Mathematics

Dr. Manjunath Krishnapur
IISc Bangaluru

Abstract: Consider the random entire function f(z)=\\sum_{n\\ge 0}a_nz^n/\\sqrt{n!}, where a_n are independent and identically distributed complex Gaussian random variables. This particular random function has the remarkable property that its zeros have a translation-invariant distribution on the plane. An analogous random function on the disk is f(z)=a_0 + a_1 z + a_2 z^2 + ... whose zeros are invariant in distribution under  automorphisms of the unit disk. We shall give an overview of some of very interesting properties of the zero sets of these functions. The lecture should be accessible to anyone with some familiarity with basic probability and basic complex analysis.