diff --git a/notes/notes.tex b/notes/notes.tex
index d8673109..38e47685 100644
--- a/notes/notes.tex
+++ b/notes/notes.tex
@@ -56,9 +56,7 @@
 The benefit of this Gaussian model is that this can be solved exactly.
 The minimum is obtained by setting the derivative with respect to $p_j$ to 0.
 \begin{equation}
-    \partial_j logL(\{p_i\} | \{n_i\}, \{k_i\}) = n_j^2\frac{(k_j - p_j n_j)}{(k_j+p_c)(n_j-k_j+p_c)}
-%    - \lambda (p_j - p_{j-1}) + \lambda (p_{j+1} - p_{j})
-- \lambda (2p_j - p_{j-1}-p_{j+1}) = 0
+    \partial_j logL(\{p_i\} | \{n_i\}, \{k_i\}) = n_j^2\frac{(k_j - p_j n_j)}{(k_j+p_c)(n_j-k_j+p_c)} - \lambda (2p_j - p_{j-1}-p_{j+1}) = 0
 \end{equation}
 For $j=0$ the last term is just $\lambda (p_1 - p_0)$, for $p_m$, the  $-\lambda (p_m - p_{m-1})$
 This is a very sparse linear system of equations that owing to the stiffness constraint should non-degenerate.
diff --git a/web/config/next/next.config.ts b/web/config/next/next.config.ts
index 34cdd290..e7601860 100644
--- a/web/config/next/next.config.ts
+++ b/web/config/next/next.config.ts
@@ -7,6 +7,7 @@ import remarkMath from 'remark-math'
 import remarkToc from 'remark-toc'
 import remarkSlug from 'remark-slug'
 import remarkImages from 'remark-images'
+import rehypeKatex from 'rehype-katex'
 
 import { findModuleRoot } from '../../lib/findModuleRoot'
 import { getGitBranch } from '../../lib/getGitBranch'
@@ -134,7 +135,7 @@ const withMDX = getWithMDX({
       //   },
       // ],
     ],
-    rehypePlugins: [],
+    rehypePlugins: [rehypeKatex],
   },
 })
 
diff --git a/web/package.json b/web/package.json
index dcbeda48..def66fe0 100644
--- a/web/package.json
+++ b/web/package.json
@@ -83,6 +83,7 @@
     "intersection-observer": "0.12.2",
     "is-absolute-url": "4.0.1",
     "iso-3166-1-alpha-2": "1.0.1",
+    "katex": "0.16.8",
     "lodash-es": "4.17.21",
     "luxon": "3.2.1",
     "marked": "4.2.12",
@@ -116,6 +117,7 @@
     "recoil-persist": "4.2.0",
     "reflect-metadata": "0.1.13",
     "regenerator-runtime": "0.13.11",
+    "rehype-katex": "7.0.0",
     "resize-observer-polyfill": "1.5.1",
     "route-parser": "0.0.5",
     "serialize-javascript": "6.0.1",
diff --git a/web/src/components/Home/HomePage.tsx b/web/src/components/Home/HomePage.tsx
index f33f2932..3753f6e7 100644
--- a/web/src/components/Home/HomePage.tsx
+++ b/web/src/components/Home/HomePage.tsx
@@ -8,6 +8,7 @@ import { MdxContent } from 'src/i18n/getMdxContent'
 import { getDataRootUrl } from 'src/io/getDataRootUrl'
 import { PageContainerNarrow } from 'src/components/Layout/PageContainer'
 import { PathogenCard } from 'src/components/Home/PathogenCard'
+import 'katex/dist/katex.min.css'
 
 export interface IndexJson {
   pathogens: Pathogen[]
@@ -31,7 +32,7 @@ export function HomePage() {
       </Row>
       <Row noGutters>
         <Col>
-          <MdxContent filepath="Home.md" />
+          <MdxContent filepath="Notes.md" />
         </Col>
       </Row>
     </PageContainerNarrow>
diff --git a/web/src/content/en/Notes.md b/web/src/content/en/Notes.md
new file mode 100644
index 00000000..a50e17d8
--- /dev/null
+++ b/web/src/content/en/Notes.md
@@ -0,0 +1,71 @@
+---
+author:
+- Richard A. Neher
+title: Estimating frequencies
+---
+
+Estimating frequency trajectories from count data is a recurring problem
+in many of our analyses. We previously fitted to a sampling likelihood
+with a stiffness prior of the type
+$$L(\{p_i\} | \{n_i\}, \{k_i\}) \sim \prod_i p_i^{k_i} (1-p_i)^{n_i-k_i} \times \prod_i e^{ - \lambda (p_i - p_{i-1})^2/2}$$
+where the stiffness $\lambda$ might want to depend on $p_i$ itself. If
+regularization is diffusive, we expect the parameter $\lambda$ the scale
+as $\lambda = 1/2D(t_i-t_{i-1})$. This non-linear likelihood is
+difficult to optimize. Since we don't really think that the data we have
+is well described by iid sampling, we might just as well approximate the
+binomial sampling LH by a gaussian with mean $\hat{p}_i = k_i/n_i$ and
+variance $$\sigma^2 = \frac{k_i(n_i-k_i)}{n_i^3}$$ Not that this
+variance diverges when $n_i=0$, corresponding to the case when there is
+no information. The log-likelihood then looks like this
+$$logL(\{p_i\} | \{n_i\}, \{k_i\}) = C + \sum_i n_i\frac{(k_i - p_i n_i)^2}{2k_i(n_i-k_i)}  + \sum_i \lambda (p_i - p_{i-1})^2/2$$
+We still need to guard this against cases where $k_i=0$ or $k_i=n_i$, or
+more generally cases where the binomial is not well approximated by a
+Gaussian. This can be achieved by simply inflating the variance -- the
+variance is anyway underestimated by the idd approximation. Here, we add
+pseudo-counts $p_c$ to both terms of the variance.
+$$logL(\{p_i\} | \{n_i\}, \{k_i\}) = C + \sum_i n_i\frac{(k_i - p_i n_i)^2}{2(k_i+p_c)(n_i-k_i+p_c)}  + \sum_i \lambda (p_i - p_{i-1})^2/2$$
+The next question is how the stiffness $\lambda$ should scale with the
+number of samples and the time discretization. If our null is that the
+true frequencies change diffusively, $\lambda \sim 1/D\Delta t$. The
+absolute value of the stiffness should be related to how much we expect
+frequencies to change from one month to the next. For influenza, this is
+typically on the order of $\sqrt{Dt} \sim 0.1$, which suggests that $Dt$
+should be on the order of 100. A month is 30 days, so a value around
+1000 seems sensible.
+
+The benefit of this Gaussian model is that this can be solved exactly.
+The minimum is obtained by setting the derivative with respect to $p_j$
+to 0.
+$$\partial_j logL(\{p_i\} | \{n_i\}, \{k_i\}) = n_j^2\frac{(k_j - p_j n_j)}{(k_j+p_c)(n_j-k_j+p_c)} - \lambda (2p_j - p_{j-1}-p_{j+1}) = 0$$
+For $j=0$ the last term is just $\lambda (p_1 - p_0)$, for $p_m$, the
+$-\lambda (p_m - p_{m-1})$ This is a very sparse linear system of
+equations that owing to the stiffness constraint should non-degenerate.
+Rearranging this to separate constant and $p_i$ dependent parts yields
+$$\frac{n_j^3}{(k_j+p_c)(n_j-k_j+p_c)}p_j  + \lambda (2p_j - p_{j-1} - p_{j+1}) =  \frac{n_j^2 k_j}{(k_j+p_c)(n_j-k_j+p_c)}$$
+
+The second derivative of the objective function with respect to
+frequencies would have the diagonal elements
+$$\frac{n_i^3}{(k_i+p_c)(n_i-k_i+p_c)} + 2\lambda$$ and off-diagonal
+elements $(i, i+1)$ of $-\lambda$. The confidence intervals would then
+be the square root of the diagonal of the inverse of this Hessian.
+
+Hierarchial frequencies {#hierarchial-frequencies .unnumbered}
+=======================
+
+Different countries in a region are pretty well mixed but deviate from
+the region-average frequency. It thus makes sense to tie them together.
+If we express the frequency in a country as
+$p_c(t_i) = p_r(t_i) + \Delta p_c(t_i)$ we can regularize the
+$\Delta p_c$ to with a quadratic penalty. The Gaussian logLH is thus
+$$logL(\{p_i\} | \{n_i\}, \{k_i\}) = C + \sum_i \sum_c n^c_i\frac{(k^c_i - (p_i + \Delta p^c) n^c_i)^2}{2(k^c_i+p_c)(n^c_i-k^c_i+p_c)}  + \sum_i \lambda (p_i - p_{i-1})^2/2 + \frac{1}{2}\sum_i\sum_c \left(\lambda (\Delta p^c_i - \Delta p^c_{i-1})^2 +\mu (\Delta p^c_{i})^2\right)$$
+The derivative with respect to the $p_i$ and $\Delta p_i^c$ now look
+somewhat different. The derivative with respect to $p_j$ is
+$$-\sum_c (n^c_j)^2\frac{(k^c_j - (p_j + \Delta p^c_j) n^c_j)}{(k^c_j+p_c)(n^c_j-k^c_j+p_c)}  + \lambda (2p_j - p_{j-1}- p_{j+1}) = 0$$
+and can be rearranged to
+$$\sum_c \frac{(n^c_j)^3}{(k^c_j+p_c)(n^c_j-k^c_j+p_c)} (p_j + \Delta p^c)   + \lambda (2p_j - p_{j-1}- p_{j+1}) = \sum_c \frac{(n^c_j)^2 k^c_j}{(k^c_j+p_c)(n^c_j-k^c_j+p_c)}$$
+This will generate a number of off-diagonal terms, but it should still
+be a pretty sparse matrix.
+
+The derivative with respect to $\Delta p^d_j$ is
+$$-(n^d_j)^2\frac{(k^d_j - (p_j + \Delta p^d_j) n^d_j)}{(k^d_j+p_c)(n^d_j-k^d_j+p_c)}  + \lambda (2\Delta p^d_j - \Delta p^d_{j-1}- \Delta p^d_{j+1}) + \mu \Delta p_j^d = 0$$
+$$\frac{(n^d_j)^3}{(k^d_j+p_c)(n^d_j-k^d_j+p_c)}(p_j + \Delta p^d_j)   + \lambda (2\Delta p^d_j - \Delta p^d_{j-1}- \Delta p^d_{j+1}) + \mu \Delta p_j^d = \frac{k^d_j(n^d_j)^2}{(k^d_j+p_c)(n^d_j-k^d_j+p_c)}$$
diff --git a/web/src/pages/_app.tsx b/web/src/pages/_app.tsx
index e35161fe..aed5f239 100644
--- a/web/src/pages/_app.tsx
+++ b/web/src/pages/_app.tsx
@@ -26,6 +26,7 @@ import { RegionPage } from 'src/components/Regions/RegionPage'
 import { VariantPage } from 'src/components/Variants/VariantPage'
 import { localeAtom } from 'src/state/locale.state'
 import 'src/styles/global.scss'
+import 'katex/dist/katex.css'
 
 const NotFoundPage = dynamic(() => import('src/pages/404'))
 
diff --git a/web/yarn.lock b/web/yarn.lock
index 03c3acda..9ae9e123 100644
--- a/web/yarn.lock
+++ b/web/yarn.lock
@@ -3918,6 +3918,13 @@
   dependencies:
     "@types/unist" "*"
 
+"@types/hast@^3.0.0":
+  version "3.0.1"
+  resolved "https://registry.yarnpkg.com/@types/hast/-/hast-3.0.1.tgz#e1705ec9258ac4885659c2d50bac06b4fcd16466"
+  integrity sha512-hs/iBJx2aydugBQx5ETV3ZgeSS0oIreQrFJ4bjBl0XvM4wAmDjFEALY7p0rTSLt2eL+ibjRAAs9dTPiCLtmbqQ==
+  dependencies:
+    "@types/unist" "*"
+
 "@types/hoist-non-react-statics@*":
   version "3.3.1"
   resolved "https://registry.yarnpkg.com/@types/hoist-non-react-statics/-/hoist-non-react-statics-3.3.1.tgz#1124aafe5118cb591977aeb1ceaaed1070eb039f"
@@ -3983,6 +3990,11 @@
   resolved "https://registry.yarnpkg.com/@types/katex/-/katex-0.11.1.tgz#34de04477dcf79e2ef6c8d23b41a3d81f9ebeaf5"
   integrity sha512-DUlIj2nk0YnJdlWgsFuVKcX27MLW0KbKmGVoUHmFr+74FYYNUDAaj9ZqTADvsbE8rfxuVmSFc7KczYn5Y09ozg==
 
+"@types/katex@^0.16.0":
+  version "0.16.3"
+  resolved "https://registry.yarnpkg.com/@types/katex/-/katex-0.16.3.tgz#a341c89705145b7dd8e2a133b282a133eabe6076"
+  integrity sha512-CeVMX9EhVUW8MWnei05eIRks4D5Wscw/W9Byz1s3PA+yJvcdvq9SaDjiUKvRvEgjpdTyJMjQA43ae4KTwsvOPg==
+
 "@types/lodash-es@4.17.6":
   version "4.17.6"
   resolved "https://registry.yarnpkg.com/@types/lodash-es/-/lodash-es-4.17.6.tgz#c2ed4c8320ffa6f11b43eb89e9eaeec65966a0a0"
@@ -4224,6 +4236,11 @@
   resolved "https://registry.yarnpkg.com/@types/unist/-/unist-2.0.6.tgz#250a7b16c3b91f672a24552ec64678eeb1d3a08d"
   integrity sha512-PBjIUxZHOuj0R15/xuwJYjFi+KZdNFrehocChv4g5hu6aFroHue8m0lBP0POdK2nKzbw0cgV1mws8+V/JAcEkQ==
 
+"@types/unist@^3.0.0":
+  version "3.0.0"
+  resolved "https://registry.yarnpkg.com/@types/unist/-/unist-3.0.0.tgz#988ae8af1e5239e89f9fbb1ade4c935f4eeedf9a"
+  integrity sha512-MFETx3tbTjE7Uk6vvnWINA/1iJ7LuMdO4fcq8UfF0pRbj01aGLduVvQcRyswuACJdpnHgg8E3rQLhaRdNEJS0w==
+
 "@types/url-join@4.0.1":
   version "4.0.1"
   resolved "https://registry.yarnpkg.com/@types/url-join/-/url-join-4.0.1.tgz#4989c97f969464647a8586c7252d97b449cdc045"
@@ -5587,7 +5604,7 @@ commander@^7.2.0:
   resolved "https://registry.yarnpkg.com/commander/-/commander-7.2.0.tgz#a36cb57d0b501ce108e4d20559a150a391d97ab7"
   integrity sha512-QrWXB+ZQSVPmIWIhtEO9H+gwHaMGYiF5ChvoJ+K9ZGHG/sVsa6yiesAD1GC/x46sET00Xlwo1u49RVVVzvcSkw==
 
-commander@^8.0.0:
+commander@^8.0.0, commander@^8.3.0:
   version "8.3.0"
   resolved "https://registry.yarnpkg.com/commander/-/commander-8.3.0.tgz#4837ea1b2da67b9c616a67afbb0fafee567bca66"
   integrity sha512-OkTL9umf+He2DZkUq8f8J9of7yL6RJKI24dVITBmNfZBmri9zYZQrKkuXiKhyfPSu8tUhnVBB1iKXevvnlR4Ww==
@@ -6148,6 +6165,13 @@ detect-newline@^3.0.0:
   resolved "https://registry.yarnpkg.com/detect-newline/-/detect-newline-3.1.0.tgz#576f5dfc63ae1a192ff192d8ad3af6308991b651"
   integrity sha512-TLz+x/vEXm/Y7P7wn1EJFNLxYpUD4TgMosxY6fAVJUnJMbupHBOncxyWUG9OpTaH9EBD7uFI5LfEgmMOc54DsA==
 
+devlop@^1.0.0, devlop@^1.1.0:
+  version "1.1.0"
+  resolved "https://registry.yarnpkg.com/devlop/-/devlop-1.1.0.tgz#4db7c2ca4dc6e0e834c30be70c94bbc976dc7018"
+  integrity sha512-RWmIqhcFf1lRYBvNmr7qTNuyCt/7/ns2jbpp1+PalgE/rDQcBT0fioSMUpJ93irlUhC5hrg4cYqe6U+0ImW0rA==
+  dependencies:
+    dequal "^2.0.0"
+
 diff-sequences@^29.4.2:
   version "29.4.2"
   resolved "https://registry.yarnpkg.com/diff-sequences/-/diff-sequences-29.4.2.tgz#711fe6bd8a5869fe2539cee4a5152425ff671fda"
@@ -7879,6 +7903,37 @@ hast-to-hyperscript@^9.0.0:
     unist-util-is "^4.0.0"
     web-namespaces "^1.0.0"
 
+hast-util-from-dom@^5.0.0:
+  version "5.0.0"
+  resolved "https://registry.yarnpkg.com/hast-util-from-dom/-/hast-util-from-dom-5.0.0.tgz#d32edd25bf28f4b178b5ae318f8d05762e67bd16"
+  integrity sha512-d6235voAp/XR3Hh5uy7aGLbM3S4KamdW0WEgOaU1YoewnuYw4HXb5eRtv9g65m/RFGEfUY1Mw4UqCc5Y8L4Stg==
+  dependencies:
+    "@types/hast" "^3.0.0"
+    hastscript "^8.0.0"
+    web-namespaces "^2.0.0"
+
+hast-util-from-html-isomorphic@^2.0.0:
+  version "2.0.0"
+  resolved "https://registry.yarnpkg.com/hast-util-from-html-isomorphic/-/hast-util-from-html-isomorphic-2.0.0.tgz#b31baee386a899a2472326a3c5692f29f86d1d3c"
+  integrity sha512-zJfpXq44yff2hmE0XmwEOzdWin5xwH+QIhMLOScpX91e/NSGPsAzNCvLQDIEPyO2TXi+lBmU6hjLIhV8MwP2kw==
+  dependencies:
+    "@types/hast" "^3.0.0"
+    hast-util-from-dom "^5.0.0"
+    hast-util-from-html "^2.0.0"
+    unist-util-remove-position "^5.0.0"
+
+hast-util-from-html@^2.0.0:
+  version "2.0.1"
+  resolved "https://registry.yarnpkg.com/hast-util-from-html/-/hast-util-from-html-2.0.1.tgz#9cd38ee81bf40b2607368b92a04b0905fa987488"
+  integrity sha512-RXQBLMl9kjKVNkJTIO6bZyb2n+cUH8LFaSSzo82jiLT6Tfc+Pt7VQCS+/h3YwG4jaNE2TA2sdJisGWR+aJrp0g==
+  dependencies:
+    "@types/hast" "^3.0.0"
+    devlop "^1.1.0"
+    hast-util-from-parse5 "^8.0.0"
+    parse5 "^7.0.0"
+    vfile "^6.0.0"
+    vfile-message "^4.0.0"
+
 hast-util-from-parse5@^6.0.0:
   version "6.0.1"
   resolved "https://registry.yarnpkg.com/hast-util-from-parse5/-/hast-util-from-parse5-6.0.1.tgz#554e34abdeea25ac76f5bd950a1f0180e0b3bc2a"
@@ -7891,6 +7946,20 @@ hast-util-from-parse5@^6.0.0:
     vfile-location "^3.2.0"
     web-namespaces "^1.0.0"
 
+hast-util-from-parse5@^8.0.0:
+  version "8.0.1"
+  resolved "https://registry.yarnpkg.com/hast-util-from-parse5/-/hast-util-from-parse5-8.0.1.tgz#654a5676a41211e14ee80d1b1758c399a0327651"
+  integrity sha512-Er/Iixbc7IEa7r/XLtuG52zoqn/b3Xng/w6aZQ0xGVxzhw5xUFxcRqdPzP6yFi/4HBYRaifaI5fQ1RH8n0ZeOQ==
+  dependencies:
+    "@types/hast" "^3.0.0"
+    "@types/unist" "^3.0.0"
+    devlop "^1.0.0"
+    hastscript "^8.0.0"
+    property-information "^6.0.0"
+    vfile "^6.0.0"
+    vfile-location "^5.0.0"
+    web-namespaces "^2.0.0"
+
 hast-util-has-property@^1.0.0:
   version "1.0.4"
   resolved "https://registry.yarnpkg.com/hast-util-has-property/-/hast-util-has-property-1.0.4.tgz#9f137565fad6082524b382c1e7d7d33ca5059f36"
@@ -7901,11 +7970,25 @@ hast-util-heading-rank@^1.0.0:
   resolved "https://registry.yarnpkg.com/hast-util-heading-rank/-/hast-util-heading-rank-1.0.1.tgz#28dfd8b0724cfb0da48308e2a794b1d9f24fd80d"
   integrity sha512-P6Hq7RCky9syMevlrN90QWpqWDXCxwIVOfQR2rK6P4GpY4bqjKEuCzoWSRORZ7vz+VgRpLnXimh+mkwvVFjbyQ==
 
+hast-util-is-element@^3.0.0:
+  version "3.0.0"
+  resolved "https://registry.yarnpkg.com/hast-util-is-element/-/hast-util-is-element-3.0.0.tgz#6e31a6532c217e5b533848c7e52c9d9369ca0932"
+  integrity sha512-Val9mnv2IWpLbNPqc/pUem+a7Ipj2aHacCwgNfTiK0vJKl0LF+4Ba4+v1oPHFpf3bLYmreq0/l3Gud9S5OH42g==
+  dependencies:
+    "@types/hast" "^3.0.0"
+
 hast-util-parse-selector@^2.0.0:
   version "2.2.5"
   resolved "https://registry.yarnpkg.com/hast-util-parse-selector/-/hast-util-parse-selector-2.2.5.tgz#d57c23f4da16ae3c63b3b6ca4616683313499c3a"
   integrity sha512-7j6mrk/qqkSehsM92wQjdIgWM2/BW61u/53G6xmC8i1OmEdKLHbk419QKQUjz6LglWsfqoiHmyMRkP1BGjecNQ==
 
+hast-util-parse-selector@^4.0.0:
+  version "4.0.0"
+  resolved "https://registry.yarnpkg.com/hast-util-parse-selector/-/hast-util-parse-selector-4.0.0.tgz#352879fa86e25616036037dd8931fb5f34cb4a27"
+  integrity sha512-wkQCkSYoOGCRKERFWcxMVMOcYE2K1AaNLU8DXS9arxnLOUEWbOXKXiJUNzEpqZ3JOKpnha3jkFrumEjVliDe7A==
+  dependencies:
+    "@types/hast" "^3.0.0"
+
 hast-util-raw@6.0.1:
   version "6.0.1"
   resolved "https://registry.yarnpkg.com/hast-util-raw/-/hast-util-raw-6.0.1.tgz#973b15930b7529a7b66984c98148b46526885977"
@@ -7959,6 +8042,16 @@ hast-util-to-string@^1.0.0:
   resolved "https://registry.yarnpkg.com/hast-util-to-string/-/hast-util-to-string-1.0.4.tgz#9b24c114866bdb9478927d7e9c36a485ac728378"
   integrity sha512-eK0MxRX47AV2eZ+Lyr18DCpQgodvaS3fAQO2+b9Two9F5HEoRPhiUMNzoXArMJfZi2yieFzUBMRl3HNJ3Jus3w==
 
+hast-util-to-text@^4.0.0:
+  version "4.0.0"
+  resolved "https://registry.yarnpkg.com/hast-util-to-text/-/hast-util-to-text-4.0.0.tgz#7f33a45d0bf7981ead44e82d9d8d75f511b3642f"
+  integrity sha512-EWiE1FSArNBPUo1cKWtzqgnuRQwEeQbQtnFJRYV1hb1BWDgrAlBU0ExptvZMM/KSA82cDpm2sFGf3Dmc5Mza3w==
+  dependencies:
+    "@types/hast" "^3.0.0"
+    "@types/unist" "^3.0.0"
+    hast-util-is-element "^3.0.0"
+    unist-util-find-after "^5.0.0"
+
 hast-util-whitespace@^2.0.0:
   version "2.0.1"
   resolved "https://registry.yarnpkg.com/hast-util-whitespace/-/hast-util-whitespace-2.0.1.tgz#0ec64e257e6fc216c7d14c8a1b74d27d650b4557"
@@ -7975,6 +8068,17 @@ hastscript@^6.0.0:
     property-information "^5.0.0"
     space-separated-tokens "^1.0.0"
 
+hastscript@^8.0.0:
+  version "8.0.0"
+  resolved "https://registry.yarnpkg.com/hastscript/-/hastscript-8.0.0.tgz#4ef795ec8dee867101b9f23cc830d4baf4fd781a"
+  integrity sha512-dMOtzCEd3ABUeSIISmrETiKuyydk1w0pa+gE/uormcTpSYuaNJPbX1NU3JLyscSLjwAQM8bWMhhIlnCqnRvDTw==
+  dependencies:
+    "@types/hast" "^3.0.0"
+    comma-separated-tokens "^2.0.0"
+    hast-util-parse-selector "^4.0.0"
+    property-information "^6.0.0"
+    space-separated-tokens "^2.0.0"
+
 he@^1.2.0:
   version "1.2.0"
   resolved "https://registry.yarnpkg.com/he/-/he-1.2.0.tgz#84ae65fa7eafb165fddb61566ae14baf05664f0f"
@@ -9455,6 +9559,13 @@ jws@^4.0.0:
     jwa "^2.0.0"
     safe-buffer "^5.0.1"
 
+katex@0.16.8, katex@^0.16.0:
+  version "0.16.8"
+  resolved "https://registry.yarnpkg.com/katex/-/katex-0.16.8.tgz#89b453f40e8557f423f31a1009e9298dd99d5ceb"
+  integrity sha512-ftuDnJbcbOckGY11OO+zg3OofESlbR5DRl2cmN8HeWeeFIV7wTXvAOx8kEjZjobhA+9wh2fbKeO6cdcA9Mnovg==
+  dependencies:
+    commander "^8.3.0"
+
 katex@^0.13.0:
   version "0.13.24"
   resolved "https://registry.yarnpkg.com/katex/-/katex-0.13.24.tgz#fe55455eb455698cb24b911a353d16a3c855d905"
@@ -11864,6 +11975,19 @@ regjsparser@^0.9.1:
   dependencies:
     jsesc "~0.5.0"
 
+rehype-katex@7.0.0:
+  version "7.0.0"
+  resolved "https://registry.yarnpkg.com/rehype-katex/-/rehype-katex-7.0.0.tgz#f5e9e2825981175a7b0a4d58ed9816c33576dfed"
+  integrity sha512-h8FPkGE00r2XKU+/acgqwWUlyzve1IiOKwsEkg4pDL3k48PiE0Pt+/uLtVHDVkN1yA4iurZN6UES8ivHVEQV6Q==
+  dependencies:
+    "@types/hast" "^3.0.0"
+    "@types/katex" "^0.16.0"
+    hast-util-from-html-isomorphic "^2.0.0"
+    hast-util-to-text "^4.0.0"
+    katex "^0.16.0"
+    unist-util-visit-parents "^6.0.0"
+    vfile "^6.0.0"
+
 rehype-slug@^4.0.1:
   version "4.0.1"
   resolved "https://registry.yarnpkg.com/rehype-slug/-/rehype-slug-4.0.1.tgz#313274501cffa997bd52dd57bf2da5851959747a"
@@ -13303,6 +13427,14 @@ unist-builder@2.0.3, unist-builder@^2.0.0:
   resolved "https://registry.yarnpkg.com/unist-builder/-/unist-builder-2.0.3.tgz#77648711b5d86af0942f334397a33c5e91516436"
   integrity sha512-f98yt5pnlMWlzP539tPc4grGMsFaQQlP/vM396b00jngsiINumNmsY8rkXjfoi1c6QaM8nQ3vaGDuoKWbe/1Uw==
 
+unist-util-find-after@^5.0.0:
+  version "5.0.0"
+  resolved "https://registry.yarnpkg.com/unist-util-find-after/-/unist-util-find-after-5.0.0.tgz#3fccc1b086b56f34c8b798e1ff90b5c54468e896"
+  integrity sha512-amQa0Ep2m6hE2g72AugUItjbuM8X8cGQnFoHk0pGfrFeT9GZhzN5SW8nRsiGKK7Aif4CrACPENkA6P/Lw6fHGQ==
+  dependencies:
+    "@types/unist" "^3.0.0"
+    unist-util-is "^6.0.0"
+
 unist-util-generated@^1.0.0:
   version "1.1.6"
   resolved "https://registry.yarnpkg.com/unist-util-generated/-/unist-util-generated-1.1.6.tgz#5ab51f689e2992a472beb1b35f2ce7ff2f324d4b"
@@ -13323,6 +13455,13 @@ unist-util-is@^5.0.0:
   resolved "https://registry.yarnpkg.com/unist-util-is/-/unist-util-is-5.2.0.tgz#37eed0617b76c114fd34d44c201aa96fd928b309"
   integrity sha512-Glt17jWwZeyqrFqOK0pF1Ded5U3yzJnFr8CG1GMjCWTp9zDo2p+cmD6pWbZU8AgM5WU3IzRv6+rBwhzsGh6hBQ==
 
+unist-util-is@^6.0.0:
+  version "6.0.0"
+  resolved "https://registry.yarnpkg.com/unist-util-is/-/unist-util-is-6.0.0.tgz#b775956486aff107a9ded971d996c173374be424"
+  integrity sha512-2qCTHimwdxLfz+YzdGfkqNlH0tLi9xjTnHddPmJwtIG9MGsdbutfTc4P+haPD7l7Cjxf/WZj+we5qfVPvvxfYw==
+  dependencies:
+    "@types/unist" "^3.0.0"
+
 unist-util-position-from-estree@^1.0.0, unist-util-position-from-estree@^1.1.0:
   version "1.1.2"
   resolved "https://registry.yarnpkg.com/unist-util-position-from-estree/-/unist-util-position-from-estree-1.1.2.tgz#8ac2480027229de76512079e377afbcabcfcce22"
@@ -13357,6 +13496,14 @@ unist-util-remove-position@^4.0.0:
     "@types/unist" "^2.0.0"
     unist-util-visit "^4.0.0"
 
+unist-util-remove-position@^5.0.0:
+  version "5.0.0"
+  resolved "https://registry.yarnpkg.com/unist-util-remove-position/-/unist-util-remove-position-5.0.0.tgz#fea68a25658409c9460408bc6b4991b965b52163"
+  integrity sha512-Hp5Kh3wLxv0PHj9m2yZhhLt58KzPtEYKQQ4yxfYFEO7EvHwzyDYnduhHnY1mDxoqr7VUwVuHXk9RXKIiYS1N8Q==
+  dependencies:
+    "@types/unist" "^3.0.0"
+    unist-util-visit "^5.0.0"
+
 unist-util-remove@^2.0.0:
   version "2.1.0"
   resolved "https://registry.yarnpkg.com/unist-util-remove/-/unist-util-remove-2.1.0.tgz#b0b4738aa7ee445c402fda9328d604a02d010588"
@@ -13378,6 +13525,13 @@ unist-util-stringify-position@^3.0.0:
   dependencies:
     "@types/unist" "^2.0.0"
 
+unist-util-stringify-position@^4.0.0:
+  version "4.0.0"
+  resolved "https://registry.yarnpkg.com/unist-util-stringify-position/-/unist-util-stringify-position-4.0.0.tgz#449c6e21a880e0855bf5aabadeb3a740314abac2"
+  integrity sha512-0ASV06AAoKCDkS2+xw5RXJywruurpbC4JZSm7nr7MOt1ojAzvyyaO+UxZf18j8FCF6kmzCZKcAgN/yu2gm2XgQ==
+  dependencies:
+    "@types/unist" "^3.0.0"
+
 unist-util-visit-parents@^3.0.0:
   version "3.1.1"
   resolved "https://registry.yarnpkg.com/unist-util-visit-parents/-/unist-util-visit-parents-3.1.1.tgz#65a6ce698f78a6b0f56aa0e88f13801886cdaef6"
@@ -13394,6 +13548,14 @@ unist-util-visit-parents@^5.0.0, unist-util-visit-parents@^5.1.1:
     "@types/unist" "^2.0.0"
     unist-util-is "^5.0.0"
 
+unist-util-visit-parents@^6.0.0:
+  version "6.0.1"
+  resolved "https://registry.yarnpkg.com/unist-util-visit-parents/-/unist-util-visit-parents-6.0.1.tgz#4d5f85755c3b8f0dc69e21eca5d6d82d22162815"
+  integrity sha512-L/PqWzfTP9lzzEa6CKs0k2nARxTdZduw3zyh8d2NVBnsyvHjSX4TWse388YrrQKbvI8w20fGjGlhgT96WwKykw==
+  dependencies:
+    "@types/unist" "^3.0.0"
+    unist-util-is "^6.0.0"
+
 unist-util-visit@2.0.3, unist-util-visit@^2.0.0:
   version "2.0.3"
   resolved "https://registry.yarnpkg.com/unist-util-visit/-/unist-util-visit-2.0.3.tgz#c3703893146df47203bb8a9795af47d7b971208c"
@@ -13412,6 +13574,15 @@ unist-util-visit@^4.0.0:
     unist-util-is "^5.0.0"
     unist-util-visit-parents "^5.1.1"
 
+unist-util-visit@^5.0.0:
+  version "5.0.0"
+  resolved "https://registry.yarnpkg.com/unist-util-visit/-/unist-util-visit-5.0.0.tgz#a7de1f31f72ffd3519ea71814cccf5fd6a9217d6"
+  integrity sha512-MR04uvD+07cwl/yhVuVWAtw+3GOR/knlL55Nd/wAdblk27GCVt3lqpTivy/tkJcZoNPzTwS1Y+KMojlLDhoTzg==
+  dependencies:
+    "@types/unist" "^3.0.0"
+    unist-util-is "^6.0.0"
+    unist-util-visit-parents "^6.0.0"
+
 universalify@^2.0.0:
   version "2.0.0"
   resolved "https://registry.yarnpkg.com/universalify/-/universalify-2.0.0.tgz#75a4984efedc4b08975c5aeb73f530d02df25717"
@@ -13536,6 +13707,14 @@ vfile-location@^4.0.0:
     "@types/unist" "^2.0.0"
     vfile "^5.0.0"
 
+vfile-location@^5.0.0:
+  version "5.0.2"
+  resolved "https://registry.yarnpkg.com/vfile-location/-/vfile-location-5.0.2.tgz#220d9ca1ab6f8b2504a4db398f7ebc149f9cb464"
+  integrity sha512-NXPYyxyBSH7zB5U6+3uDdd6Nybz6o6/od9rk8bp9H8GR3L+cm/fC0uUTbqBmUTnMCUDslAGBOIKNfvvb+gGlDg==
+  dependencies:
+    "@types/unist" "^3.0.0"
+    vfile "^6.0.0"
+
 vfile-message@^2.0.0:
   version "2.0.4"
   resolved "https://registry.yarnpkg.com/vfile-message/-/vfile-message-2.0.4.tgz#5b43b88171d409eae58477d13f23dd41d52c371a"
@@ -13552,6 +13731,14 @@ vfile-message@^3.0.0:
     "@types/unist" "^2.0.0"
     unist-util-stringify-position "^3.0.0"
 
+vfile-message@^4.0.0:
+  version "4.0.2"
+  resolved "https://registry.yarnpkg.com/vfile-message/-/vfile-message-4.0.2.tgz#c883c9f677c72c166362fd635f21fc165a7d1181"
+  integrity sha512-jRDZ1IMLttGj41KcZvlrYAaI3CfqpLpfpf+Mfig13viT6NKvRzWZ+lXz0Y5D60w6uJIBAOGq9mSHf0gktF0duw==
+  dependencies:
+    "@types/unist" "^3.0.0"
+    unist-util-stringify-position "^4.0.0"
+
 vfile@^4.0.0:
   version "4.2.1"
   resolved "https://registry.yarnpkg.com/vfile/-/vfile-4.2.1.tgz#03f1dce28fc625c625bc6514350fbdb00fa9e624"
@@ -13572,6 +13759,15 @@ vfile@^5.0.0:
     unist-util-stringify-position "^3.0.0"
     vfile-message "^3.0.0"
 
+vfile@^6.0.0:
+  version "6.0.1"
+  resolved "https://registry.yarnpkg.com/vfile/-/vfile-6.0.1.tgz#1e8327f41eac91947d4fe9d237a2dd9209762536"
+  integrity sha512-1bYqc7pt6NIADBJ98UiG0Bn/CHIVOoZ/IyEkqIruLg0mE1BKzkOXY2D6CSqQIcKqgadppE5lrxgWXJmXd7zZJw==
+  dependencies:
+    "@types/unist" "^3.0.0"
+    unist-util-stringify-position "^4.0.0"
+    vfile-message "^4.0.0"
+
 victory-vendor@^36.6.8:
   version "36.6.8"
   resolved "https://registry.yarnpkg.com/victory-vendor/-/victory-vendor-36.6.8.tgz#5a1c555ca99a39fdb66a6c959c8426eb834893a2"
@@ -13732,6 +13928,11 @@ web-namespaces@^1.0.0:
   resolved "https://registry.yarnpkg.com/web-namespaces/-/web-namespaces-1.1.4.tgz#bc98a3de60dadd7faefc403d1076d529f5e030ec"
   integrity sha512-wYxSGajtmoP4WxfejAPIr4l0fVh+jeMXZb08wNc0tMg6xsfZXj3cECqIK0G7ZAqUq0PP8WlMDtaOGVBTAWztNw==
 
+web-namespaces@^2.0.0:
+  version "2.0.1"
+  resolved "https://registry.yarnpkg.com/web-namespaces/-/web-namespaces-2.0.1.tgz#1010ff7c650eccb2592cebeeaf9a1b253fd40692"
+  integrity sha512-bKr1DkiNa2krS7qxNtdrtHAmzuYGFQLiQ13TsorsdT6ULTkPLKuu5+GsFpDlg6JFjUTwX2DyhMPG2be8uPrqsQ==
+
 webidl-conversions@^3.0.0:
   version "3.0.1"
   resolved "https://registry.yarnpkg.com/webidl-conversions/-/webidl-conversions-3.0.1.tgz#24534275e2a7bc6be7bc86611cc16ae0a5654871"