Many changes

1 files changed, 328 insertions, 113 deletions

diff --git a/zif.tex b/zif.tex
index 396e908..ea69e4f 100644
--- a/zif.tex
+++ b/zif.tex
@@ -1,4 +1,4 @@
-\documentclass[aspectratio=169,dvipsnames]{beamer}
+\documentclass[fleqn,aspectratio=169,dvipsnames]{beamer}
 
 \setbeamerfont{title}{family=\bf}
 \setbeamerfont{frametitle}{family=\bf}
@@ -126,11 +126,11 @@
 
       \bigskip
 
-      High-dimensional landscapes differ from cartoons in important ways
+      \alert<2-3>{High-dimensional landscapes differ from cartoons in important ways}
 
       \bigskip
 
-      Generic rugged landscapes covered mostly by basins attached to marginal optima
+      \alert<4-6>{Generic rugged landscapes covered mostly by basins attached to marginal optima}
 
       \bigskip
 
@@ -138,17 +138,19 @@
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
-        \onslide<1>\includegraphics[width=\textwidth]{figs/complex_landscapes.png}\centering\\\tiny C Cammarota
-        \onslide<2>\includegraphics[width=\textwidth]{figs/fake_landscape.png}\\\tiny S Martiniani, YouTube
-        \onslide<3>\includegraphics[width=\textwidth]{figs/real_landscape.png}\smallskip\\\tiny\fullcite{Suryadevara_2024_The}
-        \onslide<4>\includegraphics[width=\textwidth]{figs/eigenvalue_convergence.pdf}\smallskip\\\tiny\fullcite{Kent-Dobias_2024_Algorithm-independent}
+        \onslide<1>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/complex_landscapes.png}\centering\\\tiny C Cammarota\end{minipage}
+        \onslide<2>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/fake_landscape.png}\\\tiny S Martiniani, YouTube\end{minipage}
+        \onslide<3>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/real_landscape.png}\smallskip\\\tiny\fullcite{Suryadevara_2024_The}\end{minipage}
+        \onslide<4>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/basin_1.png}\end{minipage}
+        \onslide<5>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/basin_2.png}\end{minipage}
+        \onslide<6>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/eigenvalue_convergence.pdf}\smallskip\\\tiny\fullcite{Kent-Dobias_2024_Algorithm-independent}\end{minipage}
       \end{overprint}
     \end{column}
   \end{columns}
 \end{frame}
 
 \begin{frame}
-  \frametitle{How to count: Kac--Rice}
+  \frametitle{Typical complexity}
 
       Number of stationary points with $\nabla H(\boldsymbol x)=0$ given by integral
       over Kac--Rice measure
@@ -171,10 +173,16 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Conditioning on the type of minimum: the spherical models}
+  \frametitle{Conditioning on the type of point: spherical models}
   \begin{columns}
     \begin{column}{0.5\textwidth}
-      In spherical spin glasses, all points have the same Hessian spectral density: semicircle with radius $\mu_\text m$, but different shifts
+      In spherical spin glasses with
+      \[
+        H(\boldsymbol x)=\sum_pa_pH_p(\boldsymbol x)
+        \qquad
+        H_p(\boldsymbol x)=\frac1{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}x_{i_1}\cdots x_{i_p}
+      \]
+      all points have the same Hessian spectral density: semicircle with radius $\mu_\text m$, but different shifts
       \[
         \mu=\frac1N\operatorname{Tr}\operatorname{Hess} H(\boldsymbol x)
       \]
@@ -200,8 +208,7 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Less simple mean-field models}
-  \framesubtitle{The mixed spherical spin glasses}
+  \frametitle{Out of equilibrium dynamics and complexity: spherical models}
 
   \begin{columns}
     \begin{column}{0.41\textwidth}
@@ -372,8 +379,8 @@
       {
       \small
       \begin{align*}
-        \hspace{0em}&\delta(\lambda_\text{min}(A)) \\
-        \hspace{0em}&=\lim_{\beta\to\infty}\int
+        \hspace{-3em}&\delta(\lambda_\text{min}(A)) \\
+        \hspace{-3em}&=\lim_{\beta\to\infty}\int
           \frac{d\boldsymbol s\,\delta(N-\|\boldsymbol s\|^2)e^{-\beta\boldsymbol s^TA\boldsymbol s}}
             {\int d\boldsymbol s'\,\delta(N-\|\boldsymbol s'\|^2)e^{-\beta\boldsymbol s'^TA\boldsymbol s'}}
         \delta\left(\frac{\boldsymbol s^TA\boldsymbol s}N\right)
@@ -464,9 +471,9 @@
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
-        \onslide<1>\includegraphics[width=\textwidth]{figs/neural_landscape_1.png}\smallskip\\\tiny\fullcite{Liu_2022_Loss}
-        \onslide<2>\includegraphics[width=\textwidth]{figs/neural_landscape_2.png}\smallskip\\\tiny\fullcite{Liu_2022_Loss}
-        \onslide<3>\includegraphics[width=\textwidth]{figs/draxler_2018.png}\smallskip\\\tiny\fullcite{Draxler_2018_Essentially}
+        \onslide<1>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/neural_landscape_1.png}\smallskip\\\tiny\fullcite{Liu_2022_Loss}\end{minipage}
+        \onslide<2>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/neural_landscape_2.png}\smallskip\\\tiny\fullcite{Liu_2022_Loss}\end{minipage}
+        \onslide<3>\begin{minipage}[c][0.3\height][c]{\textwidth}\includegraphics[width=\textwidth]{figs/draxler_2018.png}\smallskip\\\tiny\fullcite{Draxler_2018_Essentially}\end{minipage}
       \end{overprint}
     \end{column}
   \end{columns}
@@ -600,8 +607,7 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{The Euler characteristic \boldmath{$\chi$}}
-  \framesubtitle{Computing the Euler characteristic}
+  \frametitle{Computing the Euler characteristic}
   \begin{columns}
     \begin{column}{0.5\textwidth}
       Morse theory: gradient flow on an arbitrary ``height'' function $h$ makes a complex
@@ -664,8 +670,6 @@
         \Omega=\{\boldsymbol x\in S^{N-1}\subset\mathbb R^N\mid f_{\boldsymbol x}(\boldsymbol a^\mu)=y^\mu,1\leq\mu\leq M\}
       \]
 
-      \bigskip
-
       Lagrange multipliers replace differential geometry:
       \[
         L(\boldsymbol  x,\boldsymbol \omega)=h(\boldsymbol  x)+\omega^0(\|\boldsymbol  x\|^2-N)+\sum_{\mu=1}^M\omega^\mu(f_{\boldsymbol x}(\boldsymbol a^\mu)-y^\mu)
@@ -679,6 +683,7 @@
       \end{overprint}
     \end{column}
   \end{columns}
+  \vspace{-1em}
       \[
         \chi(\Omega)
         =\int_{\mathbb R^{N+M+1}} d\boldsymbol  x\,d\boldsymbol \omega\,\delta\big(\begin{bmatrix}\frac{\partial L}{\partial\boldsymbol  x}&\frac{\partial L}{\partial\boldsymbol \omega}\end{bmatrix}\big)
@@ -687,13 +692,13 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Computing the Euler characteristic}
+  \frametitle{Computing the Euler characteristic: example}
   \begin{columns}
     \begin{column}{0.5\textwidth}
-      $M$ data points, $N$ parameters, $\alpha=M/N$, $\boldsymbol J^\mu\in\mathbb R^{N^p}$, $\overline{J^2}=\frac{p!}{2N^p}$
+      $M$ data points, $N$ parameters, $\alpha=M/N$, $y^\mu=V_0$, $\boldsymbol a^\mu$ spherical $p$-spin coefficients,
       \[
-        V_0=f_{\boldsymbol x}(\boldsymbol J^\mu)
-        =\frac1{p!}\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ ^\mu_{i_1,\ldots,i_p}x_{i_1}\cdots x_{i_p}
+        f_{\boldsymbol x}(\boldsymbol a^\mu)
+        =H^{(\mu)}_p(\boldsymbol x)
       \]
       Average Euler characteristic reduced to integral over $m=\frac1N\boldsymbol x\cdot\boldsymbol x_0$,
       \[
@@ -701,6 +706,10 @@
         =\left(\frac N{2\pi}\right)^{\frac12}
         \int dm\,g(m)e^{N\mathcal S_\chi(m)}
       \]
+
+      \smallskip\tiny
+
+      \fullcite{Kent-Dobias_2025_On}
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
@@ -712,26 +721,25 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{A simple model of nonlinear least squares}
-  \framesubtitle{Results}
+  \frametitle{Computing the Euler characteristic: example}
   \begin{columns}
     \begin{column}{0.5\textwidth}
-      $M$ data points, $N$ parameters, $\alpha=M/N$, $\boldsymbol J^\mu\in\mathbb R^{N^p}$, $\overline{J^2}=\frac{p!}{2N^p}$
+      $M$ data points, $N$ parameters, $\alpha=M/N$, $y^\mu=V_0$, $\boldsymbol a^\mu$ spherical $p$-spin coefficients,
       \[
-        V_0=f_{\boldsymbol x}(\boldsymbol J^\mu)
-        =\frac1{p!}\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ ^\mu_{i_1,\ldots,i_p}x_{i_1}\cdots x_{i_p}
+        f_{\boldsymbol x}(\boldsymbol a^\mu)
+        =H^{(\mu)}_p(\boldsymbol x)
       \]
       Simplest case: $p=1$, $\Omega$ is intersection of random hyperplanes with the sphere
 
       \medskip
 
-      Results in $\chi(\Omega)=2$ or $\chi(\Omega)=0$ depending on whether solutions exist
+      Results in $\overline{\chi(\Omega)}=0$ or $\overline{\chi(\Omega)}=2$ depending on whether solutions exist
 
       \medskip
 
-      \hspace{2em}\includegraphics[width=0.33\textwidth]{figs/connected.pdf}
+      \hspace{2em}\includegraphics[width=0.33\textwidth]{figs/gone.pdf}
       \hfill
-      \includegraphics[width=0.33\textwidth]{figs/gone.pdf}\hspace{2em}
+      \includegraphics[width=0.33\textwidth]{figs/connected.pdf}\hspace{2em}
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
@@ -748,8 +756,14 @@
 \end{frame}
 
 \begin{frame}
+  \frametitle{Computing the Euler characteristic: example}
   \begin{columns}
     \begin{column}{0.5\textwidth}
+      $M$ data points, $N$ parameters, $\alpha=M/N$, $y^\mu=V_0$, $\boldsymbol a^\mu$ spherical $p$-spin coefficients,
+      \[
+        f_{\boldsymbol x}(\boldsymbol a^\mu)
+        =H^{(\mu)}_p(\boldsymbol x)
+      \]
       For $p>1$, new phases possible
 
       \medskip
@@ -785,34 +799,39 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{A simple model of nonlinear least squares}
-  \framesubtitle{Results}
+  \frametitle{Computing the Euler characteristic: example}
+
   \begin{columns}
-    \begin{column}{0.5\textwidth}
-      $M$ data points, $N$ parameters, $\alpha=M/N$
-      \[
-        V_0=\hat f(J\mid \pmb a)
-        =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p}
-      \]
+    \begin{column}{1.1\textwidth}
+      \hspace{1em}Phases for inhomogeneous models: $1-\lambda$ parts linear ($p=1$) plus $\lambda$ quadratic ($p=2$)
+
+      \medskip
 
-      For $p\geq2$, different phases with $|\chi(\Omega)|\gg1$ with varying sign
+      \includegraphics[width=0.21\columnwidth]{figs/phases_12_0.pdf}
+      \nolinebreak\hspace{-2.5em}
+      \includegraphics[width=0.21\columnwidth]{figs/phases_12_1.pdf}
+      \nolinebreak\hspace{-2.5em}
+      \includegraphics[width=0.21\columnwidth]{figs/phases_12_2.pdf}
+      \nolinebreak\hspace{-2.5em}
+      \includegraphics[width=0.21\columnwidth]{figs/phases_12_3.pdf}
+      \nolinebreak\hspace{-2.5em}
+      \includegraphics[width=0.21\columnwidth]{figs/phases_12_4.pdf}
+      \nolinebreak\hspace{-2.5em}
+      \includegraphics[width=0.21\columnwidth]{figs/phases_12_5.pdf}
 
       \medskip
 
-      \includegraphics[width=0.23\textwidth]{figs/middle.pdf}
-      \includegraphics[width=0.23\textwidth]{figs/complex.pdf}
-      \includegraphics[width=0.23\textwidth]{figs/shattered.pdf}
-      \includegraphics[width=0.23\textwidth]{figs/gone.pdf}
-    \end{column}
-    \begin{column}{0.5\textwidth}
-      \begin{overprint}
-        \onslide<1>\centering\includegraphics[width=0.8\textwidth]{figs/middle.pdf}\\$\chi(\Omega)\ll0$
-        \onslide<2>\centering\includegraphics[width=0.8\textwidth]{figs/complex.pdf}\\$\chi(\Omega)\ll0$
-        \onslide<3>\centering\includegraphics[width=0.8\textwidth]{figs/shattered.pdf}\\$\chi(\Omega)\gg0$
-        \onslide<4>\includegraphics[width=\textwidth]{figs/phases_2.pdf}
-        \onslide<5>\includegraphics[width=\textwidth]{figs/phases_3.pdf}
-        \onslide<6>\includegraphics[width=\textwidth]{figs/phases_4.pdf}
-      \end{overprint}
+      \hspace{5em}
+      \includegraphics[width=0.13\textwidth]{figs/connected.pdf}
+      \hfill
+      \includegraphics[width=0.13\textwidth]{figs/middle.pdf}
+      \hfill
+      \includegraphics[width=0.13\textwidth]{figs/complex.pdf}
+      \hfill
+      \includegraphics[width=0.13\textwidth]{figs/shattered.pdf}
+      \hfill
+      \includegraphics[width=0.13\textwidth]{figs/gone.pdf}
+      \hspace{5em}
     \end{column}
   \end{columns}
 \end{frame}
@@ -825,50 +844,43 @@
       \[
         h^\mu(\boldsymbol x)\geq0 \qquad 1\leq\mu\leq M
       \]
-      Spherical perceptron: $M$ patterns $\boldsymbol\xi^\mu\in\mathbb R^N$,
-      \[
-        h^\mu(\boldsymbol x)=\boldsymbol\xi^\mu\cdot\boldsymbol x-\kappa
-      \]
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
         \onslide<1>\includegraphics[width=\textwidth]{figs/nothing_zoom.pdf}
-        \onslide<2>\includegraphics[width=\textwidth]{figs/fixed_size_zoom_points.pdf}
-        \onslide<3>\includegraphics[width=\textwidth]{figs/max_size_zoom.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/max_size_zoom.pdf}
       \end{overprint}
     \end{column}
   \end{columns}
 \end{frame}
 
 \begin{frame}
-  \frametitle{Wedged spheres}
+  \frametitle{Non manifolds: wedged spheres}
   \begin{columns}
     \begin{column}{0.5\textwidth}
       Sphere of radius $r$ uniquely defined by
       $h^\mu(\boldsymbol x)=r$ for $D$ constraints,
-      $h^\mu(\boldsymbol x)\geq r$ for other $M-D$ constraints
-      \[
-        \begin{aligned}
-          \#_r
-          =
-          \int_{\mathbb R^D} d\boldsymbol x
-          \sum_{\substack{\sigma\subset[M]\\|\sigma|=D}}
-          \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
-          \\
-          \times\bigg(\prod_{\mu\in\sigma}\delta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
-          \\
-          \times\left|
-            \det\begin{bmatrix}
-              \frac\partial{\partial\boldsymbol x}h^{\sigma_1}(\boldsymbol x)
-              &
-              \cdots
-              &
-              \frac\partial{\partial\boldsymbol x}h^{\sigma_D}(\boldsymbol x)
-            \end{bmatrix}
-          \right|
-        \end{aligned}
-      \]
-      With margin, $\#_r(\kappa)=\#_0(r+\kappa)$
+      $h^\mu(\boldsymbol x)\geq r$ for the other $M-D$ constraints
+      \begin{align*}
+        \hspace{-2em}\#_r
+        =
+        \int_{\mathbb R^D} d\boldsymbol x
+        \sum_{\substack{\sigma\subset[M]\\|\sigma|=D}}
+        \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+        \\
+        \times\bigg(\prod_{\mu\in\sigma}\delta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+        \\
+        \times\left|
+          \det\begin{bmatrix}
+            \frac\partial{\partial\boldsymbol x}h^{\sigma_1}(\boldsymbol x)
+            &
+            \cdots
+            &
+            \frac\partial{\partial\boldsymbol x}h^{\sigma_D}(\boldsymbol x)
+          \end{bmatrix}
+        \right|
+      \end{align*}
+      \alert<3>{With margin, $\#_r(\kappa)=\#_0(r+\kappa)$}
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
@@ -881,35 +893,33 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Inscribed spheres}
+  \frametitle{Non manifolds: inscribed spheres}
   \begin{columns}
     \begin{column}{0.5\textwidth}
       Sphere of maximal radius uniquely defined by
       $h^\mu(\boldsymbol x)=r$ for $D+1$ constraints,
       $h^\mu(\boldsymbol x)\geq r$ for other $M-D-1$ constraints
-      \[
-        \begin{aligned}
-          \#_\text{insc}
-          =
-          \int_0^\infty dr\int_{\mathbb R^D} d\boldsymbol x
-          \sum_{\substack{\sigma\subset[M]\\|\sigma|=D+1}}
-          \\
-          \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
-          \bigg(\prod_{\mu\in\sigma}\delta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
-          \\
-          \times\left|
-            \det\begin{bmatrix}
-              \frac\partial{\partial\boldsymbol x}h^{\sigma_1}(\boldsymbol x)
-              &
-              \cdots
-              &
-              \frac\partial{\partial\boldsymbol x}h^{\sigma_{D+1}}(\boldsymbol x)
-              \\
-              -1 & \cdots & -1
-            \end{bmatrix}
-          \right|
-        \end{aligned}
-      \]
+      \begin{align*}
+        \hspace{-2em}&\#_\text{insc}
+        =
+        \int_0^\infty dr\int_{\mathbb R^D} d\boldsymbol x
+        \sum_{\substack{\sigma\subset[M]\\|\sigma|=D+1}}
+        \\
+                     \hspace{-2em}&\times\bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+        \bigg(\prod_{\mu\in\sigma}\delta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
+        \\
+                     \hspace{-2em}&\hspace{4em}\times\left|
+          \det\begin{bmatrix}
+            \frac\partial{\partial\boldsymbol x}h^{\sigma_1}(\boldsymbol x)
+            &
+            \cdots
+            &
+            \frac\partial{\partial\boldsymbol x}h^{\sigma_{D+1}}(\boldsymbol x)
+            \\
+            -1 & \cdots & -1
+          \end{bmatrix}
+        \right|
+      \end{align*}
     \end{column}
     \begin{column}{0.5\textwidth}
       \begin{overprint}
@@ -920,7 +930,7 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Treating the determinant}
+  \frametitle{Practical considerations: treating the determinant}
   \begin{columns}
     \begin{column}{0.9\textwidth}
       \[
@@ -952,7 +962,7 @@
 \end{frame}
 
 \begin{frame}
-  \frametitle{Treating the sum over subsets}
+  \frametitle{Practical considerations: treating the sum over subsets}
   \begin{columns}
     \begin{column}{0.9\textwidth}
       \begin{align*}
@@ -993,7 +1003,7 @@
   \end{columns}
 \end{frame}
 \begin{frame}
-  \frametitle{Treating the sum over subsets}
+  \frametitle{Practical considerations: treating the sum over subsets}
   \begin{columns}
     \begin{column}{0.9\textwidth}
       Why does this work? Expand the product:
@@ -1034,4 +1044,209 @@
   \end{columns}
 \end{frame}
 
+\begin{frame}
+  \frametitle{Practical considerations: treating the sum over subsets}
+  \begin{columns}
+    \begin{column}{0.9\textwidth}
+      \includegraphics[width=\textwidth]{figs/annealed_compare.pdf}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Application to the spherical perceptron}
+  \begin{columns}
+    \begin{column}{0.6\textwidth}
+      $M$ data points, $N$ parameters, $\alpha=M/N$
+      \[
+        h^\mu(\boldsymbol x)=\boldsymbol\xi^\mu\cdot\boldsymbol x-\kappa
+      \]
+      $\boldsymbol\xi^\mu$ normally distributed
+
+      \bigskip
+
+      \alert<2>{$\kappa < 0$ : spherical obstacles have positive curvature}
+
+      \medskip
+
+      \alert<3>{$\kappa = 0$ : spherical obstacles have zero curvature}
+
+      \medskip
+
+      \alert<4>{$\kappa > 0$ : spherical obstacles have negative curvature}
+
+      \bigskip
+
+      $N\to\infty$ gives asymptotically zero curvature for all $\kappa$
+    \end{column}
+    \begin{column}{0.27\textwidth}
+      \begin{overprint}
+        \onslide<1-2>\includegraphics[width=\textwidth]{figs/curvature_demo_1.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/curvature_demo_2.pdf}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/curvature_demo_3.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Application to the spherical perceptron}
+  \begin{columns}
+    \begin{column}{0.95\textwidth}
+      Typical statistics of wedged spheres given by
+      \[
+        \frac1N\overline{\log\#_0}
+        =\lim_{n\to0}\frac\partial{\partial n}\operatorname{extr}_{Q\alert<2>{,\rho}}\mathcal S_0(Q\alert<2>{,\rho})
+      \]
+      for effective action
+      \[
+        \hspace{-2em}
+        \mathcal S_0(Q\alert<2>{,\rho})=
+        \frac12\log\det Q\alert<2>{-\frac n2\log\frac{\rho^2}{2\pi}}+\alpha\log\bigg(
+          e^{\frac12\sum_{ab}
+            Q_{ab}\frac{\partial^2}{\partial y_a\partial y_b}
+        }
+        \prod_{a=1}^n
+            \big[
+              \theta(y_a)
+              \alert<2>{+
+              \rho\delta(y_a)}
+            \big]
+            \bigg|_{y_a=-\kappa}
+            \bigg)
+      \]
+      depending on
+      \[
+        Q_{ab}=\frac1N\boldsymbol x_a\cdot\boldsymbol x_b
+        \hspace{3em}
+        \rho
+        =\frac1{\sqrt N}\lim_{\omega\to\infty}\omega^{-\alpha}\sqrt{\boldsymbol s_a^T\boldsymbol s_a}
+        =\frac1{\sqrt N}\lim_{\omega\to\infty}\omega^{-\alpha}\sqrt{\boldsymbol\eta^T_a\bar{\boldsymbol\eta}_a}
+      \]
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Application to the spherical perceptron}
+  \begin{columns}
+    \begin{column}{0.7\textwidth}
+      \includegraphics[width=\textwidth]{figs/phase_diagram_rs.pdf}
+    \end{column}
+    \begin{column}{0.3\textwidth}
+      \includegraphics[width=\textwidth]{figs/fixed_size_zoom_points.pdf}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Application to the spherical perceptron}
+  \begin{columns}
+    \begin{column}{0.7\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/phase_diagram_inscribed.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/phase_diagram_detail_1.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/phase_diagram_detail_2.pdf}
+      \end{overprint}
+    \end{column}
+    \begin{column}{0.3\textwidth}
+      \vspace{-5.5em}
+
+      \[\hspace{-2em}\overline{\log\#_\text{insc}}=\max_{r\geq0}\overline{\log\#_r(\kappa)}\]
+
+      \includegraphics[width=\textwidth]{figs/max_size_zoom.pdf}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Application to the spherical perceptron}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Why does the structure of the distribution of wedged points differ from that of all solutions?
+
+      \medskip
+
+      `Equilibrium' calculations averages over solutions at all larger margin with diverse, inequivalent geometries
+
+      \medskip
+
+      \alert<2>{Sphere-counting isolates properties of the solution set at each specific margin}
+
+      \medskip
+
+      \alert<3>{Relationship between $\#_r$, $\#_\text{insc}$, and solution topology}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/margin_rainbow.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/margin_rainbow_points.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/margin_rainbow_points-2.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Understanding the flat parts of random landscapes}
+  \begin{columns}
+    \begin{column}{0.33\textwidth}
+      \textbf{Nearly flat parts}
+
+      \vspace{-0.75em}
+
+      \rule{\columnwidth}{1pt}
+
+      Conditioning stationary point complexity on marginal optima
+
+      \bigskip
+
+      \centering
+      \includegraphics[height=10pc]{figs/msg_marg_spectra.pdf}
+
+      \smallskip\raggedright\tiny
+
+      \fullcite{Kent-Dobias_2024_Conditioning}
+    \end{column}
+    \begin{column}{0.66\textwidth}
+      \textbf{Really flat parts}
+
+      \vspace{-0.75em}
+
+      \rule{\columnwidth}{1pt}
+
+      \medskip
+
+      \begin{minipage}{0.49\columnwidth}
+        \raggedright
+        Topology of solution manifolds via the average Euler characteristic
+
+        \bigskip
+
+        \centering
+        \includegraphics[height=10pc]{figs/function_2.png}
+
+        \smallskip\raggedright\tiny
+
+        \fullcite{Kent-Dobias_2025_On}
+      \end{minipage}
+      \hfill
+      \begin{minipage}{0.49\columnwidth}
+        \raggedright
+        Geometry of solution sets via the statistics of wedged and inscribed spheres
+
+        \bigskip
+
+        \centering
+        \includegraphics[height=10pc]{figs/max_size_zoom.pdf}
+
+      \smallskip\raggedright\tiny
+
+        Work in progress, expect something on the arXiv in the coming weeks!\\
+        \vspace{10pt}
+      \end{minipage}
+    \end{column}
+  \end{columns}
+\end{frame}
+
 \end{document}