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In this lecture series, we will address the related issues on Sasakian geometry including :
     (I) Lecture One : Geometrization Problems in Sasakian 5-Manifolds
    Sasakian geometry is very rich as the odd-dimensional analogous of Kaehler geometry. For instance, a Sasaki-Einstein 5-manifold is to say that its Kaehler cone is a Calabi-Yau 3-fold. It provides interesting examples of the AdS/CFT correspondence. On the other hand, the class of simply connected, closed, oriented, smooth 5-manifolds is classifiable under diffeomorphism due to Smale-Barden. It is our goal to address the issue on the geometrization and uniformization problem of Sasakian manifolds in this first lecture series.
     (II) Lecture Two : Foliation Minimal Model Program on Sasakian Five-Manifolds
    In 1982, R. Hamilton introduced the Ricci flow and then by studying the singularity models of Ricci flow, G. Perelman completely solved Thurston geometrization conjecture and Poincare conjecture for a closed 3-manifold in 2002 and 2003. On the other hand, Mori minimal model program in birational geometry can be viewed as the complex analogue of Thurston’s geometrization conjecture. In 1985, H.-D. Cao introduced the Kaehler-Ricci flow and then recaptured the Calabi-Yau Conjecture. Recently, there is a conjecture picture by Song-Tian that the Kaehler-Ricci flow should carry out the minimal model program with scaling on projective varieties. Song-Weinkove established the above conjecture on a projective algebraic surface.
     The Sasaki-Ricci flow is introduced by Smoczyk-Wang-Zhang to study the existence of Sasaki-Einstein metrics on Sasakian manifolds. It can be viewed as a Sasaki analogue of Cao’s result for the Kaehler-Ricci flow. It is natural to conjecture that the Sasaki-Ricci flow will carry out the foliation minimal model program with scaling on quasi-regular Sasakian 5-manifolds as well. Indeed, Chang-Lin-Wu proved the Sasaki analogue of minimal model program on closed quasi-regular Sasakian 5-manifolds of foliation cyclic quotient singularities.
     (III) Lecture Three : Topology and Geometry of Legendrian Submanifolds of Sasakian Manifolds
    Legendrian submanifolds of contact manifolds and Lagrangian submanifolds of symplectic manifolds are related by symplectization. Furthermore, there is a 1-1 correspondence between minimal Lagrangian cones in a complex Euclidean (n+1)-space and minimal Legendrian submanifolds in (2n+1)-sphere with the canonical contact metric structure. In the SYZ Conjecture, in order to deal with the difficulty which states that most of the special Lagrangian tori fibration have singularities, one can model them locally as special Lagrangian cones in a complex Euclidean 3-space. Such a cone can be characterized by its link of 5-sphere which is a minimal Legendrian surface.
    In this lecture, we will address the related issue such as isotopic Legendrian submanifolds, the Smale conjecture  and existence of minimal Legendrian submanifolds via the Legendrian mean curvature flow.
     (IV) Lecture Four : Yau Uniformization Conjecture on Complete Noncompact Sasakian Manifolds
    The CR analogue of Yau uniformization conjecture states that any complete noncompact Sasakian manifold of positive CR holomorphic bisectional curvature is CR biholomorphic to the standard Heisenberg group. The first key step is to show that there exists a nonconstant CR holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative CR holomorphic bisectional curvature with the CR maximal volume growth property which is due to Chang-Han-Li-Lin. In this lecture, among the others, we will address this issue via the the Sasaki-Ricci flow.