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# Jihun Yum, Stochastic Bergman Geometry

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

For a bounded domain Ω in **C**^{n}, 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.