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\title{Relative Critical Sets:  Structure and application}
%\subtitle{2006 Sigma Xi Researcher of the Year}
\author[[email protected]]{Dr. Jason Miller}
\institute{CSU Channel Islands}
\date{18 September 2017}

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 \frametitle{About the Talk}
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\section{Introduction}

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 The concept of $d$-dimensional relative critical set generalizes the concept of (zero dimensional) critical point of a differentiable function.
 
 \onslide<2>{Let $U \subset \mathbb{R}^n$ and $f: U \longrightarrow \mathbb{R}$ be a smooth function.  Let $x  \in U$.}
 
 \onslide<3>{Let $H(f)$ be the Hessian of $f$, $\lambda_i \leq \lambda_{i+1}$ its eigenvalues and $e_i$ a unit eigenvector for $\lambda_i$ so that $\{e_i\}_{i=1}^{n}$ an orthonormal basis of $\mathbb{R}^n$.}

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 \begin{block}<1>{Critical Set, v.1}
 The $x$ is a critical point iff $\nabla f=0$ at $x$.
 \end{block}
 
 \onslide<3>{Alternatively...}
 
 \begin{block}<4>{Critical Set, v.2}
 The $x$ is a critical point iff, at $x$, $\nabla f \cdot e_i=0$ for all $i$.
 \end{block}
 
 \onslide<2>{If we specify that $\lambda_n<0>{Structure}
 Generically, a function's critical set is a set of isolated points.
 \end{block}

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\section{Definition}

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 Let $0 < d>{$0$-dimensional Relative Critical Set}
 The $x$ is a critical point iff, at $x$, $\nabla f \cdot e_i=0$ for all $i$. 
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 Let $0 < d>{$d$-dimensional Relative Critical Set}
 The $x$ is a critical point iff, at $x$, $\nabla f \cdot e_i=0$ for $i \leq n-d$.
 \end{block}
 
 \onslide<2>{If we specify that $\lambda_{n-d}<0>{Structure Question}
 What is the local generic structure of a function's $d$-dimensional ridge in $\mathbb{R}^n$ (esp. near partial umbilics)?
 \end{alertblock}
 
 \onslide<4>{The $d=1$ dimensional height ridge has applications in image analysis, so knowing its generic structure is important.  ~\cite{pemfp:acbv, ron:hrcaffv, de:riiada}}

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\section[Structure]{What's known}

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 \begin{theorem}[\cite{jnd:porac, jm:rcsiia}]
 Generically, the closure of the $1$-dimensional ridge is 
 \begin{itemize}
  \item<2> a discrete set of smooth embedded curves, that
  \item<3> has boundary points at partial umbilic points ($\lambda_{n-1}= \lambda_n$) or at singular points ($\lambda_{n-1}=0$) of the Hessian.
 % \item<4> passes smoothly through critical points of $f$.
 \end{itemize}
 \end{theorem}
 
 \begin{theorem}<4>[\cite{jm:rcsiia}]
 Generically,  the closure of the $2$-dimensional ridge is 
 \begin{itemize}
  \item<5> a discrete set of smooth embedded surfaces surfaces, that
  \item<6> has boundary curves at partial umbilic points ($\lambda_{n-2}= \lambda_{n-1}$) or at singular points ($\lambda_{n-2}=0$) of the Hessian, and its
  \item<7> boundary is smooth except at a corner where $\lambda_{n-2}= \lambda_{n-1}=0$.
 \end{itemize}
 \end{theorem}
 
 %The boundary of a $2$-dimensional ridge is a smooth curve except at corners which occur at singular partial umbilics.

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 \onslide<1>{This and related genericity result is established by 
 \begin{itemize}
  \item<3> collecting closed submanifolds and stratified sets of jet space and then 
  \item<4> using a set of mappings,
  \item<5> applying Thom's Transversality Theorem to get the result.
 \end{itemize}
 }
 
 \onslide<2>{
 \begin{theorem}[Thom's Transversality Theorem]\label{thm:TTT}
    For $M$ and $N$ smooth manifolds with $\Gamma$ a submanifold of 
    $J^{k}(M,N)$, let
    \begin{align*}
            T_{\Gamma} & = \{ f \in \mathcal{C}^{\infty}(M,N) \, \lvert \,
            j^k(f) \mbox{ is transverse to } \Gamma\}.
    \end{align*}
    Then $T_{\Gamma}$ is a residual subset of 
    $\mathcal{C}^{\infty}(M,N)$ in the Whitney 
    $\mathcal{C}^{\infty}$-topology.  If $\Gamma$ is closed, then 
    $T_{\Gamma}$ is open.
 \end{theorem}
 }

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 The boundary of the $d$-dimensional ridge inherits its geometry from that of the
 \begin{itemize}
  \item<2> geometry of the set of partial umbilic matrices (semialgebraic)
  \item<3> geometry of singular (algebraic)
 \end{itemize}
 \onslide<4>{as subsets of in $\mathcal{S}^2\mathbb{R}^n$.}
 
 \vspace{.125in}
 
 \begin{theorem}[The "$\ell$ chose two" Test]<5>  There is a closed semialgebraic set $V^{(\ell)} \subset J^2(n,1)$ with the property that if $d<\binom{\ell}{2}$ and another transversality condition holds, then the closure of a $d$-dimensional ridge of $f$ misses the partial umbilics of order $\ell$.
 \end{theorem}
 
 \begin{example}<6>
  The $3$-dimensional ridge fails this test for the partial umbilics of order $\ell=3$.
 \end{example}

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\section{Question}

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 Part of the boundary of the $3$-dimensional ridge will coincide with partial umbilics of order 2 where $\lambda_{n-3}=\lambda_{n-2}$
 
 \onslide<2>{The "$\ell$ chose two" Test implies the possibility that this part of the boundary also contains partial umbilics of order 3.}
 
 \begin{alertblock}{Question}<4>
  Knowing the geometry of the set of umbilics of order 2 in a normal slice to the set of umbilics of order 3 will illuminate the boundary structure of the 3-dimensional ridge.  \onslide<5>{Analogous information for higher umbilics could complete the structure theorem for ridges of all dimension.}
 \end{alertblock}
 
 \onslide<6>{In~\cite{via:maq}, Arnol'd remarks without proof that the structure is of a cone over projective space.}

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\section{References}

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