Let$f$and$g$be nonlinear entire functions. The relations between the dynamics of$f⊗g$and$g⊗f$are discussed. Denote byℐ (·) and$F$(·) the Julia and Fatou sets. It is proved that if$z$∈$C$, then$z$∈ℐ8464 ($f⊗g$) if and only if$g(z)$∈ℐ8464 ($g⊗f$); if$U$is a component of$F$($f$○$g$) and$V$is the component of$F$($g$○$g$) that contains$g(U)$, then$U$is wandering if and only if$V$is wandering; if$U$is periodic, then so is$V$and moreover,$V$is of the same type according to the classification of periodic components as$U$. These results are used to show that certain new classes of entire functions do not have wandering domains.

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We survey our recent new results on the geometry of Teichmuller and moduli spaces of Riemann surfaces and Calabi-Yau manifolds.

We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|-KY| is a smooth divisor with D^2=d , then X=Y\D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y=P^2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type I_d fiber appearing as a singular fiber in a rational elliptic surface π: Y →P^1.

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