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\begin{document}

\title{Igusa zeta functions and the non-archimedean SYZ fibration}

\author{Johannes Nicaise}

\address{Imperial College,
Department of Mathematics, South Kensington Campus,
London SW72AZ, UK, and KU Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Heverlee, Belgium} \email{j.nicaise@imperial.ac.uk}
\thanks{Johannes Nicaise is supported by the ERC Starting Grant MOTZETA (project 306610) of the European Research Council.}

\maketitle

\begin{abstract}
 We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions.
  The proof technique is based on the Minimal Model Program in birational geometry, but the proof was heavily inspired by ideas coming from non-archimedean geometry and mirror symmetry; we will outline these relations at the end of the paper. This text is intended to be a low-tech introduction to these topics; we only assume that the reader has a basic knowledge of algebraic geometry.
\end{abstract}

\section{Introduction}
Let $f$ be a non-constant polynomial in $\ensuremath{\mathbb{Z}}[x_1,\ldots,x_n]$, for some $n\geq 1$, with $f(\mathbf{0})=0$. For every prime number $p$, Igusa's $p$-adic zeta function $Z_{f,p}(s)$ of $f$ at $\mathbf{0}$ is a meromorphic function on $\ensuremath{\mathbb{C}}$ that encodes the numbers of solutions of the congruences $f\equiv 0$ modulo powers of $p$ that reduce to the origin $\mathbf{0}$ modulo $p$. The precise definition will be recalled in Section \ref{sec:igusa}. Igusa has proven that $Z_{f,p}(s)$ is a rational function in $p^{-s}$ by performing a local computation of $Z_{f,p}(s)$ on a log-resolution of $f$ over $\ensuremath{\mathbb{Q}}$.

This proof provides a relation between the poles of $Z_{f,p}(s)$ and the geometry of a log-resolution; see Section \ref{sec:denef}. However, many fundamental questions about these poles are still unanswered, the principal one being Igusa's Monodromy Conjecture, which relates the poles of $Z_{f,p}(s)$ to roots of the Bernstein polynomial of $f$ -- see \eqref{sss:monconj}. Igusa's proof also implies that the order of a pole of $Z_{f,p}(s)$ is at most $n$, and that, if $p$ is sufficiently large, the real part of a pole is at most $-\mathrm{lct}_{\mathbf{0}}(f)$, the opposite of the log canonical threshold of $f$ at $\mathbf{0}$. In 1999, Veys conjectured that the real part of every pole of order $n$ is equal to $-\mathrm{lct}_{\mathbf{0}}(f)$. This implies, in particular, that such poles of maximal order always satisfy the Monodromy Conjecture.

We have proven Veys's conjecture in collaboration with Chenyang Xu in \cite{NicXu2}. The main lines of the proof are explained in Section \ref{sec:veys}. Although the proof makes heavy use of the Minimal Model Program in birational geometry, an important source of inspiration was provided by a combination of non-archimedean geometry and the theory of mirror symmetry, more precisely Kontsevich and Soibelman's non-archimedean interpretation of the SYZ conjecture. This will be explained in Section \ref{sec:SYZ}.

\subsection*{Acknowledgements}
I would like to thank the organizers of the VIASM Annual Meeting 2017 for the invitation to deliver a lecture at the meeting and to write this survey article for the proceedings. The results presented here are a result of joint work with Mircea Musta\c{t}\u{a} and Chenyang Xu, and it is a pleasure to thank them both for the pleasant and interesting collaboration. I am also indebted to Wim Veys for sharing his ideas on the conjecture that constitutes the main subject of this text.

\section{Igusa's $p$-adic zeta functions}\label{sec:igusa}
\subsubsection{} We fix a non-constant polynomial $f$ in $\ensuremath{\mathbb{Z}}[x_1,\ldots,x_n]$, for some $n\geq 1$, with $f(\mathbf{0})=0$. For every prime number $p$ and every integer $d\geq 0$, we set
$$N_{f,p}(d)=\sharp \{x\in (p\ensuremath{\mathbb{Z}}/p^{d+1}\ensuremath{\mathbb{Z}})^n\,|\,f(x)\equiv 0\mod p^{d+1} \}.$$
 Thus $N_{f,p}(d)$ is the number of solutions of the congruence $f\equiv 0$ modulo $p^{d+1}$ that reduce to the origin modulo $p$. To study the asymptotic behaviour of these numbers as $d\to \infty$, we introduce the generating series
 $$P_{f,p}(T)=\sum_{d\geq 0}N_{f,p}(d)T^d\quad \in \ensuremath{\mathbb{Z}}[[T]].$$
 Now it is natural to ask the following question.

\begin{question}[Borevich-Shafarevich, Problem 9 in Ch.1\S5 of \cite{BorSha}]
Is $P_{f,p}(T)$ a rational function? That is, can we write it as a quotient of two polynomials in $\ensuremath{\mathbb{Z}}[T]$?
\end{question}

\noindent Recall that the rationality of $P_{f,p}(T)$ is equivalent to the existence of a linear recursion pattern in the sequence of coefficients $(N_{f,p}(d))_{d\geq 0}$.

\subsubsection{} The series $P_{f,p}(T)$ is easy to compute if the zero locus of $f$ in $\ensuremath{\mathbb{A}}^n_{\ensuremath{\mathbb{Z}}}$ is smooth over $\ensuremath{\mathbb{Z}}$ at the origin of $\ensuremath{\mathbb{A}}^n_{\mathbb{F}_p}$: by Hensel's lemma,
every solution modulo $p^{d+1}$ lifts to precisely $p^{n-1}$ solutions modulo $p^{d+2}$, for all $d\geq 0$, so that $N_{f,p}(d+1)=p^{n-1}N_{f,p}(d)$. This leads to the formula
 $$P_{f,p}(T)=\frac{1}{1-p^{n-1}T}.$$ However, the computation is substantially more difficult if $f$ has a singularity, already in the case of the plane cusp.

\begin{example}\label{exam:cuspgen}
If  $f=(x_1)^2-(x_2)^3$ and $p>3$, the generating series $P_{f,p}(T)$ is given by the formula
$$P_{f,p}(p^{-2}T)=
\frac{1+(p-1)p^{-1}T+(p-1)p^{-4}T^5-p^{-5}T^6}{(1-p^{-1}T)(1-p^{-5}T^{6})}.$$
\end{example}

 The reason for rescaling the variable $T$ by a factor $p^{-2}$ will become apparent below (spoiler: the exponents $5$ and $6$ that appear in the denominator are related to the so-called log canonical threshold $5/6$ of the plane cusp singularity).
 We can expect in general that $P_{f,p}(T)$ will reflect some interesting properties of the singularity of $f$ at $\mathbf{0}$.

 \subsubsection{} Igusa has proven in \cite{Igu} that $P_{f,p}(T)$ is always rational. A key step in the proof is to rewrite the generating series $P_{f,p}(T)$ as a $p$-adic integral
 $$Z_{f,p}(s)=\int_{(p\ensuremath{\mathbb{Z}}_p)^n}|f|_p^s|dx|$$ where $s$ is a complex variable and $|dx|$ is the normalized Haar measure on $\ensuremath{\mathbb{Z}}_p^n$. This $p$-adic integral converges if $\Re(s)>0$ and defines a analytic function on the complex right half-plane, which is called Igusa's $p$-adic zeta function of $f$ at the origin $\mathbf{0}$. It is an easy exercise in $p$-adic integration to express $Z_{f,p}(s)$ in terms of $P_{f,p}(T)$: one has  \begin{equation}\label{eq:tranform}
 P_{f,p}(p^{-n-s})=\frac{1 - p^{n}Z_{f,p}(s) }{1-p^{-s}}.\end{equation}
 Thus it is enough to prove that $Z_{f,p}(s)$ is a rational function in $p^{-s}$ (and, in particular, extends to a meromorphic function on $\ensuremath{\mathbb{C}}$). Igusa proved this by taking a resolution of singularities for $f$ over $\ensuremath{\mathbb{Q}}$ and using the change of variables formula for $p$-adic integrals to reduce to the case where $f$ is a monomial, which can be solved by a simple computation.

  \subsubsection{} Igusa's proof not only establishes the rationality of $Z_{f,p}(s)$ but also provides some interesting information about the possible poles of
$Z_{f,p}(s)$ and their expected orders in terms of the geometry of a resolution of singularities for $f$. This is a striking result, because it relates arithmetic properties of $f$ (the poles of the zeta function) with geometric properties of $f$ (the geometry of a resolution of singularities).
  A completely explicit formula for $Z_{f,p}(s)$ in terms of a resolution of $f$ was later given by Denef, for $p\gg 0$. In the next section, we will review the precise formulations of Igusa's theorem and Denef's formula.

\section{Igusa's theorem and Denef's formula}\label{sec:denef}
\subsubsection{} First, we need to introduce some notation. Let $h:Y\to \ensuremath{\mathbb{A}}^n_{\ensuremath{\mathbb{Q}}}$ be a log-resolution for the morphism $f:\ensuremath{\mathbb{A}}^n_{\ensuremath{\mathbb{Q}}}\to \ensuremath{\mathbb{A}}^1_{\ensuremath{\mathbb{Q}}}$ defined by the polynomial $f$. This means that  $Y$ is a smooth $\ensuremath{\mathbb{Q}}$-variety, $h$ is a projective morphism of $\ensuremath{\mathbb{Q}}$-varieties that is an isomorphism over $\ensuremath{\mathbb{A}}^n_{\ensuremath{\mathbb{Q}}}\setminus \mathrm{div}(f)$, and $\mathrm{div}(f\circ h)$ is a strict normal crossings divisor on $Y$. Such a morphism $h$ always exists, by Hironaka's embedded resolution of singularities in characteristic zero.

\subsubsection{} To every log-resolution $h$, we associate the following numerical invariants. We write
$$\mathrm{div}(f\circ h)=\sum_{i\in I}N_iE_i$$
 where $E_i,\,i\in I$ are the prime components of the divisor $\mathrm{div}(f\circ h)$, and the $N_i$ are their multiplicities. Since $h$ is an isomorphism over the complement of $\mathrm{div}(f)$, we can write the relative canonical divisor of $h$ as
 $$K_{Y/X}=\sum_{i\in I}(\nu_i-1)E_i.$$ The number $\nu_i$ is called the log-discrepancy of $X$ at the divisor $E_i$; these are fundamental invariants in birational geometry. Roughly speaking, the multiplicities $N_i$ measure the complexity of $f$ and the log-discrepancies $\nu_i$ measure the complexity of the resolution $h$.
  For every non-empty subset $J$ of $I$, we set
$$E_J=\cap_{j\in J}E_j,\quad E_J^o=E_J\setminus(\cup_{i\notin J}E_i).$$
 The sets $E_J^o$ form a stratification of $\mathrm{div}(f\circ h)$ into locally closed subsets.

 \begin{theorem}[Igusa \cite{Igu}]
 For every prime number $p$, the zeta function $Z_{f,p}(s)$ lies in the ring $$\ensuremath{\mathbb{Q}}\left[\frac{1}{p^{as+b}-1}\right]_{a,b\in \ensuremath{\mathbb{Z}}_{>0}}.$$ If $s_0$ is a pole of order $m$ of $Z_{f,p}(s)$, then there exists a subset $J$ of $I$ of cardinality $m$ such that $E_J^o(\ensuremath{\mathbb{Q}}_p)\cap h^{-1}(p\ensuremath{\mathbb{Z}}_p^n)\neq \emptyset$ and $\Re(s_0)=-\nu_j/N_j$ for every $j$ in $J$.
 \end{theorem}

 \begin{theorem}[Denef \cite{Den}]
If the prime number $p$ is sufficiently large, then
$$Z_{f,p}(s)=p^{-n}\sum_{\emptyset \neq J\subset I}\sharp(\overline{E}_J^o(\mathbb{F}_p)\cap \overline{h}^{-1}(\mathbf{0}))\prod_{j\in J}\frac{p-1}{p^{N_js+\nu_j}-1}$$
where $\overline{(\cdot)}$ denotes reduction modulo $p$.
 \end{theorem}
\noindent  To be precise, Denef's formula is valid when the resolution $h$ has ``good reduction modulo $p$'' in a certain technical sense; for our purposes, it suffices to know that this condition is always satisfied for $p\gg 0$.

\begin{example}\label{exam:cuspzeta}
A log-resolution $h$ for $f=(x_1)^2-(x_2)^3$ can be constructed by means of three consecutive point blow-ups (see Figure \ref{fig:cuspres}). Then $\mathrm{div}(f\circ h)$ has four irreducible components:
 the strict transform $E_0$ of $\mathrm{div}(f)$, with numerical data $N_0=\nu_0=1$, and three exceptional components $E_1,\,E_2,\,E_3$ whose numerical data $(N_i,\nu_i)$ are given by
 $(2,2)$, $(3,3)$ and $(6,5)$, respectively. The log resolution $h$ has good reduction modulo $p$ for all primes $p>3$; in those cases, Denef's formula yields
 \begin{eqnarray*}
 \frac{p^2}{p-1}Z_{f,p}(s)&=& p\frac{p^{-2-2s}}{1-p^{-2-2s}}+p\frac{p^{-3-3s}}{1-p^{-3-3s}}+(p-2)\frac{p^{-5-6s}}{1-p^{-5-6s}}
 \\ & & +(p-1)\frac{p^{-5-6s}}{1-p^{-5-6s}}
 \left(\frac{p^{-1-s}}{1-p^{-1-s}}+\frac{p^{-2-2s}}{1-p^{-2-2s}}+\frac{p^{-3-3s}}{1-p^{-3-3s}}  \right)
 \end{eqnarray*}
 which is compatible with the comparison formula \eqref{eq:tranform} and our expression for $P_{f,p}(p^{-2}T)$ in Example \ref{exam:cuspgen}.
\end{example}

\begin{figure*}[h!]
  \includegraphics[width=1\textwidth]{cusp.pdf}
\caption{Log-resolution of the plane cusp}
\label{fig:cuspres}
\end{figure*}

\subsubsection{} \label{sss:monconj} We can draw some immediate consequences from Igusa and Denef's results. Igusa's theorem implies that the real parts of the poles of $Z_{f,p}(s)$ are all contained
in the finite set
$$\{-\frac{\nu_i}{N_i}\,|\,i\in I,\,E_i(\ensuremath{\mathbb{Q}}_p)\cap h^{-1}(p\ensuremath{\mathbb{Z}}_p^n)\neq \emptyset \}.$$
 In practice, most of these candidates will not be real parts of actual poles of $Z_{f,p}(s)$. For one thing, the list of candidates strongly depends on the
 choice of the resolution $h$, whereas $Z_{f,p}(s)$ only depends on $f$ and $p$. But even if $n=2$, when there exists a minimal log-resolution $h$ of $f$, most of
 the candidates will not appear as real parts of poles of the zeta function. A partial explanation of this phenomenon would be given by the so-called Monodromy Conjecture, which puts an additional restriction on the poles of $Z_{f,p}(s)$.

 \begin{conjecture}[Igusa's Monodromy Conjecture]\label{conj:igusa}
If $p$ is sufficiently large and $s_0$ is a pole of $Z_{f,p}(s)$, then $\Re(s_0)$ is a root of the Bernstein polynomial of $f$ at $\mathbf{0}$. In particular, $\exp(2\pi i \Re(s_0))$ is a local monodromy eigenvalue of the complex hypersurface defined by the equation $f=0$.
 \end{conjecture}

\noindent The Monodromy Conjecture quantifies how the singularities of the complex hypersurface defined by $f=0$ influence the poles of the $p$-adic zeta function $Z_{f,p}(s)$, and thus the asymptotic behaviour of the coefficients of the generating series $P_{f,p}(T)$.
 This conjecture has been solved if $n=2$ by Loeser \cite{Loe}, and also for some special classes of singularities, but the general case is wide open. We refer to \cite{Ni:japan} for an overview of results that were known around 2010; a significant breakthrough since that time was the proof for Newton non-degenerate polynomials in three variables -- see \cite{LemPro} and \cite{BorVey}.

\subsubsection{} Igusa's theorem also implies that the order of a pole $s_0$ of $Z_{f,p}(s)$ is at most
$$\max\{\sharp J\,|\,J\subset I,\,\Re(s_0)=-\nu_j/N_j\mbox{ for all }j\in J,\,E_J(\ensuremath{\mathbb{Q}}_p)\cap h^{-1}(p\ensuremath{\mathbb{Z}}_p^n)\neq \emptyset\}.$$
In particular, it is at most $n$, since $E_J$ is empty if $J$ has more than $n$ elements ($n+1$ different prime components of a strict normal crossings divisor on a variety of dimension $n$ can never intersect in a point).

\subsubsection{} Finally, from Denef's formula, it also follows that the real part of a pole of $Z_{f,p}(s)$ is at most
$$-\min\{\frac{\nu_i}{N_i}\,|\,i\in I,\,E_i\cap h^{-1}(\mathbf{0})\neq \emptyset\}$$
 when $p$ is sufficiently large (if $E_i\cap h^{-1}(\mathbf{0})$ is empty, then $\overline{E}_i\cap \overline{h}^{-1}(\mathbf{0})$ is empty for $p\gg 0$).
 The number
$$\mathrm{lct}_{\mathbf{0}}(f)=\min\{\frac{\nu_i}{N_i}\,|\,i\in I,\,E_i\cap h^{-1}(\mathbf{0})\neq \emptyset\}$$ is an important invariant in birational geometry, called the log-canonical
 threshold of $f$ at $\mathbf{0}$. It is independent of the choice of a log-resolution $h$. It is used to measure the degree of the singularity of $f$ at $\mathbf{0}$, and to divide
 the singularities into different types in the Minimal Model Program.

 \subsubsection{} The main subject of this survey is the following conjecture.
 \begin{conjecture}[Veys 1999]
 Assume that $p$ is sufficiently large. If $s_0$ is a pole of $Z_{f,p}(s)$ of order $n$, then $\Re(s_0)=-\mathrm{lct}_{\mathbf{0}}(f).$
 \end{conjecture}
\noindent  Thus if  $Z_{f,p}(s)$ has a pole of the largest possible order (namely, $n$), then its real part is also as large as possible. Veys's conjecture was originally stated in \cite{LaeVey} for
  a different type of zeta function (the so-called topological zeta function), but the proof we will present is valid for the topological and motivic zeta functions, as well; see \cite[\S3]{NicXu}. Veys's conjecture implies, in particular, that poles of order $n$ satisfy Igusa's Monodromy Conjecture, because it is known that $-\mathrm{lct}_{\mathbf{0}}(f)$ is the largest root of the Bernstein polynomial of $f$ at $\mathbf{0}$ (see for instance Theorem 10.6 in \cite{Kol}).

\section{Proof of Veys's conjecture}\label{sec:veys}
\subsubsection{} We will deduce Veys's conjecture from a more general result about the geometry of log-resolutions. Let $k$ be a field of characteristic zero, $X$ a smooth $k$-variety, $f:X\to \ensuremath{\mathbb{A}}^1_k$ a dominant morphism and $h:Y\to X$ a log-resolution of $f$ as above. We fix a closed point $x$ on $X$ such that $f(x)=0$. The situation we have studied so far corresponds to the case $k=\ensuremath{\mathbb{Q}}$, $X=\ensuremath{\mathbb{A}}^n_{\ensuremath{\mathbb{Q}}}$, $x=\mathbf{0}$. We will continue to use the notations $N_i$, $\nu_i$, $E_J$ etc. The result that we will prove is local on $X$ at the point $x$; shrinking $X$ around $x$, we may assume that $E_J\cap h^{-1}(x)\neq \emptyset$ as soon as $E_J\neq \emptyset$, for every non-empty subset $J$ of $I$. This assumption simplifies some of the notations we will use.

\subsubsection{} \label{sss:weight} For every $i\in I$, we set $\mathrm{wt}_{f}(E_i)=\nu_i/N_i$ and we call this value the {\em weight} of $f$ at  $E_i$. The log-canonical threshold of $f$ at $x$ is then given by
$$\mathrm{lct}_x(f)=\min\{\mathrm{wt}_f(E_i)\,|\,i\in I\}.$$
We write $Y_0$ for the divisor $$\mathrm{div}(f\circ h)=\sum_{i\in I}N_iE_i,$$
and we denote by $\Delta(Y_0)$ its dual intersection complex. This is a finite $\Delta$-complex whose vertices correspond bijectively to the components $E_i$, $i\in I$, and such that for every subset $J$ of $I$, the number of faces spanned by the vertices $v_j$ with $j\in J$ is equal to the number of connected components of $E_J$.
 We introduce a function
$$\mathrm{wt}_{f}:\Delta(Y_0)\to \ensuremath{\mathbb{R}}$$ that is completely characterized by the following properties:
\begin{itemize}
\item for every $i\in I$, the value of $\mathrm{wt}_f$ at the vertex of $\Delta(Y_0)$ corresponding to $E_i$ is given by $\mathrm{wt}_f(E_i)=\nu_i/N_i$;
\item the function $\mathrm{wt}_f$ is affine on every face of $\Delta(Y_0)$.
\end{itemize}

\begin{example}\label{exam:weightcusp}
In the case of the minimal log-resolution of the plane cusp (Example \ref{exam:cuspzeta}) we can immediately recover the dual intersection complex $\Delta(Y_0)$ and the
weight function $\mathrm{wt}_f$ from Figure \ref{fig:cuspres}. Since the dimension of $Y_0$ equals one, the dual intersection complex is nothing else than the {\em dual graph} of $Y_0$, with four vertices $v_0,\ldots,v_3$ corresponding to the irreducible components $E_0,\ldots,E_3$ and three edges corresponding to the intersection points of these components. The weight function $\mathrm{wt}_f$ is characterized by its values $\mathrm{wt}_f(v_0)=\mathrm{wt}_f(v_1)=\mathrm{wt}_f(v_2)=1$ and $\mathrm{wt}_f(v_3)=5/6$, and the property that $\mathrm{wt}_f$ is affine on every edge.
\end{example}

\begin{figure*}[h!]
  \includegraphics[width=0.9\textwidth]{weight_cusp.pdf}
\caption{The weight function of the minimal log-resolution of the plane cusp}
\label{fig:cuspweight}
\end{figure*}

\subsubsection{} \label{sss:observ} The key observation is that $\mathrm{wt}_f$ seems to induce a {\em flow} on $\Delta(Y_0)$ in the direction of decreasing weights, and
 that this flow collapses $\Delta(Y_0)$ onto the locus where $\mathrm{wt}_f$ is minimal, that is, equal to $\mathrm{lct}_x(f)$. Recall that a {\em collapse} if a particular type of
 strong deformation retraction that can be defined in a combinatorial way for $\Delta$-complexes.
 This observation can be made precise in the case $n=2$ (see for instance Example \ref{exam:weightcusp} and Figure \ref{fig:cuspweight}) but it is not clear what the correct notion of a flow
 on a $\Delta$-complex would be in higher dimensions.  We will explain the origin of this observation in Section \ref{sec:SYZ}: it has roots in Kontsevich and Soibelman's non-archimedean SYZ fibration for degenerations of Calabi-Yau varieties in the theory of mirror symmetry.

\subsubsection{}   In \cite{NicXu}, Chenyang Xu and the author have proven two results that provide compelling evidence for the above observation.
  We denote by $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$ the locus where $\mathrm{wt}_f$ reaches its minimal value $\mathrm{lct}_x(f)$; this is a union of faces of $\Delta(Y_0)$.
 The first result can be understood as follows: if we place ourselves at any point of $\Delta(Y_0)$, then the weight function should somehow select a path along which the weight decreases and which ends in $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$, the locus of minimal weight. If our starting point lies in the interior of a maximal face $\sigma$ of $\Delta(Y_0)$ and the weight function $\mathrm{wt}_f$ is constant on $\sigma$, then $\mathrm{wt}_f$ does not single out any direction at all. So, if our observation is valid, this can only happen when $\sigma$ is already contained in the locus of minimal weight. This property is confirmed by the following theorem.

 \begin{theorem}[Nicaise--Xu, Theorem 2.4 in \cite{NicXu}]\label{thm:maxface}
 If $\mathrm{wt}_{f}$ is constant on a maximal face $\sigma$ of $\Delta(Y_0)$, then its value on $\sigma$ is the minimal value $\mathrm{lct}_x(f)$ of $\mathrm{wt}_{f}$.
\end{theorem}

\noindent The proof of the theorem uses sophisticated techniques from the Minimal Model Program (MMP) in birational geometry; the main idea is to run a well-chosen MMP algorithm on the pair $(Y,Y_0)$ and to study the behaviour of $Y$ near the stratum of $Y_0$ corresponding to the face $\sigma$.

\subsubsection{} Now we consider the homotopy types of the intermediate level sets of the function $\mathrm{wt}_f$.
  A small technical complication (which will disappear in the SYZ picture in Section \ref{sec:SYZ}) is that the components of $Y_0$ come in two flavours: the exceptional components of $h$ and the components of the strict transform of $\mathrm{div}(f)$. The following theorem refines a fundamental result of de Fernex, Koll\'ar and Xu \cite{dFKoXu}.

 \begin{theorem}[Nicaise--Xu, Theorem 4.10 in \cite{NicXu}]\label{thm:collapse}
 Assume that $\mathrm{div}(f)$ is reduced. We denote by $\Delta(Y_0)_{\mathrm{exc}}$ the subcomplex of $\Delta(Y_0)$ spanned by the vertices that correspond to exceptional components of $h$.
 For every $w\in \ensuremath{\mathbb{R}}$, we denote by $\Delta(Y_0)^{\leq w}$ the subcomplex of $\Delta(Y_0)$ spanned by the vertices $v$ such that $\mathrm{wt}_{f}(v)\leq w$, and we define $\Delta(Y_0)^{\leq w}_{\mathrm{exc}}$ accordingly.
  \begin{itemize}
 \item Assume that $\mathrm{lct}_x(f)=1$ (that is, $(X,\mathrm{div}(f))$ is log-canonical at $x$). There exists a collapse of $\Delta(Y_0)$ onto $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$ that simultaneously collapses $\Delta(Y_0)^{\leq w}$ onto $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$ for every $w\geq \mathrm{lct}_x(f)$. In particular, the inclusion maps of $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$
     and $\Delta(Y_0)^{\leq w}$ into $\Delta(Y_0)$ are homotopy equivalences.
  \item Assume that $\mathrm{lct}_x(f)\neq 1$. There exists a collapse of $\Delta(Y_0)_\mathrm{exc}$ onto $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$ that simultaneously collapses $\Delta(Y_0)^{\leq w}_\mathrm{exc}$ onto $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$ for every $w\geq \mathrm{lct}_x(f)$. In particular, the inclusion maps of $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$
     and $\Delta(Y_0)^{\leq w}_\mathrm{exc}$ into $\Delta(Y_0)_\mathrm{exc}$ are homotopy equivalences.
  \end{itemize}
    \end{theorem}

Note that in the second case of Theorem \ref{thm:collapse}, we have $\mathrm{lct}_x(f)<1$ so that $\Delta(Y_0)^{=\mathrm{lct}_x(f)}$ is contained in $\Delta(Y_0)_{\mathrm{exc}}$ (components $E_i$ in the strict transform of $\mathrm{div}(f)$ all have $N_i=\nu_i=1$ because we assumed that $\mathrm{div}(f)$ is reduced).

\subsubsection{} Now let us explain how Theorem \ref{thm:maxface} implies Veys' conjecture. Let $f$ be a non-constant polynomial in $\ensuremath{\mathbb{Z}}[x_1,\ldots,x_n]$ and let $h:Y\to \ensuremath{\mathbb{A}}^n_{\ensuremath{\mathbb{Q}}}$ be a log-resolution for the morphism $f:\ensuremath{\mathbb{A}}^n_{\ensuremath{\mathbb{Q}}}\to \ensuremath{\mathbb{A}}^1_{\ensuremath{\mathbb{Q}}}$. Suppose that $s_0$ is a pole of order $n$ of $Z_{f,p}(s)$, for some sufficiently large prime number $p$. Denef's formula then implies that there exists a subset $J$ of $I$ of cardinality $n$ such that $E_J\cap h^{-1}(\mathbf{0})\neq \emptyset$ and $\Re(s_0)=-\nu_j/N_j$ for every $j\in J$.
 Then $E_J$ is a finite set of points, and each of these points corresponds to a face of $\Delta(Y_0)$ of dimension $n-1$ on which $\mathrm{wt}_f$ is constant with value $-\Re(s_0)$. Since the dimension of $\Delta(Y_0)$ is at most $n-1$, such a face is always maximal. Now Theorem \ref{thm:maxface} implies that $\Re(s_0)=-\mathrm{lct}_{\mathbf{0}}(f)$, which confirms Veys' conjecture.

\section{The non-archimedean SYZ fibration}\label{sec:SYZ}
\subsubsection{} To conclude this survey, we explain how the observation in \eqref{sss:observ} originated from the non-archimedean SYZ fibration of Kontsevich and Soibelman.
  The starting point of this story is the theory of {\em mirror symmetry}, which has grown out of mathematical physics (string theory) and has led to fascinating developments in algebraic geometry. For lack of competence, we will only superficially discuss the physical background. According to string theory, our universe can be described as a 10-dimensional real manifold: the 4 dimensions of space-time and an additional 6 dimensions coming from a complex Calabi-Yau  3-fold. Towards the end of the 80s, physicists realized that the Calabi-Yau manifold was not uniquely determined by the physical theory; rather, Calabi-Yau 3-folds seemed to come in {\em mirror pairs} giving rise to equivalent physical theories. One famous property of mirror 3-folds $(X,\check{X})$ is the following symmetry of their Hodge diamonds:
$$h^{p,q}(X)=h^{3-p,q}(\check{X}).$$ Systematic searches in databases of Calabi-Yau 3-folds supplied empirical evidence for the mirror phenomenon.

\subsubsection{} Mirror symmetry caught the attention of algebraic geometers through a spectacular application to enumerative geometry by Candelas, de la Ossa, Green and Parkes around 1990.
 They managed to compute numbers of rational curves of fixed degrees on the quintic threefold $X$; these are now known as Gromov-Witten invariants.
 Their calculation relied on the insight that the complex and symplectic geometries of $X$ and its mirror partner $\check{X}$ are swapped: the Gromov-Witten invariants
 of $X$ (which are symplectic in nature) can be recovered from certain period integrals on $\check{X}$ (which live in complex geometry and are easier to compute). At that time,
 there was no mathematical framework available to explain these results; a rigorous mathematical confirmation was given only around 1995.

 \subsubsection{} An important challenge for algebraic geometers was then to give an exact definition of what it means to be a mirror pair and to devise techniques to construct such pairs.
 In recent years, much progress has been made, in particular by Kontsevich--Soibelman and Gross--Siebert. Both of these programs are based on a conjectural
 geometric explanation of mirror symmetry due to Strominger, Yau and Zaslow, known as the SYZ conjecture (1996).  Let $\mathcal{X}^{\ast}$ be a projective family of $n$-dimensional complex Calabi-Yau manifolds
 over a punctured disk $D^{\ast}$ in $\ensuremath{\mathbb{C}}$, and assume that this family has semistable reduction over the unpunctured disk $D$ and that it is maximally degenerate. The latter condition means that
 the monodromy transformation on the degree $n$ cohomology of the general fiber $\mathcal{X}_t$ of $\mathcal{X}^{\ast}$ has a Jordan block of rank $n+1$. Then $\mathcal{X}_t$ should admit a fibration $\rho:\mathcal{X}_t\to S$ where $S$ is an $n$-dimensional topological manifold and the fibers of $\rho$ are so-called speical Lagrangian tori in $\mathcal{X}_t$.
   As the parameter $t\in D^{\ast}$ tends to $0$, the fibers of $\rho$ are contracted (in the sense of Gromov-Hasudorff convergence of metric spaces) and we are left with the base $S$.
     The mirror partner of $\mathcal{X}_t$ is then constructed by dualizing the torus fibration $\rho$ over the locus where it is smooth and compactifying the result in a suitable way (this involves deforming the dual fibration by so-called quantum corrections, which is the most subtle part of the picture). We refer to the excellent survey paper \cite{Gro} for a more precise statement and additional background on the SYZ conjecture, as well as the Gross--Siebert program.

 \subsubsection{} The SYZ conjecture seems very difficult to solve in its original differential-geometric set-up.
 A fundamental insight of Kontsevich and Soibelman in \cite{KoSo} is that a close analog of the SYZ fibration can be found in the world of non-archimedean geometry, more precisely in the context of Berkovich spaces. Here, the base $S$ of the fibration arises as a so-called {\em skeleton} and it is intimately related to the geometry of minimal models of the degeneration $\mathcal{X}^{\ast}$ in the sense of the Minimal Model Program. We will now explain the construction of the non-archimedean SYZ fibration, as well as its relation with the results in Section \ref{sec:veys}. Let us emphasize right away that the non-archimedean SYZ fibration is not merely an analog of the conjectural structure in a different context; it can effectively be used to realize the original goal of defining an constructing mirror partners over the complex numbers, by means of non-archimedean GAGA and algebraization techniques. The case of $K3$ surfaces was discussed in \cite{KoSo}, but substantial technical difficulties are still obstructing this approach in higher dimensions.

 \subsubsection{} We will work over an algebraic model of the complex disk $D$, namely, the spectrum $\ensuremath{\mathrm{Spec}\,} R$ of the ring $R=k[\negthinspace[ t]\negthinspace]$ of formal power series over an algebraically closed  field $k$ of characteristic zero. We write $K=k(\negthinspace( t)\negthinspace)$ for the quotient field of $R$; then $\ensuremath{\mathrm{Spec}\,} K$ is the algebraic analog of the punctured disk $D^*$.
  The $t$-adic absolute value on $K$, defined by $|a|=\exp(-\mathrm{ord}_t a)$ for all non-zero elements $a$ in $K$, endows $K$ with an analytic structure and allows one to develop a theory of analytic functions and analytic spaces over $K$. The theory that is most suited for our purposes is Berkovich's theory of $K$-analytic spaces, first presented in \cite{Ber}. To every $K$-scheme of finite type $X$, one can attach a $K$-analytic space $X^{\mathrm{an}}$ which is called the {\em analytification} of $X$, in analogy with the situation over the complex numbers. If $X$ is proper over $K$, then the non-archimedean GAGA principle guarantees that $X^{\mathrm{an}}$ completely determines the geometry of $X$; the upshot is that we can apply additional tools to analyze $X^{\mathrm{an}}$ coming from analytic geometry.

  \subsubsection{} Now let $Z$ be a smooth, projective, geometrically connected $K$-scheme with trivial canonical line bundle. For instance, if $\ensuremath{\mathscr{X}}^{\ast}$ is the family of Calabi-Yau manifolds considered above, we get such an object $Z$ over $K=\ensuremath{\mathbb{C}}(\negthinspace( t)\negthinspace)$ by base change to the quotient field of the completed local ring of $D$ at the origin.
  An {\em snc model} for $Z$ is a regular flat projective $R$-scheme $\ensuremath{\mathscr{Z}}$ endowed with an isomorphism of $K$-schemes $\ensuremath{\mathscr{Z}}_K\to Z$ such that the special fiber $\ensuremath{\mathscr{Z}}_k$ is a divisor with strict normal crossings. Then we can attach to $\ensuremath{\mathscr{Z}}_k$ a dual intersection complex $\Delta(\ensuremath{\mathscr{Z}}_k)$ in the same way as before. A fundamental theorem of Berkovich implies that there exists a canonical topological embedding of $\Delta(\ensuremath{\mathscr{Z}}_k)$ into $Z^{\mathrm{an}}$, whose image is called the {\em skeleton} of $\ensuremath{\mathscr{Z}}$ and is denoted by $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$. Moreover, this embedding has a canonical retraction $\rho_{\ensuremath{\mathscr{Z}}}:Z^{\mathrm{an}}\to \mathrm{Sk}(\ensuremath{\mathscr{Z}})$ that extends to a strong deformation retraction of $Z^{\mathrm{an}}$ onto $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$. We refer to \cite[\S2.5]{Ni:simons} for an explicit example. Thus $Z^{\mathrm{an}}$ is homotopy equivalent to $\Delta(\ensuremath{\mathscr{Z}}_k)$, for every snc model $\ensuremath{\mathscr{Z}}$ of $Z$. We can use the retraction maps $\rho_{\ensuremath{\mathscr{Z}}}$ to organize the skeleta $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$ into a projective system, indexed by the snc models $\ensuremath{\mathscr{Z}}$ ordered by the dominance relation. Then one can show that the natural continuous map
  $$Z^{\mathrm{an}}\to \lim_{\stackrel{\longleftarrow}{\ensuremath{\mathscr{Z}}}}\mathrm{Sk}(\ensuremath{\mathscr{Z}}) $$ induced by the retractions $\rho_{\ensuremath{\mathscr{Z}}}$
  is a homeomorphism. This result gives a complete description of the topology of $Z^{\mathrm{an}}$ in terms of the birational geometry of its snc models. It also provides a natural bridge between birational geometry and non-archimedean geometry.

\subsubsection{} Let $\omega$ be a volume form on $Z$.
 In a joint work with Mircea Musta\c{t}\u{a} \cite{MuNi}, we defined a {\em weight function}
 $$\mathrm{wt}_{\omega}:X^{\mathrm{an}}\to \ensuremath{\mathbb{R}}\cup \{+\infty\}$$
  which refines a construction of Kontsevich and Soibelman in \cite{KoSo} and is also related to work of Boucksom, Favre and Jonsson in \cite{BoFaJo}.
  Let $\ensuremath{\mathscr{Z}}$ be an snc model of $Z$. Then we can view $\omega$ as a rational section of the relative log canonical line bundle $\omega_{\ensuremath{\mathscr{Z}}/R}(\ensuremath{\mathscr{Z}}_{k,\mathrm{red}})$; we denote the associated Cartier divisor by $\mathrm{div}_{\ensuremath{\mathscr{Z}}}(\omega)$.
    The restriction of $\mathrm{wt}_{\omega}$ to the skeleton $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$ of  $\ensuremath{\mathscr{Z}}$ can be computed in the following way. It is affine on every face, so that
  it is completely determined by its values at the vertices of $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$. Each vertex $v$ corresponds to a prime component $E$ of the special fiber $\ensuremath{\mathscr{Z}}_k$; the value of $\mathrm{wt}_{\omega}$ at $v$ equals $\nu/N$, where $N$ is the multiplicity of $E$ in $\ensuremath{\mathscr{Z}}_k$ and $\nu$ is the multiplicity of $E$ in $\mathrm{div}_{\ensuremath{\mathscr{Z}}}(\omega)$.
     Note the similarity with the definition of the weight function $\mathrm{wt}_f$ in \eqref{sss:weight}. The weight function $\mathrm{wt}_f$ can also be interpreted as the weight function of a volume form on a suitable $K$-analytic space; see \cite[\S6]{MuNi}.

\subsubsection{}
The {\em essential skeleton} $\mathrm{Sk}(Z)$ of $Z$ is the set of points in $Z^{\mathrm{an}}$ where $\mathrm{wt}_{\omega}$ reaches its minimal value. This is a non-empty compact subspace of $Z^{\mathrm{an}}$. One can show that the weight function $\mathrm{wt}_{\omega}$ is strictly decreasing under the retraction map $\rho_{\ensuremath{\mathscr{Z}}}$, for every snc model $\ensuremath{\mathscr{Z}}$ of $Z$, so that $\mathrm{Sk}(Z)$ is always contained in $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$. More precisely, since $\mathrm{wt}_{\omega}$ is affine on every face of $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$, the essential skeleton $\mathrm{Sk}(Z)$ is a union of faces of $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$. The crucial point is that, whereas $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$ obviously depends on $\ensuremath{\mathscr{Z}}$, the essential skeleton $\mathrm{Sk}(Z)$ does not depend on the choice of an snc model. It is also independent of $\omega$, because multiplying $\omega$ with a constant in $K^{\ast}$ shifts the weight function by a constant integer.
 In Kontsevich and Soibelman's vision of the non-archimedean SYZ fibration, the base of the fibration is precisely the essential skeleton $\mathrm{Sk}(Z)$. In general, there does not exist any snc model $\ensuremath{\mathscr{Z}}$ of $Z$ such that the essential skeleton $\mathrm{Sk}(Z)$ coincides with $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$. However, it was shown by Chenyang Xu and the author in \cite{NicXu2} that we can identify $\mathrm{Sk}(Z)$ with the skeleton of a suitable model of $Z$ if we enlarge the class of models by considering so-called {\em divisorially log terminal (dlt)} models.
  For such (good) dlt models, one can generalize the construction of the skeleton $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$ (which is still homeomorphic to the dual intersection complex of $\ensuremath{\mathscr{Z}}_k$) and the retraction map $\rho_{\ensuremath{\mathscr{Z}}}:Z^{\mathrm{an}}\to \mathrm{Sk}(\ensuremath{\mathscr{Z}})$.
  If $\ensuremath{\mathscr{Z}}$ is a minimal dlt model of $Z$ in the sense of the Minimal Model Program, then $\mathrm{Sk}(Z)=\mathrm{Sk}(\ensuremath{\mathscr{Z}})$.

  \subsubsection{} This interpretation of $\mathrm{Sk}(Z)$ has two important advantages: it provides us with a map $\rho_{\ensuremath{\mathscr{Z}}}:Z^{\mathrm{an}}\to \mathrm{Sk}(Z)$, which plays the role of the SYZ fibration in the non-archimedean context (beware that it depends on the choice of a minimal dlt model, which is not unique in general). Moreover, we can use tools from the Minimal Model Program to study the properties of the essential skeleton $\mathrm{Sk}(Z)$ and the map $\rho_{\ensuremath{\mathscr{Z}}}$. According the the SYZ heuristic, the fibers of the non-archimedean SYZ fibration $\rho_{\ensuremath{\mathscr{Z}}}$ should be contracted in the limit $t \to 0$, in a suitable sense, so that
  $Z^{\mathrm{an}}$ collapses onto the locus $\mathrm{Sk}(Z)$ where the weight function $\mathrm{wt}_{\omega}$ reaches its minimal value.
      Then it is tempting to speculate that the weight function $\mathrm{wt}_{\omega}$ induces a flow on $Z^{\mathrm{an}}$ towards $\mathrm{Sk}(Z)$, along paths of decreasing weight.
      It is not yet clear what an appropriate definition of flow would be in this context (except if $\mathrm{dim}(Z)=1$; see Example \ref{exam:syz}).
 However, this heuristic has led us to formulate and prove the following theorems, which at their turn were the source of Theorems \ref{thm:maxface} and \ref{thm:collapse} above.

 \begin{theorem}[Nicaise--Xu, Theorem 5.4 in \cite{NicXu}]\label{thm:maxface2} Let $\ensuremath{\mathscr{Z}}$ be an snc model of $Z$.
  If $\mathrm{wt}_{\omega}$ is constant on a maximal face $\sigma$ of $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$, then its value on $\sigma$ is the minimal value of $\mathrm{wt}_{\omega}$. In other words,
 $\sigma$ is contained in the essential skeleton $\mathrm{Sk}(Z)$ of $Z$.
\end{theorem}

 \begin{theorem}[Nicaise--Xu, Theorem 5.6 in \cite{NicXu}]\label{thm:collapse2}
 Denote by $w_{\min}$ the minimal value of $\mathrm{wt}_{\omega}$ on $Z^{\mathrm{an}}$, and let $\ensuremath{\mathscr{Z}}$ be an snc model of $Z$.
 For every $w\in \ensuremath{\mathbb{R}}$, we denote by $\mathrm{Sk}(\ensuremath{\mathscr{Z}})^{\leq w}$ the subcomplex of $\ensuremath{\mathscr{Z}}$ spanned by the vertices $v$ such that $\mathrm{wt}_{\omega}(v)\leq w$.
  There exists a collapse of $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$ onto $\mathrm{Sk}(Z)$ that simultaneously collapses $\mathrm{Sk}(\ensuremath{\mathscr{Z}})^{\leq w}$ for every $w\geq w_{\min}$. In particular, the inclusion maps of $\mathrm{Sk}(Z)$ and $\mathrm{Sk}(\ensuremath{\mathscr{Z}})^{\leq w}$ into $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$ are homotopy equivalences.
\end{theorem}

\noindent Combining Theorem \ref{thm:collapse2} with Berkovich's strong deformation retraction of $Z^{\mathrm{an}}$ onto $\mathrm{Sk}(\ensuremath{\mathscr{Z}})$, one can then show that
 for every minimal dlt model $\ensuremath{\mathscr{Z}}$ of $Z$, the retraction map $\rho_{\ensuremath{\mathscr{Z}}}:Z^{\mathrm{an}}\to \mathrm{Sk}(Z)$ can be extended to a strong deformation retraction of $Z^{\mathrm{an}}$ onto the essential skeleton $\mathrm{Sk}(Z)$, which is in line with the SYZ heuristic.

\subsubsection{} There are some interesting open questions related to the shape of the essential skeleton $\mathrm{Sk}(Z)$. Assume that $Z$ has semi-stable reduction over $R$ (that is, it has an snc model with reduced special fiber) and that it is maximally degenerate (meaning that the monodromy transformation on the $\ell$-adic cohomology of $Z$ in degree $n=\mathrm{dim}(Z)$ has a Jordan block of size $n+1$). Then it was shown in \cite[4.2.4]{NicXu2} that $\mathrm{Sk}(Z)$ is a closed pseudo-manifold of dimension $n$.
If $Z$ is an abelian variety, then $\mathrm{Sk}(Z)$ is homeomorphic to a real torus of dimension $n$, by \cite[4.3.3]{HalNic}.
 Now assume that $Z$ is Calabi-Yau, in the sense that it is geometrically simply connected and the Hodge numbers $h^{0,i}(Z)$ vanish for
 $0<i<n$. Then $\mathrm{Sk}(Z)$ has the rational homology of the $n$-sphere $S^n$ by \cite[4.2.4]{NicXu2}, and its fundamental group has trivial profinite completion \cite[6.1.3]{HalNic}. It is expected that $\mathrm{Sk}(Z)$ is homeomorphic to $S^n$, but
  this appears to be a very hard question; it has been solved by Koll\'ar and Xu in \cite{KolXu} for $n\geq 3$, and for $n=4$ assuming that $Z$ has a minimal dlt model that is snc.
 A crucial difficulty is proving that $\mathrm{Sk}(Z)$ is a topological manifold.

\begin{example}\label{exam:syz}
The easiest example of a non-archimedean SYZ fibration is provided by a Tate elliptic curve over $K$. This is an elliptic curve $E$ over $K$ with Kodaira--N\'eron reduction type $I_n$, for some $n>0$. Every smooth and proper $K$-curve of positive genus has a unique minimal snc model; the condition of having reduction type $I_n$ with $n\geq 2$ is then equivalent with the property that the special fiber of the minimal snc model $\ensuremath{\mathscr{E}}$ of $E$ is a reduced cycle of $n$ rational curves (for simplicity we exclude the case $n=1$: then the minimal {\em normal crossings} model has a reduced special fiber which is a rational curve intersecting itself at one point).

In this case, $\ensuremath{\mathscr{E}}$ is also a minimal dlt model of $E$. Thus the essential skeleton $\mathrm{Sk}(E)$ equals the skeleton $\mathrm{Sk}(\ensuremath{\mathscr{E}})$ of $\ensuremath{\mathscr{E}}$, which we can identify with the dual graph of $\ensuremath{\mathscr{E}}_k$; this is a circle subdivided by $n$ vertices (see Figure \ref{fig:SYZ} for an example where $n=3$).

 To reconstruct the entire $K$-analytic space $E^{\mathrm{an}}$, we take the projective limit of the dual graphs $\mathrm{Sk}(\ensuremath{\mathscr{E}}')$ where $\ensuremath{\mathscr{E}}'\to \ensuremath{\mathscr{E}}$ is a composition of blow-ups of points in the special fiber. The effect of such a point blow-up can take two forms: either the center of the blow-up is an intersection point of components in the special fiber, in which case the corresponding edge of the skeleton is subdivided by adding a vertex in its interior; or the center lies on a unique component of the special fiber, in which case a new edge is attached to the vertex of the dual graph corresponding to this unique component. This is illustrated in Figure \ref{fig:SYZ}.

 \begin{figure*}[h!]
  \includegraphics[width=1\textwidth]{syz.pdf}
\caption{Effect of point blow-ups on the dual graphs}
\label{fig:SYZ}
\end{figure*}

 Repeatedly blowing up points and passing to the projective limit,
 what appears is a circle $\mathrm{Sk}(E)$ with infinitely many trees growing out of it, each tree splitting into infinitely many branches at infinitely many points (Figure \ref{fig:tate}). The non-archimedean SYZ fibration $\rho_{\ensuremath{\mathscr{E}}}$ contracts all of these trees onto the circle. For every volume form $\omega$ on $E$, the weight function $\mathrm{wt}_{\omega}$ is constant along the circle $\mathrm{Sk}(E)$ and strictly increasing as we move away from $\mathrm{Sk}(E)$ along one of the branches.

 \begin{figure*}[h!]
\begin{center}
  \includegraphics[width=0.3\textwidth]{tate.pdf}
  \end{center}
\caption{Analytification of a Tate elliptic curve; the essential skeleton is the circle in the middle}
\label{fig:tate}
\end{figure*}

\end{example}

redpajama_set_name: RedPajamaArXiv


text:
Not This Foolishness Again: Franklin Graham says Obama seen as 'Son of Islam'
Posted By:Unknown | At:5:25 PM 1 Comment so far
It is people like this that give Chrisitians a bad name. People like Franklin Graham and Rick Santorum are the personification of what Marvin Winans was talking about when he relied the message his muslim friend said to him about how he 'wished he'd met Christ before he met a Christian.' This innate need to paint President Obama as other is out of control.
(MSNBC) Rev. Franklin Graham, the son of evangelist Billy Graham and a prominent evangelical leader in his own right, waded into contentious waters Tuesday when asked for his views on the religious beliefs of President Obama and the GOP hopefuls.
Graham, the CEO and president of the Billy Graham Evangelistic Association, told a Morning Joe panel he couldn't say for certain that Obama is a Christian.
"You have to ask him. I cannot answer that question for anybody. All I know is I'm a sinner, and that God has forgiven me of my sins," Graham said. "You have to ask every person. He has said he's a Christian, so I just have to assume that he is."
But Graham also said he couldn't "categorically" say Obama wasn't a Muslim, in part, because Islam has gotten a "free pass" under Obama. Graham also said the Muslim world sees Obama as a "son of Islam," because the president's father and grandfather were Muslim.
According to Edina Lekovic, director of policy at the Muslim Public Affairs Council, being born in a Muslim family doesn't make one a Muslim. A person has to make an active choice to become a Muslim, Lekovic said.
Obama has said again and again that he is a Christian, both as a presidential candidate and as president.
"I'm a Christian by choice," Obama told a group of New Mexico voters last September, answering a question from a member of the audience. He said he has embraced his faith even though growing up, "my family didn't, frankly. They weren't folks who went to church every week."
In Chicago, Obama was a member of Trinity United Church of Christ for years, but he quit in May 2008 after videos of the Rev. Jeremiah Wright's racially-divisive sermons surfaced on the Web.
"Our relations with Trinity have been strained by the divisive statements of Reverend Wright, which sharply conflict with our own views," Obama and his wife Michelle wrote at the time.
The debate over the president's faith was brought up again on the campaign trail this Saturday, when Rick Santorum told a Tea Party crowd in Columbus, Ohio, that Obama's agenda is "not about you. It's not about you. It's not about your quality of life. It's not about your job. It's about some phony ideal, some phony theology. Oh, not a theology based on the Bible, a different theology, but no less a theology."
When pressed by reporters after Saturday's comments, the former Pennsylvania senator said he did not imply the president is not a Christian, but said the president was trumping religious freedoms.
Graham told the Morning Joe panel that he and Santorum share the same moral beliefs, and that he's confident Santorum is a fellow Christian.
"His values are so clear on moral issues, no question about it," he told the Morning Joe panel.
Graham spoke with a little less confidence about Gingrich's faith, and cast doubt on whether Romney's Mormonism is compatible with Christianity.
"I think Newt is a Christian, at least he told me he is," Graham said. He added that Romney's Mormon faith is not recognized as part of the Christian faith by most Christians, but he wouldn't give his own view.
Romney has stood by his faith, saying Mormonism's values are "as American as motherhood and apple pie."
"I believe in my Mormon faith," Romney said in a 2007 speech, "and I endeavor to live by it. My faith is the faith of my fathers. I'll be true to them and to my beliefs."
Categories: Islamophobia, Obama as Other, You Call This Christianity
Johanna February 21, 2012 at 10:10 PM
This is ridiculous! I would l halfway respect him if he held the other candidates to the same standard as he held the President. He cannot clearly state thef President is a Christian but can clearly stat Santorum and Gingrich are? My quote of the day is from Al Sharpton...the Christian right needs to meet the right Christians.

redpajama_set_name: RedPajamaCommonCrawl


text:
BIAFRA: ARBITRARY ARREST AND DETENTION OF BIAFRANS
BIAFRA: VIOLATIONS OF THE RIGHTS OF BIAFRANS TO FREEDOM OF EXPRESSION, PEACEFUL ASSEMBLY AND ASSOCIATION
The Important Role of the Church in Liberation Movements
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June 15 2020 | Radio Biafra A publication by Vanguard on Saturday June 13, titled "Twenty-One Years Of Democracy: Rule Of Law Suffering In Nigeria, SANs Lament" is something commendable, even though it came a little too late. Vanguard went on to interview some of the Senior Advocates of Nigeria who fearlessly spoke…
Sorry You Got Your Facts Wrong – Cleric Tells Edochie
June 15 2020 | Radio Biafra A cleric and advocate of Biafra restoration, Aiden Dillon, has said that veteran actor, Pete Edochie, got his facts wrong on Igbo land being landlocked. The cleric was reacting to Edochie's statement in which he expressed dissatisfaction over IPOB insulting the president of the proscribed Ohanaeze Ndigbo, John…
The Name "Nigeria" Represents Backwardness – Mazi Nnamdi Kanu
June 15 2020 | Radio Biafra In the light of the irredeemable situation Nigeria has found itself, the Supreme Leader of the Indigenous People of Biafra, Mazi Nnamdi Kanu, in his quest to emancipate the Biafran and black man in general, has dug deep into the root of Nigeria's problem. As the Igbo…
Africa Continent Will Struggle To Be Developed Without Biafra – Nnamdi Kanu Thunders
June 15 2020 | Radio Biafra The quest for Biafra seems to be fundamentally more visionary than most people can possibly grasp. This is more so with the Supreme Leader of the Indigenous People of Biafra, Mazi Nnamdi Kanu, asserting that the nation Biafra holds the key to Africa's development. Reflective of the…
June 12: The Shambolic Snail Democratic Resurgence In Nigeria Politics
June 15 2020 | Radio Biafra The dubious rebirth of snail democracy in Nigeria politics over military rule in 1999 may have cause Abraham Lincoln, founder of democracy to cry out loud in his grave for extending the modern system of government to Nigeria. Yes! Europe no doubt has their own share of blame…
Biafrans Are Going Home Soon – Mazi Nnamdi Kanu
June 14 2020 | Radio Biafra Following the recent incidents being reported in the Presidential Villa, leader of the Indigenous People of Biafra, IPOB, Nnamdi Kanu, has told Biafrans to get ready to leave Nigeria sooner than later. Nnamdi Kanu was reacting to the reports of gunshots at the Presidential Villa involving Nigeria's…
Aso Rock Shooting: Buhari's Health Goes Downhill, As Cabal Gang Up Against Aisha Buhari
June 13 2020 | Radio Biafra More details have emerged amid the squabble among family members and officials of the Muhammadu Buhari Presidency. According to new findings by SR, the current crisis in the villa is likely fuelled by an underlying issue within the seat of power, but covid-19 is being used as enough excuse…
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Bugis Hotpot & BBQ is the sister restaurant of the popular Jiu Gong Ge Hot Pot (九宫格火锅).
Bugis Hotpot & BBQ is currently offering a "4th person dines for free" promotion, so we decided to check it out.
Located at the ground floor of Bugis Cube behind the bus stop opposite Bugis Junction, Bugis BBQ & Hotpot Buffet should be pretty easy to find.
The spacious arrangement of the long tables allows 6-8 people to seat together, making it the perfect place for large group gatherings.
Most of the popular barbecue and hotpot ingredients can be found on these two shelves located at the back of the restaurant. The shelf on the left contains various vegetables, mushrooms and meats. The shelf on the right contains various balls, noodles and fresh fruits.
There are 6 soup bases to choose from: Spicy/Wild Mushroom/Tomato/Chicken Soup/The Pig Bone/Dongyan. You can also order additional meats, cold dishes, staple food such as rice and noodles and drinks from this menu.
You can make your own dipping sauce concoction from the 16 sauces available.
The Seafood Platter comes with 5 big prawns, 4 pieces of scallops with garlic sauce, 5 mussels and a flower crab.
The Lamb and Streaky Pork are not as thinly-sliced as I would have liked but at least they are fresh.
The Fat Cow beef slices are draped on a Barbie figurine like a dress, which is really cute.
$18.80++ at lunch and $22.90++ during dinner is pretty good price for both barbecue and hotpot. The ingredients are fresh and plentiful and there is space for large groups of people.


Brentwood is a neighborhood of approximately 3,582 residents located in the northeast section of the city. It is borders are formed by New York Avenue N.E., the rail yard to the east of North Carolina Street N.E., Rhode Island Avenue N.E. and Montana Avenue N.E.. In 2010, the average price for homes sold in this neighborhood was $207,176, with only about 33% of the homes owner occupied. Like a number of other neighborhood, this neighborhood faces a number of challenges.
The U. S. Senate Page School is the only school within Brentwood. Shopping is better than many neighborhoods as there is a Home Depot and a Giant Foods located here along with many other shopping opportunities. The Brentwood Recreation Center offers residents entertainment opportunities as well. Residents will also find a library here as well as the museum of the Architect of the Capitol. There are numerous Metro bus stops in the neighborhood and many will take advantage of the Rhode Island Metro station as well. There is only one place of worship in the neighborhood. Noise may issue as Brentwood features a major rail yard where both Amtrak and Metro trains regularly travel thru.
As far as crime is concerned data seems to indicate that the area is relatively safe and most crime seems to be focused on crimes against property, although there are crimes against people as well. As always you should check the recent statistics and make an informed decision before you move into any neighborhood.
"moderately priced and close to work"
Brentwood is an area mostly controlled by the government. Like much of DC, there are government buildings everywhere you turn, and a major rail yard lies within its borders as well. Housing is scarce, but moderately priced with houses running about 300K. Such a price is quite affordable for blue collar workers and government employees who may work there. The area is close to Gallaudet University which, because its students are Deaf and unaware of the loudness of their radios, can make the neighborhood a bit noisier at night. For this reason it is also necessary to ensure that one keeps an eye out for pedestrians who cannot hear your car coming. Naturally the rail yard doesn't help either. With that said, the neighborhood is safe relative to most other areas surrounding the Arboretum and Gallaudet University. Having the additional law enforcement presence due to the government facilities really helps in this area. People who like to walk their dogs after work may not find Brentwood to be an ideal location due to all the traffic. Of course, I cannot recommend the public schools in this area, so if you have the money it might be better to send your children to private schools.
Private ownership across several centuries has allowed the Brentwood neighborhood to develop in a way that is unique in the District of Columbia's history. Not surprisingly, much of Washington's acreage has long been controlled in one way or another by the federal government. However, until most of it was taken over under eminent domain laws during the 1940s, Brentwood was the private estate of the Winslow family. Their stately home, demolished to make way for new development, gave the area its elegant name.
While there are numerous single and multiple family homes in Brentwood, the area is more well known for its numerous businesses and industrial areas. In addition to housing the Washington Metro's extensive maintenance facility, it is also home to one of the largest mail handling and sorting facilities in the area. You may have heard of this facility a few years ago when it was the site of the anthrax poisoning event that took the lives of two postal employees.
Bordering Galludet University, Brentwood also features excellent public transportation and the nationally famous Union Market, one of the oldest continually operating farmer's markets in the country.


Introducing new tastes to baby is an exciting time for parents, but it often comes with lots of questions. Below is a general timetable of when to introduce foods from Cooking Light First Foods. By Carolyn Land Williams, M.Ed., R.D.
* We strongly recommend that you consult your child's pediatrician before modifying any diet or eating regimen, or introducing new categories of food.


Can we sustain our energy model?
NO. Do not let more laps, our current energy model is unsustainable. And so is our lifestyle linked to it. This will leave here for ecointeligentes reflection of our friends.
But what would a sustainable energy model? Can we say that would be one characterized by a pattern of production and consumption that made compatible economic, social and environmental development, meeting the energy needs of the present without compromising the ability of future generations to meet their own needs.
Energy security: must ensure continuity of supply to consumers at reasonable prices.
Competitiveness: should not endanger the competitiveness of the economy and its growth.
Environmental sustainability: production and consumption of energy should be a manageable impact on our environment.
The energy sector is responsible for 80% of emissions of greenhouse gases.
Our current energy model is characterized by a constant growth in energy consumption, based on finite fossil resources. According to forecasts by the International Energy Agency (IEA), world primary energy demand will grow at an annual rate of 1.5% until 2030, predominantly from fossil fuels, which represent 80% of the energy consumed on that date.
Regarding the economic sustainability, it is a remarkable economy based on consumption of finite fossil fuels compromised their competitiveness before the expected growth will experience rising prices of energy raw materials.
We must also consider the risks of supply disruption faced with situations of various kinds, such as those affecting several countries in the European Union, caused by the conflict between Russia and Ukraine.
On the side of the environmental criteria, the development of energy consumption baseline scenario implies an increase in emissions of greenhouse gases (GHGs) much higher than necessary to achieve the stage 450 ppm, which aims to limit the increase in the temperature to 2 ° C .
We know there is international consensus on the need to reduce global emissions by at least 50% by 2050 against 1990 levels to avoid an increase higher than the above temperature.
From the social point of view, the current energy model prevents access to advanced forms of energy , such as the electricity mainly to 2,000 million people, hindering human development and the potential for future economic growth in this part of the Site.
Concerning the offer: it consists of the implementation of technologies that enable the decarbonisation of the energy mix, the still essential promotion of renewable energies and CO2 capture and storage (CCS).
In all analyzes, the energy efficiency is considered the primary measure to meet the challenges of the energy model from the point of view of demand, estimating the IEA that about 60% reduction of GHG emissions needed by 2020 to reach the 450 ppm scenario comes from the efficiency measures in end uses.
Along with the energy efficiency, the second largest block measures and noted in offering solutions, is promoting renewable energy, which would be about a 20% decrease emissions by 2020, largely in line with the EU 20/20/20 targets.


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Save Cover Preview for Thomas Bartholin. The Anatomy House in Copenhagen
Distributed for Museum Tusculanum Press
Thomas Bartholin. The Anatomy House in Copenhagen
Edited by Niels W. Bruun
Translated by Peter Fisher and with an Introduction by Morten Fink-Jensen
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The first anatomical theater was established at the University of Copenhagen in 1644, and it was there that Thomas Bartholin first demonstrated the existence of the thoracic duct, and, later, the lymphatic vessels, an achievement that brought him immediate fame.
In 1662, Bartholin published A Short Description of the Anatomy House in Copenhagen, which meticulously describes the layout of the Anatomy House alongside the first eighteen years of its history. This book presents Bartholin's work for the first time in English, enabling a broader audience to draw on the detailed accounts of Bartholin and the other doctors who used the Anatomy House. Notes and an introduction, as well as numerous illustrations, help to make this a valuable resource for historians of medicine.
Read Moreabout Thomas Bartholin. The Anatomy House in Copenhagen Read Lessabout Thomas Bartholin. The Anatomy House in Copenhagen
225 pages | 44 color plates, 55 halftones | 6 3/4 x 9 1/2 | © 2015
View all books from Museum Tusculanum Press
Thomas Bartholin and the Anatomy House at the University of Copenhagen
An Account of the Text
Domus Anatomica Hafniensis
The Anatomy House in Copehagen
Index of Persons
Literature cited in the Commentary
The State of the Anatomy House 12 April 1676
Inventarium po Anatomiæ-Huset. 1676
Inventory of the Anatomy House 1676
Index of Illustrations


To improve the access efficiency of multidimensional spatial database, this study proposes a new index structure named Space Hypercube Grid Tree (SHG-Tree). By avoiding the problems of node split and recombination, SHG-Tree can efficiently support the common operations over spatial database containing objects with dynamic region. The main contributions of this paper include: (1) Proposes SHG-Tree of n-dimensional space with a hierarchical tree structure. It reflects the region overlapping relationship of hypercube grid units with different granularity. (2) Proposes the linearization methods to present the bounding rectangle of object as a union of variant granularity hypercube grids. (3) Gives operations of SHG-Tree. Experiments result shows the size of SHG-Tree is small enough to remain in main memory even to very large spatial database by applying proper linearization strategy and the queries on SHG-Tree are less than ten milliseconds to ensure the real-time of query.


FFT are very pleased to have been asked to speak this week at a mini conference hosted in Durham town hall about one of our most passionate topics, supporting local food.
As everyone will know sustainable locally sourced produce makes up some of our core values and ethos, and we are very honoured to share our experiences with Durham Local Authority, Durham NHS and Durham University. This is closely linked to our recent visit to Sheffield Hallam University who hosted the 'Filling the Holiday Gap' conference, discussing how we can help under-nourished children during the school holidays when access to food for is more challenging. FFT have been running pilot 'sports and diet' projects during these summer months but they do need to be 'sustainable' - not solely reliant upon volunteer time - schools also need to be open during the summer - perhaps the logical answer is to look a similar projects involving an 'extended school' philosophy - with schools accessible over 52 weeks as opposed to the current 39 week model applicable in so many areas.
For more information regarding this please do not hesitate to get in touch with either Mike Carden or Tom Lambeth at FFT.


Insurance companies often try to reduce the amount they will pay for an injury claim – or reject the claim entirely – by using the same dubious and predictable tactics that some insurance companies been using for years.
If you're like most people – meaning you have little if any experience dealing directly with insurance companies – you might become frustrated and decide to just forget the whole business of seeking compensation, even if you've been seriously injured.
Insurance companies count on a certain percentage of people walking away from legitimate claims out of exasperation. Don't be one of those people.
If you are having trouble with an insurance company that is refusing to honor an injury claim, a Pasadena wrongful death attorney can help. Below is a brief discussion of the most common ploys used by insurance companies to avoid paying injury victims who have legitimate injury claims.
HOW DO INSURANCE COMPANIES TRY TO AVOID PAYING CLAIMS?
1. By denying fault or liability: The immediate response of some insurance companies to any injury claim is a denial that any of their policyholders could be at fault. If you are injured, and the insurance company tells you that no coverage is available, have the insurance adjuster explain why – in writing. A letter of explanation should refer to the exact provisions that limit the policy's coverage.
When an adjuster informs you that there is no coverage for your claim "according to the company's attorneys," don't think you are defeated. You might even be sent a letter from a lawyer working for the insurance company. Don't let yourself be bullied.
One lawyer's opinion is only one lawyer's opinion. If the insurance company refuses to offer an acceptable settlement and your claim goes to court, the court alone will decide.
Alternatively, an insurance company might tell you to try collecting against a different policy – and from a different company – if several parties were involved in the accident.
If you are injured by negligence in a traffic collision or in any other kind of accident, it's usually best to speak first with a qualified personal injury attorney who can advise you regarding your legal rights – and avoid speaking to the insurance company entirely.
2. By using delaying tactics: An insurance company should process an injury claim within a reasonable length of time. Stalling is a typical tactic because if the company stalls long enough, some people will give up and the company pays out nothing.
The company might claim that an investigation is "ongoing" or "still pending." Some companies will even tell claimants that they are at fault for a delay because they missed a deadline, failed to send a notification, or some other technicality.
3. By alleging that your injury was pre-existing: An insurance company will sometimes make the charge that you weren't really injured as you claim, but instead, you were already injured, and you saw the accident as a chance to take advantage of your pre-existing condition.
This is why you must seek medical attention immediately after an accident, whether or not you think or feel that you've been injured. If you need to file an injury claim later, you'll need to have complete medical documentation that confirms your claim.
4. By minimizing the payment of a claim: When an insurance company cannot entirely deny a legitimate claim, it may try to pay as little as possible. The company may try to reduce what it should pay by claiming that the injury victim was – himself or herself – at least partly at fault for the accident and injury.
If you had vacation days or sick days due from your employer, an insurance company might try to avoid paying for the wages you lost while you were injured by saying those wages were already covered. In such cases, if you are the injury victim making the injury claim, let a skilled Pasadena personal injury attorney advise and represent you.
When a consumer in California purchases an automobile insurance policy, the insurance company becomes legally obligated to provide the coverage spelled out in the policy and to settle legitimate injury claims against the policy promptly and professionally. An auto insurance company that fails to meet their legal obligation may be operating in "bad faith" under California law.
A California injury victim whose bad faith lawsuit prevails against an insurance company may also in some cases be awarded part of his or her attorney fees along with the damages for the original, underlying personal injury claim.
How can an injury victim know if an insurance company's actions constitute bad faith? A personal injury attorney can probably tell you upon examining the details of your injury claim.
WHAT IF THERE REALLY IS NO COVERAGE?
In some cases, once you've read the provisions of the insurance policy and you've met with an insurance adjuster and a personal injury attorney, you might find that the company is right and the insurance policy, in fact, provides no coverage. In such cases, your injury attorney may be able to suggest other sources of compensation.
If a defective vehicle or vehicle part or a badly-maintained roadway played any part in the accident that injured you, you may be able to file a claim against the vehicle manufacturer or the entity responsible for road maintenance. If a crash involves a large commercial truck, a number of parties could share liability.
Candidly, if you have been seriously injured by someone else's negligence in a southern California traffic accident – or any other kind of accident – dealing directly and personally with the insurance company is probably not something you should be doing. "Going it alone" might be fine if your only injuries were scratches, but when your health and future may be at stake, get legal help right away.
If a negligent Southern California driver has seriously injured you, speak to a personal injury attorney before you sign any document, accept any settlement, or even make a statement to an insurance company.
When you are recovering from severe injuries, an experienced personal injury lawyer can handle the legal work on your behalf while you concentrate on something even more important – getting your health back.
What Is Insurance Bad Faith?
How Are Car Accident Settlement Amounts Determined?


BOCA RATON, Fla., March 20, 2019 /PRNewswire/ -- Campus Management Corp. announced today that Lipscomb University selected the CampusNexus platform to assist in building a data-powered campus. The faith-based liberal arts institution, located in Nashville, Tenn., will be using the cloud-based platform, which includes CampusNexus Student, CampusNexus Engage, and CampusNexus Finance, HR & Payroll to aid in the transformation of its business processes and academic programs.
"Initially we looked to Campus Management for a flexible solution to support our online programs, but once we recognized the platform's full suite of capabilities, we decided to utilize it across the entire university," said Mike Green, Senior Vice President and CIO, Lipscomb University. "It is critical to work with dynamic technology providers as higher education business models, learning strategies, and student needs continue to evolve."
uilt on Microsoft technology, Campus Management clients benefit from Microsoft Dynamics 365 and the trusted Microsoft Azure cloud platform. The strong Microsoft foundation enables Campus Management to focus solely on higher education innovations, academic delivery, and operational efficiencies.
"We are excited to work closely with Lipscomb University as they continue to envision and innovate methods to advance superior learning outcomes for students," said Cynthia Haynie, Chief Technology Officer, Campus Management. "Their growing student body requires more schedule and term flexibility, which our solutions have been supporting for years. Overall, the institution will realize greater student engagement and operational efficiency through its inclusive enterprise-wide strategy and supporting technology."
For more detail about Campus Management, visit campusmanagement.com.


FRIDAY, 9/13 -> It's been a big year for The Solution's own Lazy Eyez; traveling the country DJing with the Reminders, winning "best hip hop DJ in Denver" in the Westword Music Showcase & spinning all sorts of other dope events around CO. We're super proud of his success and on that note, we're going big to celebrate his birthday with special guests DJ Discord and DJ Deft joining DJ Low Key and Lazy on the turntables at the Meadowlark!!!
That's not all though, we're still beyond geeked to be featuring food from head chef Josh Bitz & the brand new Meadowlark Kitchen (official opening coming later this year) every Friday night. After a couple of meals by the crew, we've been blown away by the top quality dishes, made with farmer's market produce and top quality ingredients. In this week's Westword, Josh was mentioned as "the most underrated chef in Denver", and after sampling a few of his test menus, we have to agree. 21+ / 9pm / No cover until 11pm / Only $5 after.


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MANDY AMANO
REELS & CLIPS
TAPED AUDITIONS
Clips from THE SUN DEVIL & THE PRINCESS
Clip from MY GIRLFRIEND'S BACK
SUN DEVIL Film
Print Portfolio
Texas-raised Mandy Amano became "the face that launched 1,000 websites" after her Pepsi commercial premiered during the Super Bowl. This led to appearances in Maxim, countless online magazines, blogs, and comic books - most notably Shi and RoninHood.
A genre girl at heart, some of her favorite roles have been starring as "O", the ultimate modern femme fatale, in the neo-noir thriller Proxy Kill; the fierce and fiery fighter pilot "Oujo" in Squadron; and "Thalia", the little but lethal Black Ops vampire, in Blood Soldiers.
She landed a dream role when Steven Ayromlooi cast her as "The Princess" opposite the mighty Evan Parke in the film festival/comic con phenom The Sun Devil and The Princess, which racked up a show-stopping 20 awards - including 2 Best Actress wins for her portrayal of the strong, yet ethereal princess.
Described as "Audrey Hepburn... who could kick your ass" by Shi creator Billy Tucci, Amano is currently shooting a Western feature (title TBD) as "Abigail," the pampered courtesan who also happens to be a deadly gunslinger.
Other tidbits:
Mandy's other passions include All-American Mopar Muscle, French Bulldogs, & kickass boots.
The cause closest to her heart is taking care of veterans, especially the work done by The Heroes Project, Hounds & Heroes, Homes for Our Troops, Warrior Foundation Freedom Station, Rock to Recovery, Final Salute Inc., VETPAW, VME (Veterans in Media & Entertainment), Stack Up Org, and Heroes & Horses... to name a few.
And her dream is to be Princess Leia when she grows up.


Shennecossett Yacht Club's First Annual Lighthouse Regatta on August 2 will honor the memory of Bruce Lockwood of Groton Long Point, CT, a distinguished racer on Long Island Sound and many other venues. The Bruce Lockwood Cup, a perpetual trophy, will be presented to the boat in the spinnaker fleet with the best corrected time.
Bruce was an avid and enthusiastic competitor, involved in racing his entire life. He grew up on Long Island Sound and belonged to many sailing organizations including Shennecossett Yacht Club (serving as Commodore 1986-1988), Ram Island Yacht Club, Baldwin Yacht Club, Off Soundings Club, Storm Trysail Club, Woodbridge Association, and the Mystic River Mudhead Sailing Association. He won Key West Race Week, three Block Island Race Weeks, the Mumm 30 Northeast Championship, was the top boat in the Mudhead Fleet eight times, and was victorious in too many Off Soundings series to count; the last one in 2012, just four weeks before he died at age 90. In addition, he raced his boat to Bermuda and even participated in the Transatlantic Challenge in 2005 at the age of 83.
Many sailors on Long Island Sound have been inspired by Bruce as they witnessed his enthusiasm and skill on a series of sailboats including his J/36 Arrow, J/29 Tomahawk, and Farr 30 One More Time. A new Arrow, a 9-foot Dyer Dhow donated by family and friends, has joined the fleet at Mystic Seaport in Bruce's memory.
All racers are invited to participate in the Lighthouse Regatta to honor Bruce by the awarding of the trophy in his name. Cruising sailors may sign up for a non-spinnaker class with PHRF numbers assigned by the race committee. The course in Fishers Island Sound will be within view of many historic lighthouses, a unique concentration of nine in a small geographic area. Proceeds from this Eastern Connecticut Sailing Association event will be donated to the New London Maritime Society's Lighthouse fund. Visit syc-ct.com for more information.


On behalf of all Australian neuroendocrine cancer sufferers we applaud the efforts of Alexander Masters and the wonderful initiative of iCancer. This 'viral' campaign has raised awareness of the plight of our patients and I hope will stimulate interest in research not only into this project but myriad others, that may one day translate into better therapies or cure for this maligned condition.
PS Everyone needs a friend like Alexander.


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Assignment 2: New Hires
Due Week 5 and worth 125 points
Imagine you are the HR manager at a company. You need to hire three (3) new people. One employee will work at the front desk as a customer service assistant, and the other two (2) will work on the production line. Give at least five (5) places you would recruit for each position and explain why each place would be a viable option.
Write a one (1) page paper in which you:
Select at least five (5) places you would recruit for a customer service assistant and at least five (5) places you would recruit for production line workers.
Explain why each place you selected would be a viable option for recruiting employees.
Format your assignment according to the following formatting requirements:
Typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides.
Include a cover page containing the title of the assignment, your name, your professor's name, the course title, and the date. The cover page is not included in the required page length.
Analyze effective recruiting and selection strategies that can be used to meet organizational requirements.
Use technology and information resources to research issues in human resource management.
Write clearly and concisely about human resource management using proper writing mechanics.
Grading for this assignment will be based on answer quality, logic / organization of the paper, and language and writing skills, using the following rubric.
Unacceptable
Below 60% F
Meets Minimum Expectations
60-69% D
70-79% C
80-89% B
90-100% A
1. Select at least five (5) places you would recruit for a customer service assistant and at least five (5) places you would recruit for production line workers.
Weight: 45%
Did not submit or incompletely selected at least five (5) places you would recruit for a customer service assistant and at least five (5) places you would recruit for production line workers.
Insufficiently selected at least five (5) places you would recruit for a customer service assistant and at least five (5) places you would recruit for production line workers.
Partially selected at least five (5) places you would recruit for a customer service assistant and at least five (5) places you would recruit for production line workers.
Satisfactorily selected at least five (5) places you would recruit for a customer service assistant and at least five (5) places you would recruit for production line workers.
Thoroughly selected at least five (5) places you would recruit for a customer service assistant and at least five (5) places you would recruit for production line workers.
2. Explain why each place you selected would be a viable option for recruiting employees.
Did not submit or incompletely explained why each place you selected would be a viable option for recruiting employees.
Insufficiently explained why each place you selected would be a viable option for recruiting employees.
Partially explained why each place you selected would be a viable option for recruiting employees.
Satisfactorily explained why each place you selected would be a viable option for recruiting employees.
Thoroughly explained why each place you selected would be a viable option for recruiting employees.
3. Clarity, writing mechanics, and formatting requirements
More than 8 errors present
7-8 errors present
https://healthtermpapers.com/wp-content/uploads/2020/05/logoHTP-2-300x75.png 0 0 admin https://healthtermpapers.com/wp-content/uploads/2020/05/logoHTP-2-300x75.png admin2020-11-29 05:39:212020-11-29 05:39:21Assignment 2: New Hires
Continuum of individual response to change( organizational change) Christian Ethics Project


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People of Devon Project
People of Devon
These are the people of Devon and these are their stories…
"How would you like to read a life story about your neighbour or perhaps that lady on the bus, or your postie …The People of Devon Project; take a look at it!"
Pippa Quelch, BBC Radio Devon
"Ride With Me"
"Farmer's sons are expected to come home to work—it's a lot of pressure. I've always told my boys that they didn't need to come home; I was expected to, but when it comes to their lives, they know best. I always thought one of my sons would take over the farm though, but they decided to take a different path." Continue reading "Ride With Me"
30th Mar 2021 7th May 2021
"Two years ago, I received a small package from my mother. It was a Christmas present—a little recipe book that she had made herself. On the cover was a photograph of a field that my great grandfather used to work—the same field that our family home is built on today. The book was purely composed … Continue reading "Food for the Soul"
"You are my Rock"
Anna Grayson has had several careers in her life – scientist, broadcaster and now artist and photographer. But behind all her success is a remarkable 46-year-long love story… "I was at the University of St. Andrews studying Geology, but there had been a virus going around so I'd fallen Ill. I was so unwell in … Continue reading "You are my Rock"
1st Jan 2021 1st Jan 2021
"Water Mountains, Love and Pain"
"It was a huge storm—we haven't had one like it since I don't think. A few friends and I had rented a house in Ireland so we could chase big swells. We'd been there for three months at this point, but the winter had been terrible and most of them had decided to travel back … Continue reading "Water Mountains, Love and Pain"
29th Oct 2020 30th Oct 2020
"No Work…In Progress"
"There was an eerie atmosphere in the office. A lot of people were being made redundant. That day, one of my colleagues from the engineering department got called into a 'special' meeting, so we anxiously waited for him to return. He walked back through the office door and before he said anything, I could tell that … Continue reading "No Work…In Progress"
28th Sep 2020 11th Oct 2020
"A Flash of Life"
"I was born with a brain condition, but I didn't know I had it. In 2016, as I was photographing a wedding, I got a devastating headache, collapsed, and my life changed forever. "As a farmer's daughter, originally, I wanted to rear chickens. When I was eight years old, I started my own egg business. … Continue reading "A Flash of Life"
"The Final Act"
"I hadn't done any acting since A level drama, but I'd heard about the open auditions for the play Quills through a friend. I arrived out of breath at the Globe in Topsham – having first gone to the Globe in Exeter I had to race across town to get where I was supposed to be. To my surprise I got the part of the tortured priest, the Abbe de Coulmier. Continue reading "The Final Act"
14th Aug 2020 14th Aug 2020
"Rock and Netty"
"During a weekend in July, the majority of my friends went to the Phoenix Festival, but the tickets were too expensive, so I decided to go to a party on Appledore cliffs. Like every self-respecting 19-year-old I wanted to be a rock star. I'd started working on a sound crew—I was carrying gear and twiddling … Continue reading "Rock and Netty"
27th Jul 2020 27th Jul 2020
"I sat on my bed in a cramped flat in the centre of London and googled 'Shanghai'. I browsed through pictures of skyscrapers and winding tree-lined streets with washing blowing in the sunshine. The juxtaposition amazed me, and I didn't know which version of the city was the real one. Now I know… it's both. … Continue reading "The Big Leap"
"My Father and I"
"On our 21st birthday my twin sister organised a party. I had a few drinks with some friends, so I ended up arriving late. The party came to a pause and my father stood up to give a speech. He said, 'This is my beautiful daughter! I'm so proud of her…' He then turned and … Continue reading "My Father and I"
30th Jun 2020 30th Jun 2020
"Inspired from Below: Soil, Children and Other Earthy Wonders"
"Most of the ingredients were the freshest people had experienced, some were pulled out of the ground only a few minutes before. uring this community event, we gave groups of people a list of fresh ingredients and set them the task of picking… Continue reading "Inspired from Below: Soil, Children and Other Earthy Wonders"
"Love and a Farm"
"We had a long-distance relationship for four and a half years. My family and I used to stay in a holiday cottage on a farm in Devon. Then one summer when I was 16 the farmer's son asked for my number. When we later came down to celebrate my father's 50th birthday, James and I … Continue reading "Love and a Farm"
"Dancing Back to Me"
"I didn't know that missing home could be so painful. I was eighteen when I moved to London. I had a boyfriend and friends there, but my first year at the London Contemporary Dance School was a struggle. I'd been dancing my whole life. It was what I dreamt of doing and trained so hard … Continue reading "Dancing Back to Me"
5th Jun 2020 11th Jun 2020
"In Search of a Way"
"I was in Germany and I'd just finished my PhD. Having spent a year in Costa Rica, I'd become a specialist in coral reef ecology. Although there isn't much money in marine biology research, and very few advertised positions, it was what I really wanted to do. "I'd just met Gernot, so I didn't feel … Continue reading "In Search of a Way"
1st Jun 2020 1st Jun 2020
"Dark Clouds of Music"
"At 16 all I wanted to do was get on a bus and leave. I was badly bullied. I wanted to quit music school. I felt helpless. I wanted to be famous. Any famous; just so they'd leave me alone." Kat, a prodigious violinist in a popular indie-folk band, a professional internet harmful content manager, … Continue reading "Dark Clouds of Music"
26th May 2020 17th Jan 2021
We travelled that same night. Nothing with us. We'd been living in a couple of different places, but at this last one the local Mafia wanted the tiny room we had. It was a kitchen, a toilet and a bedroom in one—you couldn't move. They took me outside. 'We'll just kill you. No one cares,' they kept telling me as they beat me. Then they gave me two options: stay and die or leave. I grabbed my wife and daughter Continue reading "Safe Haven"
20th May 2020 21st Mar 2021
"An Unexpected Arrival"
"The lockdown had started. Lou was very close. During the sweep the midwife had said that the baby would probably come during the weekend. We went home and had a pretty normal evening. I did a bit of gardening, had tea and went to bed. I'm not going to lie, I was a bit worried … Continue reading "An Unexpected Arrival"
"Changing the Narrative"
"Everyone writes and everyone reads out loud. It works very democratically. I'd done work in prison's before, when I used to live in London," says Pippa as if this were as commonplace as a walk on Dartmoor. "No, I've never been scared. I´m a semi-retired teacher and you find the same dynamics in any sort … Continue reading "Changing the Narrative"
5th May 2020 21st Mar 2021
"The Precipice of Change"
"The phone rang at around 1AM. Apparently the hospital had tried me once —they only ring you twice and then move on to the next person. I'd been on the transplant list for three years at that point. It was the night before I was supposed to start dialysis. I had all the tubes in and everything. Long story short, I contracted a rare disease on a farm when I was 14. By the time I was 22 Continue reading "The Precipice of Change"
"The Weight of Dreams"
"I was doing a lot of drugs. A lot of coke. My girlfriend of five years had just broken up with me. I had a great job managing one of the best hair salons in London, but I was lost. Some of my best friends were Muslim, and they were just so kind and understanding … Continue reading "The Weight of Dreams"
5th May 2020 20th Mar 2021
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P.O.D.P on social media.
If you haven't read our last story the link is in our bio!
Series Two of our BBC Radio Devon Series will air on the 5th of April! Be sure to listen in!😊👍👂
Our new true story about the Dartmoor cowboy is now online! A heartwarming tale of true love, family and grit...it has a bit of everything!
Phil Heard is an award-winning Dartmoor farmer that tends to his cattle on horseback. In our new story "Ride with Me," we follow Phil as he finds love, rebuilds the family business and struggles with the prospect of his sons not taking over the farm.
Our new true story is now online! It is a heartwarming tale of family, beliefs, loss and faith in humanity.
"Food for the Soul," tells Laura's story. It explores the deep roots, joys, and losses of a family in an Italian village, and how a connection to Devon is born from the desire to provide healthy food to everyone.
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Big Air is Bringing Paulding County its First Trampoline Park!
HIRAM, Ga. - Jan. 15, 2019 // PRNewswire // - Big Air Trampoline Park® is excited to announce a new franchise location to be located in Hiram, Georgia opening in early 2019. The 40,000 square foot indoor park, located just 20 minutes west of Atlanta, will be Big Air's first Georgia location.
"Big Air has been absolutely exploding, with new Parks all around the United States," said president Kevin Odekirk. "So much of the country is already familiar with our unparalleled brand of active family fun. We are just thrilled to introduce ourselves to the great people of Atlanta."
Big Air Hiram will be nearly 40,000 square feet in size and will include 12 activities for guests of all ages including trampoline dodgeball, trampoline volleyball, climbing walls, slam dunk courts, ninja-warrior courses and more. The park also plans to add virtual reality gaming technology in the future.
Big Air's award-winning birthday party program will be featured at Big Air Hiram. With multiple event spaces, the park will also host team parties, corporate team-building events and more.
Other in-park experiences will include Cosmic Nights® on Friday and Saturday nights, and Toddler Time® on select weekday mornings. Parents will be able to relax in Big Air's Parents' Lounge, with access to comfortable seating, free WiFi and a comprehensive food & drink menu.
"We are so excited to be part of Hiram and Paulding County," say Ben and Sue Lackey, owners of Big Air Hiram. "When looking at potential franchises, we fell in love with the fun family atmosphere of the Big Air culture, so we wanted to bring a park to the community to help create everlasting memories."
Big Air Hiram is anticipating an April 2019 opening and will be located at 4471 Jimmy Lee Smith Parkway, Hiram, GA, 30141. For up-to-date news, follow @bigairhiram on Facebook and Instagram.
Southern California-based Big Air Trampoline Park® is making leaps and bounds in the family entertainment industry. Founded in 2012, Big Air is led by an executive team with over 30 years of experience in the entertainment industry. With trademarked attractions like Battlebeam®, events like Toddler Time®, and an award-winning birthday party program, Big Air offers a brand experience that is second-to-none. The company is now expanding across the United States through franchising, with a program that offers exceptional support in location analysis, construction, park operations, marketing strategy, and more.
Southern, California-based Big Air Trampoline Park is making leaps and bounds in the family entertainment industry.


As search numbers grow online, Search Engine Optimization (SEO) is becoming more important than ever. In March 2013, 20.4 billion searches were conducted from desktop computers alone! Mobile search, too, is growing important: 113.1 million people use mobile to search online. As search continues to grow and evolve, what will be the most important marketing goals to focus on?
Salesforce asked 9 SEO experts, including Moz CEO and co-founder Rand Fishkin, Bing Senior Product Manager Duane Forrester, and Distilled Founder Will Critchlow, about how the relationship between SEO and marketing will develop over the next few years.
When asked about the future of SEO, Fishkin suggested that all forms of inbound marketing, including organic search and social media, will become increasingly interconnected. So much so that it will be impossible for marketers to ignore any of these channels and still do a good job. "The future will bring less and less of an ability to do great SEO without embracing these other channels and perspectives," he said.
Debra Mastaler, President of Alliance-Link, said that Google will continue to give precedence to authors in search, finding better ways to use social data to determine the best results for each individual. Jimmy Yu, CEO of BrightEdge, says that cellphones and tablets will play larger roles in search.
When asked to discuss the relationship between social media and SEO, the general consensus among experts seems to be that they go hand in hand. "It is virtually the same thing now," Chris Bennet, Founder and CEO of 97th Floor, said. "As social has become a huge part of not only Google's algorithm but also how to spread content and drive traffic." If your content is designed to do well on social media, you should expect it to do well in organic search, too, suggests Critchlow.
On the topic of how content marketing has changed SEO, experts suggested that it's more important now than ever to create content people need rather than creating content just for content's sake. Now marketers will have to determine which audiences are most important to target and what exactly those audiences want to see online. "This is the era of less is more SEO," said Adam Audette, Chief Knowledge Officer at RKG.