display:block;color:#FFB86C;">$\pmb{N(r) = 3\cdot\left[N_2(r) + N_3(r) + N_6(r)\right]}$</span>

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### 3,9

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### 4,8
<img style="width:80%; height:auto;" src="./totient-count-48-1-1000000-1.png"></img>

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<div style="hyphens:auto;word-break:break-word;display:flex; justify-content: center;flex-direction: column; align-items: center; text-align: left;">
<p style="width:80%;">$\underline{\mathrm{CM}}C \cong \frac{\mathcal{D}^b(\operatorname{coh}\mathbb{X})}{\langle \tau^{-}[1]\rangle} \cong \frac{\mathcal{D}^b(\mathrm{mod}\,P)}{\langle \tau^{-}[1]\rangle}$ where $\mathbb{X}$ is a weighted projective line of weight 2,3,6 or 2,4,4.
</p>

<p style="width:80%;">Cluster tilting object $T$ in $\underline{\mathrm{CM}}C$ whose (opposite) endomorphism ring
is isomorphic to the relation extension of $P$.</p>
</div>