\contentsline {mtchap}{\numberline {2}Combinatorics}{565}
\contentsline {mtsec}{\numberline {2.1}Final Elements in The Weyl Group}{565}
\contentsline {mtsec}{\numberline {2.2}Final Types and Young Diagrams}{567}
\contentsline {mtsec}{\numberline {2.3}Canonical Types}{572}
\contentsline {mtsec}{\numberline {2.4}Admissible Elements}{575}
\contentsline {mtsec}{\numberline {2.5}Shuffles}{576}
\contentsline {mtchap}{\numberline {3}The Flag Space}{578}
\contentsline {mtsec}{\numberline {3.1}The Flag space of the Hodge bundle}{578}
\contentsline {mtsec}{\numberline {3.2}The canonical flag of an abelian variety}{580}
\contentsline {mtchap}{\numberline {4}Strata on the Flag Space}{582}
\contentsline {mtsec}{\numberline {4.1}The Stratification}{582}
\contentsline {mtsec}{\numberline {4.2}Some Important Strata}{583}
\contentsline {mtsec}{\numberline {4.3}Shuffling flags}{588}
\contentsline {mtsec}{\numberline {4.4}The E-O strata on ${{\@mathcal A}}_g \otimes {\@mathbb F}_p$}{591}
\contentsline {mtchap}{\numberline {5}Extension to the boundary}{591}
\contentsline {mtchap}{\numberline {6}Existence of boundary components}{593}
\contentsline {mtchap}{\numberline {7}Superspecial fibres}{594}
\contentsline {mtchap}{\numberline {8}Local structure of strata}{598}
\contentsline {mtsec}{\numberline {8.1}Stratified Spaces}{598}
\contentsline {mtsec}{\numberline {8.2}Height $1$-Maps}{598}
\contentsline {mtchap}{\numberline {9}Punctual flag spaces}{601}
\contentsline {mtchap}{\numberline {10}Pieri formulas}{606}
\contentsline {mtchap}{\numberline {11}Irreducibility properties}{611}
\contentsline {mtchap}{\numberline {12}The Cycle Classes}{614}
\contentsline {mtsec}{\numberline {12.1}Fulton's formulas}{615}
\contentsline {mtsec}{\numberline {12.2}The $p$-rank strata}{617}
\contentsline {mtsec}{\numberline {12.3}The $a$-number Strata}{619}
\contentsline {mtsec}{\numberline {12.4}Positivity of tautological classes}{621}
\contentsline {mtchap}{\numberline {13}Tautological rings}{621}
\contentsline {mtchap}{\numberline {14}Comparison with ${\@mathcal S}(g,p)$}{625}
\contentsline {mtchap}{\numberline {15}Appendix $g=3,4$}{626}
\contentsline {mtsec}{\numberline {15.1}Admissible Strata for $g=3$}{626}
\contentsline {mtsec}{\numberline {15.2}E-O Cycle Classes for $g=3$}{626}
\contentsline {mtsec}{\numberline {15.3}E-O Cycle Classes for $g=4$}{627}
\contentsline {mtchap}{References}{627}