Title:   Local connectivity of the Julia sets of holomorphic maps with bounded type Siegel disks
Speaker:   Fei Yang [Nanjing University]

Abstract:   Let f be a holomorphic map containing an irrationally indifferent fixed point z0. If f is locally linearizable at z0, then the maximal linearizable domain containing z0 is called the Siegel disk of f centered at z0. The topology of the boundaries of Siegel disks has been studied extensively in past 3 decades. This was motivated by the prediction of Douady and Sullivan that the Siegel disk of every non-linear rational map is a Jordan domain.

For the topology of whole Julia sets of holomorphic maps with Siegel disks, the results appear less. Petersen proved that the quadratic Julia sets with bounded type Siegel disks are locally connected. Later Yampolsky proved the same result by an alternative method based on the existence of complex bound of unicritical circle maps. A big progress was made by Petersen and Zakeri in 2004. They proved that for almost all rotation number, the quadratic Julia sets with Siegel disks are locally connected. Recently J. Yang proved a striking result that the Julia set of any polynomial (assumed to be connected) is locally connected at the boundary points of their bounded type Siegel disks.

In this talk we prove that a long iteration of a class of quasi-Blaschke models has certain expanding property near the unit circle. This leads us to prove the local connectivity of the Julia sets of a number of rational maps and transcendental entire functions with bounded type Siegel disks. This is a joint work with S. Wang, G. Zhang and Y. Zhang.
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For more information please visit: http://scgp.stonybrook.edu/archives/29359

Title:   Family Floer program and non-archimedean SYZ mirror construction
Speaker:   Hang Yuan [Stony Brook University]

Abstract:   Given a Lagrangian fibration, we provide a natural construction of a mirror Landau-Ginzburg model consisting of a rigid analytic space, a superpotential function, and a dual fibration based on Fukaya’s family Floer theory. The mirror in the B-side is constructed by the counts of holomorphic disks in the A-side together with the non-archimedean analysis and the homological algebra of the A infinity structures. It fits well with the SYZ dual fibration picture and explains the quantum/instanton corrections and the wall crossing phenomenon. Instead of a special Lagrangian fibration, we only need to assume a weaker semipositive Lagrangian fibration to carry out the non-archimedean SYZ mirror reconstruction.
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Title:   The Nirenberg Problem for Conical Singularities
Speaker:   Lisandra Hernandez-Vazquez [Stony Brook University]

Abstract:   We propose a new approach to the question of prescribing Gaussian curvature on the 2-sphere with at least three conical singularities and angles less than $2\pi$, the main result being sufficient conditions for a positive function of class at least $C^2$ to be the Gaussian curvature of such a conformal conical metric on the round sphere. Our methods particularly differ from the variational approach in that they don’t rely on the Moser-Trudinger inequality. Along the way, we also prove a general precompactness theorem for compact Riemann surfaces with at least three conical singularities and angles less than $2\pi$.
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