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DTSTART;TZID=Asia/Seoul:20230419T160000
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SUMMARY:Jihun Yum\, Stochastic Bergman Geometry
DESCRIPTION:    Speaker\n\n\nJihun Yum\nIBS-CCG\n\n\n\n\n\n\nFor a bounded domain Ω in Cn\, let P(Ω) be the set of all (real) probability distributions on Ω. Then\, in general\, P(Ω) becomes an infinite-dimensional smooth manifold and it always admit a natural Riemannian pseudo-metric\, called the Fisher information metric\, on P(Ω). Information geometry studies a finite-dimensional submanifold M\, which is called a statistical model\, in P(Ω) using geometric concepts such as Riemannian metric\, distance\, connection\, and curvature\, to better understand the properties of statistical models M and provide insights into the behavior of learning algorithms and optimization methods. \nIn this talk\, we first introduce a map Φ : Ω → P(Ω) and prove that the pull-back of the Fisher information metric on P(Ω) is exactly same as the Bergman metric of Ω. This map provides a completely new perspective that allows us to view Bergman geometry from a stochastical viewpoint. We will discuss the following 4 things. \n1. The relation between Φ and the Kobayashi map ι : Ω → CP∞. \n2. A Stochastic formula for the holomorphic sectional curvature of the Bergman metric. \n3. A Stochastic condition for injectivity of a proper holomorphic surjective map between two bounded domains. \n4. The central limit theorem on Ω. \nThis is a joint work with Gunhee Cho at UC Santa Barbara University.
URL:https://ccg.ibs.re.kr/event/2023-04-19/
LOCATION:B266\, IBS\, Korea\, Republic of
CATEGORIES:Several Complex Variables Seminar
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