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Jihun Yum, Stochastic Bergman Geometry

April 19, 2023 @ 4:00 pm - 6:00 pm KST

B266, IBS Korea, Republic of


Jihun Yum

For 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.

In 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.

1. The relation between Φ and the Kobayashi map ι : Ω → CP.

2. A Stochastic formula for the holomorphic sectional curvature of the Bergman metric.

3. A Stochastic condition for injectivity of a proper holomorphic surjective map between two bounded domains.

4. The central limit theorem on Ω.

This is a joint work with Gunhee Cho at UC Santa Barbara University.


April 19, 2023
4:00 pm - 6:00 pm KST
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IBS Korea, Republic of


Sungyeon Kim
IBS 복소기하학연구단 Center for Complex Geometry
기초과학연구원 복소기하학연구단
대전 유성구 엑스포로 55 (우) 34126
IBS Center for Complex Geometry
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
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