From f8a41fb6f541132f8d4a40f5de8435bb8135b407 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 4 Feb 2025 10:18:29 -0300
Subject: Initial commit

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+\documentclass[aspectratio=169,usenames,dvipsnames,fleqn]{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}
+
+\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{ictp-saifr_colloquium.bib}
+
+\title{
+  Structural barriers to random optimization
+}
+\author{\textbf{Jaron Kent-Dobias}\\Simons--FAPESP Young Investigator}
+\date{19 February 2025}
+
+\begin{document}
+
+\begin{frame}
+  \maketitle
+
+  \vspace{-6pc}
+  \begin{minipage}[c]{10pc}
+    \centering
+    \includegraphics[height=6pc]{figs/ift-unesp.png}
+  \end{minipage}
+  \hfill\begin{minipage}[c]{10pc}
+    \centering
+    \includegraphics[height=6pc]{figs/logo-ictp-saifr.jpg}
+  \end{minipage}
+  \vspace{2pc}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Overview}
+
+  \begin{columns}
+    \begin{column}{\textwidth}
+      \huge
+
+      \color{ictpgreen}{\textbf{Introduction}}
+
+      \bigskip
+
+      \color{ictpgreen}{\textbf{Complexity \& marginal complexity}}
+
+      \bigskip
+
+      \color{ictpgreen}{\textbf{Level set topology}}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Complexity of random landscapes}
+  \begin{columns}
+    \begin{column}{0.4\textwidth}
+      Complexity $\Sigma=\frac1N\overline{\log\#_\text{points}}$ describes typical number of stationary points
+
+      \bigskip
+
+      Complexity of marginal minima crucial for understanding dynamics in mixed $p$-spin models
+
+      \bigskip
+
+      Lucky accident: natural parameter
+      sets type of stationary point
+    \end{column}
+    \begin{column}{0.6\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/folena_2020.png}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/folena_2020_2.png}
+      \end{overprint}
+
+        \smallskip
+
+        \tiny\fullcite{Folena_2020_Rethinking}
+    \end{column}
+  \end{columns}
+\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*}
+      How can \emph{marginality} be specified?
+\end{frame}
+
+\begin{frame}
+  \frametitle{Hessian shifts and stationary point stability}
+  \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
+      \[
+        \delta\big(\operatorname{Tr}\operatorname{Hess}H(\pmb x)-N\mu_\text{m}\big)
+      \]
+
+      \medskip
+
+      \alert<7>{In generic models, spectral density depends on stationarity, energy, etc!}
+    \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}}
+        \only<7>{\includegraphics[width=0.9\columnwidth]{figs/msg_marg_spectra.pdf}}
+      \end{overlayarea}
+    \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: non-Gaussian landscapes
+      \[
+        H(\pmb x)=\frac12\sum_{i=1}^{\alpha N}V_i(\pmb x)^2
+      \]
+      for spherical $\pmb x$ and Gaussian functions $V_i$
+      \[
+        \overline{V_i(\pmb x)V_j(\pmb x')}=\delta_{ij}f\bigg(\frac{\pmb x\cdot\pmb x'}N\bigg)
+      \]
+
+      \vspace{-2em}
+
+      \begin{overprint}
+        \onslide<1-2>\[
+          f(q)=\tfrac12q^2+\tfrac12q^3
+        \]
+        \onslide<3>\[
+          f(q)=\kappa q+(1-\kappa)q^2
+        \]
+      \end{overprint}
+
+      \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}}
+        \onslide<3>\vspace{-1em}
+
+        \includegraphics[width=\textwidth]{figs/most_squares_nonzoom.pdf}
+
+        \vspace{-0.4em}
+
+        \includegraphics[width=\textwidth]{figs/most_squares_zoom.pdf}
+
+        \vspace{1em}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Which marginal minima attract the dynamics?}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      `Best case' performance: lowest marginal minima without the \emph{Overlap Gap Property}
+
+      \smallskip
+
+      \tiny
+      \fullcite{Gamarnik_2021-10_The}
+
+      \normalsize
+      \medskip
+
+      `Worst case' performance: ???
+
+      \smallskip
+      \tiny\fullcite{Folena_2023_On}
+
+      \normalsize
+      \medskip
+
+      \textcolor{mb}{\textbf{\boldmath{$E_\text{gs}$:}} ground state, energy of lowest minima}
+
+      \smallskip
+
+      \textcolor{mg}{\textbf{\boldmath{$E_\text{alg}$:}} algorithmic bound, set by OGP}
+
+      \smallskip
+
+      \textcolor{my}{\textbf{\boldmath{$E_\text{th}$:}} `threshold', marginal minima dominate}
+
+      \medskip
+
+      Gradient descent destination depends on \emph{global} property: basin of attraction size;
+      stationary point analysis is only \emph{local}
+
+      \medskip
+
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/folena_2023.png}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/folena_new_2.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{Characteristics of the Euler 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}
+  \begin{columns}
+    \begin{column}{0.16\textwidth}
+      \Large
+      \textbf{\color{ictpblue}\boldmath{$\chi$} for constant energy level sets}
+      \vspace{11em}
+    \end{column}
+    \begin{column}{0.7\textwidth}
+      \begin{overprint}
+        \onslide<1>\centering\rotatebox{90}{\includegraphics[height=\textwidth]{figs/Stillinger-0.png}}
+        \onslide<2>\centering\rotatebox{90}{\includegraphics[height=\textwidth]{figs/Stillinger-1.png}}
+        \onslide<3>\centering\rotatebox{90}{\includegraphics[height=\textwidth]{figs/Stillinger-2.png}}
+      \end{overprint}
+    \end{column}
+    \begin{column}{0.16\textwidth}
+      \begin{overprint}
+        \onslide<2>\centering High energy
+
+        \vspace{0.5em}
+
+        $\chi(\Omega)\ll0$
+
+        \vspace{0.5em}
+
+        hole\\
+        dominated
+        \onslide<3>\centering Low energy
+
+        \vspace{0.5em}
+
+        $\chi(\Omega)\gg0$
+
+        \vspace{0.5em}
+
+        component\\
+        dominated
+      \end{overprint}
+      \vspace{15em}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \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
+      \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*}
+      \[
+        \hspace{-1em}\operatorname{sgn}\big(\det\operatorname{Hess}(\pmb x)\big)
+        =
+        \operatorname{sgn}\left(\prod_{i=1}^D\lambda_i\right)
+        =(-1)^{\text{index}}
+      \]
+      \[
+        \chi(\Omega)
+        =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)
+        \,\det\operatorname{Hess}h(\pmb x)
+      \]
+      \emph{Kac--Rice without the absolute value!}
+    \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{Computing the Euler characteristic of level sets}
+
+  \begin{columns}
+    \begin{column}{0.6\textwidth}
+      Pick whatever height function $h:\Omega\to\mathbb R$ you like: $h(\pmb x)=\frac1N\pmb
+      x_0\cdot\pmb x$ for arbitrary $\pmb x_0$.
+      \[
+        \chi(\Omega)
+        =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)\,\det\operatorname{Hess}h(\pmb x)
+      \]
+      Level set $\Omega$ defined by $H(\pmb x)=EN$ and $\|\pmb x\|^2=N$
+
+      \bigskip
+
+      Lagrange multipliers replace differential geometry:
+      \[
+        L(\pmb x,\pmb\omega)=h(\pmb x)+\omega_0(\|\pmb x\|^2-N)+\omega_1(H(\pmb x)-EN)
+      \]
+    \end{column}
+    \begin{column}{0.4\textwidth}
+      \begin{overprint}
+        \onslide<1>\includegraphics[width=\textwidth]{figs/function-0.png}
+        \onslide<2>\includegraphics[width=\textwidth]{figs/function-1.png}
+        \onslide<3>\includegraphics[width=\textwidth]{figs/function-2.png}
+      \end{overprint}
+    \end{column}
+  \end{columns}
+      \[
+        \chi(\Omega)
+        =\int_{\mathbb R^{N+2}} d\pmb x\,d\pmb\omega\,\delta\big(\begin{bmatrix}\frac{\partial L}{\partial\pmb x}&\frac{\partial L}{\partial\pmb\omega}\end{bmatrix}\big)
+        \,\det\begin{bmatrix}\frac{\partial^2L}{\partial\pmb x^2}&\frac{\partial^2L}{\partial\pmb x\partial\pmb\omega}\\\frac{\partial^2L}{\partial\pmb x\partial\pmb\omega}&\frac{\partial^2L}{\partial\pmb\omega^2}\end{bmatrix}
+      \]
+\end{frame}
+
+\begin{frame}
+  \begin{columns}
+    \begin{column}{\textwidth}
+      \includegraphics[width=\textwidth]{figs/slice.png}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Results: \boldmath{$3+s$} mixed spherical models}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      \begin{align*}
+        H(\pmb x)=\lambda_s\sum_{i_1,i_2,i_3}^NJ_{i_1,i_2,i_3}x_{i_1}x_{i_2}x_{i_3} \hspace{4em} \\
+        +(1-\lambda_s)\sum_{i_1,\ldots,i_s}^NJ_{i_1,\ldots,i_s}x_{i_1}\cdots x_{i_s}
+      \end{align*}
+
+      \textcolor{mb}{\textbf{\boldmath{$E_\text{gs}$:}} ground state, energy of lowest minima}
+
+      \smallskip
+
+      \textcolor{mg}{\textbf{\boldmath{$E_\text{alg}$:}} algorithmic bound, set by OGP}
+
+      \smallskip
+
+      \textcolor{my}{\textbf{\boldmath{$E_\text{th}$:}} `threshold', marginal minima dominate}
+
+      \smallskip
+
+      \textcolor{mr}{\textbf{\boldmath{$E_\text{sh}$:}} `shattering', $\chi$ changes sign}
+
+      \bigskip
+
+      \tiny
+      \fullcite{Kent-Dobias_2024_On}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \includegraphics[width=\textwidth]{figs/folena_new.pdf}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Preliminary results: other models?}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Example: non-Gaussian landscapes
+      \[
+        H(\pmb x)=\frac12\sum_{i=1}^{\alpha N}V_i(\pmb x)^2
+      \]
+      for spherical $\pmb x$ and Gaussian functions $V_i$
+
+      \medskip
+
+      $E_\text{sh}$ consistent with gradient descent? More work needed...
+
+      \bigskip\tiny
+
+      \fullcite{Kent-Dobias_2024_Algorithm-independent}
+
+      \smallskip
+
+      \fullcite{Kent-Dobias_2024_On}
+    \end{column}
+    \begin{column}{0.5\textwidth}
+        \vspace{-1em}
+
+        \begin{overprint}
+          \onslide<1>\includegraphics[width=\textwidth]{figs/most_squares_nonzoom.pdf}
+          \onslide<2>\includegraphics[width=\textwidth]{figs/extrapolation.pdf}
+        \end{overprint}
+
+        \vspace{-0.4em}
+
+        \includegraphics[width=\textwidth]{figs/most_squares_zoom_2.pdf}
+
+        \vspace{1em}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Outlook, other applications, future directions}
+  \begin{columns}
+    \begin{column}{0.5\textwidth}
+      Euler characteristic reveals structure of problems with no energy function:\\ e.g., the set of $\pmb x$ such that
+      \[
+        V_i(\pmb x)=\sqrt NV_0 \qquad i=1,\ldots,\alpha N
+      \]
+      for independent Gaussian $V_i$
+
+      \medskip
+
+      \tiny
+      \fullcite{Kent-Dobias_2024_On}
+
+      \bigskip\normalsize
+
+      \textcolor{ictpgreen}{\textbf{To Do:}}
+
+      Resolve GD question: better DMFT, direct reasoning for relationship to topology
+
+      \medskip
+
+      Extend topological arguments beyond GD
+    \end{column}
+    \begin{column}{0.5\textwidth}
+      \includegraphics[width=\textwidth]{figs/spheres.png}
+
+      \medskip
+
+      \includegraphics[width=\textwidth]{figs/phases.png}
+    \end{column}
+  \end{columns}
+\end{frame}
+
+\end{document}
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