From 337c72cb0c71b50cad612327f292032ce8ff5933 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 18 Aug 2025 21:47:37 -0300
Subject: Initial commit

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+\documentclass[aspectratio=169,dvipsnames]{beamer}
+
+\setbeamerfont{title}{family=\bf}
+\setbeamerfont{frametitle}{family=\bf}
+\setbeamerfont{normal text}{family=\rm}
+\setbeamertemplate{navigation symbols}{}
+\setbeamercolor{titlelike}{parent=structure,fg=cyan}
+
+\usepackage{enumitem}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{pifont}
+\usepackage{graphicx}
+\usepackage{xcolor}
+\usepackage{tikz}
+
+\definecolor{ictpblue}{HTML}{0471b9}
+\definecolor{ictpgreen}{HTML}{0c8636}
+
+\definecolor{mb}{HTML}{5e81b5}
+\definecolor{my}{HTML}{e19c24}
+\definecolor{mg}{HTML}{8fb032}
+\definecolor{mr}{HTML}{eb6235}
+
+\setbeamercolor{titlelike}{parent=structure,fg=ictpblue}
+\setbeamercolor{itemize item}{fg=ictpblue}
+
+\usepackage[
+  style=phys,
+  eprint=true,
+  maxnames = 100,
+  terseinits=true
+]{biblatex}
+
+
+\addbibresource{zif.bib}
+
+\title{
+  Understanding the flat parts of random landscapes
+}
+\author{\textbf{Jaron Kent-Dobias}}
+\date{9 September 2025}
+
+\begin{document}
+
+\begin{frame}
+  \maketitle
+
+  \vspace{-8pc}
+  \begin{minipage}[c]{10pc}
+    \centering
+    \includegraphics[height=6pc]{figs/ift-unesp.png}
+
+    \vspace{2em}
+
+    \includegraphics[height=1.5pc]{figs/Simons-Foundation-Logo_blue.png}
+  \end{minipage}
+  \hfill\begin{minipage}[c]{10pc}
+    \centering
+    \includegraphics[height=6pc]{figs/logo-ictp-saifr.jpg}
+
+    \vspace{2em}
+
+    \includegraphics[height=1.5pc]{figs/fapesp.png}
+  \end{minipage}
+  \vspace{2pc}
+\end{frame}
+
+\begin{frame}
+  \frametitle{How to count: Kac--Rice}
+
+      Number of stationary points with $\nabla H(\pmb x)=0$ given by integral
+      over Kac--Rice measure
+      \begin{align*}
+        \#_\text{points}
+        &=\int_\Omega d\pmb x\,\delta\big(\nabla H(\pmb x)\big)\,\big|\det\operatorname{Hess}H(\pmb x)\big|
+      \end{align*}
+      Note absolute value of the determinant: want to account for curvature but not add $-1$
+
+      \bigskip
+
+      Can specify properties of points by inserting $\delta$-functions:
+      \begin{align*}
+        \#_\text{points}\alert<2>{(E)}
+        &=\int_\Omega d\pmb x\,\delta\big(\nabla H(\pmb x)\big)\,\big|\det\operatorname{Hess}H(\pmb x)\big|
+        \alert<2>{\,\delta\big(H(\pmb x)-NE\big)}
+      \end{align*}
+
+      \emph{How do we condition on marginal minima?}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Conditioning on the type of minimum: the 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
+      \[
+        \mu=\frac1N\operatorname{Tr}\operatorname{Hess} H(\pmb x)
+      \]
+
+      \bigskip
+
+      Condition on marginal minima by inserting
+      \[
+        \delta\big(\operatorname{Tr}\operatorname{Hess}H(\pmb x)-N\mu_\text{m}\big)
+      \]
+
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overlayarea}{\textwidth}{14.5em}
+        \only<1-2>{\includegraphics[width=\columnwidth]{figs/mu_0.75.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_0.75.pdf}}
+        \only<3>{\includegraphics[width=\columnwidth]{figs/mu_1.5.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_1.5.pdf}}
+        \only<4>{\includegraphics[width=\columnwidth]{figs/mu_2.25.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_2.25.pdf}}
+        \only<5>{\includegraphics[width=\columnwidth]{figs/mu_3.5.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_3.5.pdf}}
+        \only<6>{\includegraphics[width=\columnwidth]{figs/mu_2.pdf}\\\hphantom{ello}\includegraphics[width=0.8\columnwidth]{figs/land_2.pdf}}
+      \end{overlayarea}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Less simple mean-field models}
+  \framesubtitle{The mixed spherical spin glasses}
+
+  \begin{columns}
+    \begin{column}{0.41\textwidth}
+      \alert<1-2>{Pure: all procedures reach $E_\text{th}$}
+
+      \medskip
+
+      \alert<3-4>{Mixed: different procedures reach different energies}
+
+      \medskip
+
+      Dynamical endpoints bounded by complexity of marginal minima
+
+      \bigskip
+
+      \tiny
+      \fullcite{Folena_2020_Rethinking}
+
+      \smallskip
+
+      \fullcite{Folena_2021_Gradient}
+
+      \smallskip
+
+      \fullcite{Kent-Dobias_2023_How}
+    \end{column}
+    \begin{column}{0.59\textwidth}
+      \begin{overlayarea}{\columnwidth}{0.8\textheight}
+      \only<1>{%
+      \includegraphics[width=0.8\columnwidth]{figs/complexity_3_marginal.pdf}
+      }%
+      \only<2>{%
+        \bigskip
+
+        \begin{minipage}[b]{0.83\columnwidth}
+          \centering
+          \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+
+          \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+
+          \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf}
+
+          \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf}
+          \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf}
+
+          \includegraphics[width=0.18\columnwidth]{figs/land_3.5.pdf}
+        \end{minipage}
+        \raisebox{0.75em}{\includegraphics[width=0.15\columnwidth]{figs/energy_arrow.pdf}}
+      }
+      \only<3>{%
+      \includegraphics[width=0.8\columnwidth]{figs/complexity_34_marginal.pdf}
+      }%
+      \only<4>{%
+      \bigskip
+
+      \begin{minipage}[b]{0.83\columnwidth}
+        \centering
+        \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+
+        \includegraphics[width=0.18\columnwidth]{figs/land_0.75.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf}
+
+        \includegraphics[width=0.18\columnwidth]{figs/land_1.5.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_2.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf}
+
+        \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf}
+        \includegraphics[width=0.18\columnwidth]{figs/land_3.5.pdf}
+
+        \includegraphics[width=0.18\columnwidth]{figs/land_2.25.pdf}
+      \end{minipage}
+        \raisebox{0.75em}{\includegraphics[width=0.15\columnwidth]{figs/energy_arrow_o1.pdf}}
+      }%
+      \end{overlayarea}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Marginal minima in generic models}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \alert<1>{In spherical spin glasses, all points have the same Hessian spectral density: semicircle with radius $\mu_\text m$, but different shifts}
+
+      \bigskip
+
+      In generic models, spectral density depends on stationarity, energy, etc!
+
+      \bigskip
+
+      \alert<2>{Example: multi-spherical model
+      \[
+        H(\{\boldsymbol s_1,\boldsymbol s_2\})
+        =H_p^{(1)}(\boldsymbol s_1)+H_p^{(2)}(\boldsymbol s_2)+\epsilon\boldsymbol s_1\cdot\boldsymbol s_2
+      \]}
+      \hspace{-0.25em}In most models we don't understand the Hessian at all
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+      \onslide<1>\includegraphics[width=\columnwidth]{figs/mu_0.75.pdf}
+      \onslide<2>\includegraphics[width=0.9\columnwidth]{figs/msg_marg_spectra.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Towards generic marginal complexity}
+  \begin{columns}
+    \begin{column}{0.6\textwidth}
+      \begin{itemize}[leftmargin=4em]
+        \item[\color{ictpgreen}\bf Trick \#1:] condition stationary points on \emph{value of smallest eigenvalue}
+      \end{itemize}
+      {
+      \small
+      \begin{align*}
+        \hspace{-1em}&\delta(\lambda_\text{min}(A)) \\
+        &=\lim_{\beta\to\infty}\int
+          \frac{d\pmb s\,\delta(N-\|\pmb s\|^2)e^{-\beta\pmb s^TA\pmb s}}
+            {\int d\pmb s'\,\delta(N-\|\pmb s'\|^2)e^{-\beta\pmb s'^TA\pmb s'}}
+        \delta\left(\frac{\pmb s^TA\pmb s}N\right)
+      \end{align*}
+      }
+
+      \bigskip
+
+      Only works if you happen to have the correct shift $\mu$
+
+      \bigskip
+
+      \tiny
+      \fullcite{Kent-Dobias_2024_Conditioning}
+
+    \end{column}
+    \begin{column}{0.4\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/spectrum_less.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/spectrum_more.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/spectrum_eq.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+
+\begin{frame}
+  \frametitle{Towards generic marginal complexity}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \begin{itemize}[leftmargin=4em]
+        \item[\color{ictpgreen}\bf Trick \#1:] condition stationary points on \emph{value of smallest eigenvalue}
+      \end{itemize}
+      {
+      \small
+      \begin{align*}
+        \hspace{-3em}&\delta(\lambda_\text{min}(A)) \\
+        \hspace{-3em}&=\lim_{\beta\to\infty}\int
+          \frac{d\pmb s\,\delta(N-\|\pmb s\|^2)e^{-\beta\pmb s^TA\pmb s}}
+            {\int d\pmb s'\,\delta(N-\|\pmb s'\|^2)e^{-\beta\pmb s'^TA\pmb s'}}
+        \delta\left(\frac{\pmb s^TA\pmb s}N\right)
+      \end{align*}
+      }
+
+      \medskip
+
+      \begin{itemize}[leftmargin=4em]
+        \item[\color{ictpgreen}\bf Trick \#2:] adjust $\mu\propto\operatorname{Tr}\operatorname{Hess}H$ until order-$N$ large deviation breaks
+      \end{itemize}
+
+      \bigskip
+
+      \tiny
+      \fullcite{Kent-Dobias_2024_Conditioning}
+
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \hspace{0.9em}
+      \includegraphics[scale=0.8]{figs/spectrum_less.pdf}
+      \hspace{-1.6em}
+      \includegraphics[scale=0.8]{figs/spectrum_eq.pdf}
+      \hspace{-1.6em}
+      \includegraphics[scale=0.8]{figs/spectrum_more.pdf}
+      \\
+      \includegraphics[scale=0.8]{figs/large_deviation.pdf}
+
+      \vspace{-1em}
+
+      \small
+      \begin{align*}
+        \hspace{-0.5em}G_0(\mu)=\frac 1N\log\Big\langle\delta\big(\lambda_\text{min}(A-\lambda\mu)\big)\Big\rangle_{A\in\text{GOE}(N)}
+      \end{align*}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Marginal complexity: example}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Example: model of nonlinear least squares:
+      \[
+        H(\pmb s)=\frac12\sum_{i=1}^{M}V_i(\pmb s)^2
+      \]
+      for spherical $\pmb s\in\mathbb R^N$ and $M=\alpha N$ inhomogeneous random functions $V_i$
+      \[
+        V_i(\boldsymbol s)=H_2^{(i)}(\boldsymbol s)+H_3^{(i)}(\boldsymbol s)
+      \]
+
+      \bigskip
+
+      \tiny
+      \fullcite{Kent-Dobias_2024_Conditioning}
+
+      \smallskip
+
+      \fullcite{Kent-Dobias_2024_Algorithm-independent}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\vspace{4em}\includegraphics[width=\textwidth]{figs/most_squares_complex.pdf}
+        \onslide<2>\vspace{-1.75em}\includegraphics[width=\textwidth]{figs/most_squares_complexity.pdf}
+
+        \vspace{-1.95em}
+
+        \hspace{-0.25em}\colorbox{white}{\includegraphics[width=\textwidth]{figs/most_squares_stability.pdf}}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{The Euler characteristic \boldmath{$\chi$}}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      The Euler characteristic $\chi(\Omega)$ is a topological invariant of a manifold $\Omega$
+
+      \medskip
+
+      Defined by tiling the manifold, then taking the alternating sum
+      \begin{align*}
+        \chi(\Omega_{\text{cow}})
+        &=
+        {\only<2,5->{\color{Red}}\#_\text{vertices}}
+        &&\hspace{-1em}-
+        {\only<3,5->{\color{ictpgreen}}\#_\text{edges}}
+        &&\hspace{-1em}+
+        {\only<4,5->{\color{ictpblue}}\#_\text{faces}}
+        \\
+        &\color{White}\only<2->{\color{Black}}=
+        {\only<2,5->{\color{Red}}2904}
+        &&\hspace{-1em}\color{White}\only<3->{\color{Black}}-
+        {\only<3,5->{\color{ictpgreen}}8706}
+        &&\hspace{-1em}\color{White}\only<4->{\color{Black}}+
+        {\only<4,5->{\color{ictpblue}}5804} \\
+        &\color{White}\only<5->{\color{Black}}=2
+      \end{align*}
+      \[
+        \color{White}\only<6->{\color{Black}}\chi(\Omega_\text{football})
+        ={\only<6->{\color{Red}}60}-{\only<6->{\color{ictpgreen}}90}+{\only<6->{\color{ictpblue}}32}=2
+      \]
+
+      \color{White}\only<7>{\color{Black}}Cow is homeomorphic to a sphere
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1,5>\includegraphics[width=\textwidth]{figs/cow.png}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/cow_vert.png}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/cow_edge.png}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/cow_face.png}
+        \onslide<6->\hspace{2em}\includegraphics{figs/Football_Pallo_valmiina-cropped.jpg}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{The Euler characteristic \boldmath{$\chi$}}
+  \framesubtitle{Characteristics of the characteristic}
+
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      For closed, connected 2-dimensional manifolds, related to genus $g$ by
+      $\chi=2-2g$
+
+      \medskip
+
+      General properties:
+      \vspace{-0.5em}
+      \[
+        \chi(\Omega)=0 \text{ for odd-dimensional $\Omega$}
+      \]
+      \vspace{-1.6em}
+      \[
+        \chi(S^D)=2\text{ for even }D
+      \]
+      \[
+        \chi(\Omega_1\sqcup\Omega_2)=\chi(\Omega_1)+\chi(\Omega_2)
+      \]
+      \[
+        \chi(\Omega_1\times\Omega_2)=\chi(\Omega_1)\times\chi(\Omega_2)
+      \]
+
+      \smallskip
+
+      Examples:
+      \vspace{-0.5em}
+      \[\chi(M\text{ even-$D$ spheres})=2M\]
+      \vspace{-1.6em}
+      \[\chi(S^1\times\text{anything})=0\]
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \includegraphics[width=\textwidth]{figs/genus.png}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{The Euler characteristic \boldmath{$\chi$}}
+  \framesubtitle{Computing the Euler characteristic}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Morse theory: gradient flow on an arbitrary ``height'' function $h$ makes a complex
+      \begin{align*}
+        \chi(\Omega)
+        &=
+        {\only<2,5>{\color{Red}}\#_\text{vertices}}
+        -
+        {\only<3,5>{\color{ictpgreen}}\#_\text{edges}}
+        +
+        {\only<4,5>{\color{ictpblue}}\#_\text{faces}}
+        +\cdots \\
+        &=
+        {\only<6>{\color{ictpblue}}\#_\text{index 0}}
+        -
+        {\only<6>{\color{ictpgreen}}\#_\text{index 1}}
+        +
+        {\only<6>{\color{Red}}\#_\text{index 2}}
+        +\cdots \\
+        &=\sum_{i=0}^D(-1)^i\#_\text{index i}
+      \end{align*}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/other_sphere.png}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/other_sphere_vert.png}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/other_sphere_edge.png}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/other_sphere_face.png}
+        \onslide<5>\includegraphics[width=\textwidth]{figs/other_sphere_all.png}
+        \onslide<6>\includegraphics[width=\textwidth]{figs/other_sphere_crit.png}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{The Euler characteristic \boldmath{$\chi$}}
+  \framesubtitle{Computing the Euler characteristic}
+
+  \begin{columns}
+    \begin{column}{0.6\textwidth}
+      \[
+        \Omega=\left\{
+          \pmb a\in\mathbb R^N\mid\|\pmb a\|^2=N, V_0=\hat f(J^i\mid\pmb a)\;\forall\;1\leq i\leq M
+        \right\}
+      \]
+      Pick whatever height function $h:\Omega\to\mathbb R$ you like: $h(\pmb a)=\frac1N\pmb
+      a_0\cdot\pmb a$ for arbitrary $\pmb a_0$
+      \[
+        \#_{\substack{\text{critical}\\\text{points}}}
+        =\int_\Omega d\pmb x\,
+        \delta\big(\nabla h(\pmb x)\big)\,
+        \big|\det\operatorname{Hess}h(\pmb x)\big|
+      \]
+      \[
+        \hspace{-1em}\operatorname{sgn}\big(\det\operatorname{Hess}(\pmb x)\big)
+        =
+        \operatorname{sgn}\left(\prod_{i=1}^D\lambda_i\right)
+        =(-1)^{\text{index}}
+      \]
+      \[
+        \chi(\Omega)
+        =\sum_{i=0}^D(-1)^i\#_\text{index i}
+        =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)\,\det\operatorname{Hess}h(\pmb x)
+      \]
+    \end{column}
+    \begin{column}{0.4\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/function_1.png}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/function_2.png}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/function_3.png}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{A simple model of nonlinear least squares}
+  \framesubtitle{Results}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      $M$ data points, $N$ parameters, $\alpha=M/N$
+      \[
+        V_0=\hat f(J\mid \pmb a)
+        =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p}
+      \]
+      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
+
+      \medskip
+
+      \hspace{2em}\includegraphics[width=0.33\textwidth]{figs/connected.pdf}
+      \hfill
+      \includegraphics[width=0.33\textwidth]{figs/gone.pdf}\hspace{2em}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/intersections_1.pdf}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/intersections_2.pdf}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/intersections_3.pdf}
+        \onslide<4>\includegraphics[width=\textwidth]{figs/intersections_4.pdf}
+        \onslide<5>\includegraphics[width=\textwidth]{figs/phases_1.pdf}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{A simple model of nonlinear least squares}
+  \framesubtitle{Results}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      $M$ data points, $N$ parameters, $\alpha=M/N$
+      \[
+        V_0=\hat f(J\mid \pmb a)
+        =\sum_{i_1=1}^N\cdots\sum_{i_p=1}^NJ_{i_1,\ldots,i_p}a_{i_1}\cdots a_{i_p}
+      \]
+
+      For $p\geq2$, different phases with $|\chi(\Omega)|\gg1$ with varying sign
+
+      \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}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\end{document}
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