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\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/logo-ictp-saifr.jpg}

    \vspace{2em}

    \includegraphics[height=1.5pc]{figs/Simons-Foundation-Logo_blue.png}
  \end{minipage}
  \hfill\begin{minipage}[c]{10pc}
    \centering
    \includegraphics[height=6pc]{figs/ift-unesp.png}

    \vspace{2em}

    \includegraphics[height=1.5pc]{figs/fapesp.png}
  \end{minipage}
  \vspace{2pc}
\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}

      \medskip

      Conditioning stationary point complexity on marginal optima

      \bigskip

      \centering
      \includegraphics[height=10pc]{figs/msg_marg_spectra.pdf}

    \end{column}
    \begin{column}{0.66\textwidth}
      \textbf{Really flat parts}

      \vspace{-0.75em}

      \rule{\columnwidth}{1pt}

      \bigskip

      \begin{minipage}{0.49\columnwidth}
        \raggedright
        Topology of solution manifolds via the average Euler characteristic

        \bigskip

        \centering
        \includegraphics[height=10pc]{figs/function_2.png}
      \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}
      \end{minipage}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Marginal complexity}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      Statistics of local optima imagined to govern behavior in rugged optimisation

      \bigskip

      \alert<2-3>{High-dimensional landscapes differ from cartoons in important ways}

      \bigskip

      \alert<4-6>{Generic rugged landscapes covered mostly by basins attached to marginal optima}

      \bigskip

      Understanding marginal optima is more important for typical dynamics than understanding typical optima
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \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{Typical complexity}

      Number of stationary points with $\nabla H(\boldsymbol x)=0$ given by integral
      over Kac--Rice measure
      \begin{align*}
        \#_\text{points}
        &=\int_\Omega d\boldsymbol x\,\delta\big(\nabla H(\boldsymbol x)\big)\,\big|\det\operatorname{Hess}H(\boldsymbol 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\boldsymbol x\,\delta\big(\nabla H(\boldsymbol x)\big)\,\big|\det\operatorname{Hess}H(\boldsymbol x)\big|
        \alert<2>{\,\delta\big(H(\boldsymbol x)-NE\big)}
      \end{align*}

      \emph{How do we condition on marginal minima?}
\end{frame}

\begin{frame}
  \frametitle{Conditioning on the type of point: spherical models}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      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
      \[
        \mu=\frac1N\operatorname{Tr}\operatorname{Hess} H(\boldsymbol x)
      \]

      \bigskip

      Condition on marginal minima by inserting
      \[
        \delta\big(\operatorname{Tr}\operatorname{Hess}H(\boldsymbol x)-N\mu_\text{m}\big)
      \]

    \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{Out of equilibrium dynamics and complexity: spherical models}

  \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
      \[
        H(\{\boldsymbol x_1,\boldsymbol x_2\})
        =H_p^{(1)}(\boldsymbol x_1)+H_p^{(2)}(\boldsymbol x_2)+\epsilon\boldsymbol x_1\cdot\boldsymbol x_2
      \]}
      \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{0em}&\delta(\lambda_\text{min}(A)) \\
        &=\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)
      \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\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)
      \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-\mu I)\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:
      \[
        H(\boldsymbol x)=\frac12\sum_{i=1}^{M}V_i(\boldsymbol x)^2
      \]
      for spherical $\boldsymbol x\in\mathbb R^N$ and $M=\alpha N$ inhomogeneous random functions $V_i$
      \[
        V_i(\boldsymbol x)=H_2^{(i)}(\boldsymbol x)+H_3^{(i)}(\boldsymbol x)
      \]

      \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{Really flat parts: zero-cost sets}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      Diverse optimization problems exactly fit data or satisfy constraints, reach zero cost

      \bigskip

      Stationary point technology useless for describing zero-cost sets

      \bigskip

      Mostly understood by pointwise sampling or interpolation between pairs of
      sampled points
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \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}
\end{frame}

\begin{frame}
  \frametitle{Zero cost manifolds vs.\ zero cost non-manifolds}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      \textbf{Manifolds}

      \medskip

      Defined by sets of \emph{analytic equalities}

      \medskip

      Network $f_{\boldsymbol x}:\mathbb R^P\to\mathbb R$, $\boldsymbol x\in S^{N-1}\subset\mathbb R^N$, with analytic activations (e.g.\ $\tanh$):
      \[
        y^\mu=f_{\boldsymbol x}(\boldsymbol a^\mu)\qquad\mu=1,\ldots,M
      \]

      \smallskip

      \textbf{(Usual) Non-manifolds}

      \medskip

      Defined by sets of \emph{inequalities}

      \medskip

      Jamming of disks in $d$ dimensions:
      \[
        \|\boldsymbol x^i-\boldsymbol x^j\|\geq 2R\qquad i,j=1,\ldots,P
      \]
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/intersections_3.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/nothing_zoom.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{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}

  \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 $\pmb x_0$.
      \[
        \chi(\Omega)
        =\int_\Omega d\pmb x\,\delta\big(\nabla h(\pmb x)\big)\,\det\operatorname{Hess}h(\pmb x)
      \]
      \[
        \Omega=\{\boldsymbol x\in S^{N-1}\subset\mathbb R^N\mid f_{\boldsymbol x}(\boldsymbol a^\mu)=y^\mu,1\leq\mu\leq M\}
      \]

      Lagrange multipliers replace differential geometry:
      \[
        L(\boldsymbol  x,\boldsymbol \omega)=h(\boldsymbol  x)+\omega^0(\|\boldsymbol  x\|^2-N)+\sum_{\mu=1}^M\omega^\mu(f_{\boldsymbol x}(\boldsymbol a^\mu)-y^\mu)
      \]
    \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}
  \vspace{-1em}
      \[
        \chi(\Omega)
        =\int_{\mathbb R^{N+M+1}} d\boldsymbol  x\,d\boldsymbol \omega\,\delta\big(\begin{bmatrix}\frac{\partial L}{\partial\boldsymbol  x}&\frac{\partial L}{\partial\boldsymbol \omega}\end{bmatrix}\big)
        \,\det\begin{bmatrix}\frac{\partial^2L}{\partial\boldsymbol  x^2}&\frac{\partial^2L}{\partial\boldsymbol  x\partial\boldsymbol \omega}\\\frac{\partial^2L}{\partial\boldsymbol  x\partial\boldsymbol \omega}&\frac{\partial^2L}{\partial\boldsymbol \omega^2}\end{bmatrix}
      \]
\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,
      \[
        f_{\boldsymbol x}(\boldsymbol a^\mu)
        =H^{(\mu)}_p(\boldsymbol x)
      \]
      Average Euler characteristic reduced to integral over $m=\frac1N\boldsymbol x\cdot\boldsymbol x_0$,
      \[
        \overline{\chi(\Omega)}
        =\left(\frac N{2\pi}\right)^{\frac12}
        \int dm\,g(m)e^{N\mathcal S_\chi(m)}
      \]

      \smallskip\tiny

      \fullcite{Kent-Dobias_2025_On}
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/action_1.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/action_3.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\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,
      \[
        f_{\boldsymbol x}(\boldsymbol a^\mu)
        =H^{(\mu)}_p(\boldsymbol x)
      \]
      Simplest case: $p=1$, $\Omega$ is intersection of random hyperplanes with the sphere

      \medskip

      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/gone.pdf}
      \hfill
      \includegraphics[width=0.33\textwidth]{figs/connected.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/action_phase_1.pdf}
        \onslide<4>\includegraphics[width=\textwidth]{figs/intersections_3.pdf}
        \onslide<5>\includegraphics[width=\textwidth]{figs/intersections_4.pdf}
        \onslide<6>\includegraphics[width=\textwidth]{figs/action_phase_2.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/phases_1.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\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

      \alert<2>{\textbf{\boldmath{$m^*=0$, $\mathcal S_\chi(m^*)$ complex, $\overline{\chi(\Omega)}\ll0$, local maximum exists}}}

      \medskip

      \alert<3>{\textbf{\boldmath{$m^*=0$, $\mathcal S_\chi(m^*)$ complex, $\overline{\chi(\Omega)}\ll0$, no local maximum}}}

      \medskip

      \alert<4>{\textbf{\boldmath{$m^*=0$, $\mathcal S_\chi(m^*)$ real, $\overline{\chi(\Omega)}\gg0$}}}

      \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<1-2>\includegraphics[width=\textwidth]{figs/action_phase_3.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/action_phase_4.pdf}
        \onslide<4>\includegraphics[width=\textwidth]{figs/action_phase_5.pdf}
        \onslide<5>\includegraphics[width=\textwidth]{figs/phases_2.pdf}
        \onslide<6>\includegraphics[width=\textwidth]{figs/phases_3.pdf}
        \onslide<7>\includegraphics[width=\textwidth]{figs/phases_4.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Computing the Euler characteristic: example}

  \begin{columns}
    \begin{column}{1.1\textwidth}
      \hspace{1em}Phases for inhomogeneous models: $1-\lambda$ parts linear ($p=1$) plus $\lambda$ quadratic ($p=2$)

      \medskip

      \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

      \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}

\begin{frame}
  \frametitle{Non manifolds: constraint satisfaction}
  \begin{columns}
    \begin{column}{0.5\textwidth}
      Continuous constraint satisfaction problems with $\boldsymbol x\in\Omega\subseteq\mathbb R^N$, with $\Omega$ $D$-dimensional
      \[
        h^\mu(\boldsymbol x)\geq0 \qquad 1\leq\mu\leq M
      \]
    \end{column}
    \begin{column}{0.5\textwidth}
      \begin{overprint}
        \onslide<1>\includegraphics[width=\textwidth]{figs/nothing_zoom.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/max_size_zoom.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \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 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}
        \onslide<1>\includegraphics[width=\textwidth]{figs/nothing_zoom.pdf}
        \onslide<2>\includegraphics[width=\textwidth]{figs/fixed_size_zoom_r.pdf}
        \onslide<3>\includegraphics[width=\textwidth]{figs/fixed_size_zoom_0.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \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{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}
        \onslide<1>\includegraphics[width=\textwidth]{figs/max_size_zoom.pdf}
      \end{overprint}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Practical considerations: treating the determinant}
  \begin{columns}
    \begin{column}{0.9\textwidth}
      \[
        |\det M|=\sqrt{\det MM^T}
        =\int\frac{d\boldsymbol s}{(2\pi)^{D/2}}\,d\bar{\boldsymbol\eta}\,d\boldsymbol\eta\,
        e^{-\frac12\boldsymbol s^TMM^T\boldsymbol s-\bar{\boldsymbol\eta}^TMM^T\boldsymbol\eta}
      \]
      \begin{align*}
        &\#_r
        =
        \int\frac{d\boldsymbol x\,d\boldsymbol s}{(2\pi)^{D/2}}d\bar{\boldsymbol\eta}\,d\boldsymbol\eta
        \sum_{\substack{\sigma\subset[M]\\|\sigma|=D}}
        \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
        \\
        &\hspace{8em}
        \times
        \bigg(
          \prod_{\mu\in\sigma}
          \delta\big(h^\mu(\boldsymbol x)-r\big)
          e^{
            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
          }
        \bigg)
      \end{align*}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{Practical considerations: treating the sum over subsets}
  \begin{columns}
    \begin{column}{0.9\textwidth}
      \begin{align*}
        &\#_r
        =
        \int\frac{d\boldsymbol x\,d\boldsymbol s}{(2\pi)^{D/2}}d\bar{\boldsymbol\eta}\,d\boldsymbol\eta
        \\
        &\times
        \sum_{\substack{\sigma\subset[M]\\|\sigma|=D}}
        \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)
          e^{
            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
          }
        \bigg)
      \end{align*}
      \begin{align*}
        &\#_r
        =
        \lim_{\omega\to\infty}\int\frac{d\boldsymbol x\,d\boldsymbol s}{(2\pi)^{D/2}}d\bar{\boldsymbol\eta}\,d\boldsymbol\eta
        \\
        &
        \times\prod_{\mu=1}^M\bigg(\omega\theta\big(h^\mu(\boldsymbol x)-r\big)
          +
          \omega^{1-\frac MD}\delta\big(h^\mu(\boldsymbol x)-r\big)
          e^{
            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
          }
        \bigg)
      \end{align*}
    \end{column}
  \end{columns}
\end{frame}
\begin{frame}
  \frametitle{Practical considerations: treating the sum over subsets}
  \begin{columns}
    \begin{column}{0.9\textwidth}
      Why does this work? Expand the product:
      \begin{align*}
        &\prod_{\mu=1}^M
        \bigg(
          \omega\theta\big(h^\mu(\boldsymbol x)-r\big)
          +
          \omega^{1-\frac MD}\delta\big(h^\mu(\boldsymbol x)-r\big)
          e^{
            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
          }
        \bigg)
        \\
        &=\sum_{d=0}^M\omega^{M(1-\frac dD)}
        \sum_{\substack{\sigma\subset[M]\\|\sigma|=d}}
        \bigg(\prod_{\mu\in[M]\backslash\sigma}\theta\big(h^\mu(\boldsymbol x)-r\big)\bigg)
        \\
        &
        \hspace{8em}\times\bigg(
          \prod_{\mu\in\sigma}
          \delta\big(h^\mu(\boldsymbol x)-r\big)
          e^{
            -\frac12[\boldsymbol s\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]^2
            -[\bar{\boldsymbol\eta}\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
            [\boldsymbol\eta\cdot\frac\partial{\partial\boldsymbol x}h^\mu(\boldsymbol x)]
          }
        \bigg)
      \end{align*}
      Our term is $d=D$, giving $\omega^0$. For the undesired terms:
      \begin{itemize}
        \item $d>D$: $\omega$ raised to a negative power, limit kills term
        \item $d<D$: integral over $\boldsymbol s$, $\bar{\boldsymbol\eta}$, $\boldsymbol\eta$ gives determinant of matrix with rank less than dimension
      \end{itemize}
    \end{column}
  \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}
