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October 07, 2021
Thursday, October 07
Analysis Seminar
Time: 10:30
Speaker: Blake Boudreaux (Western)
Title: "Weighted Bergman Kernels on Domains in Cn"
Room: MC 108

Abstract: Given a domain ΩCn, the space of square-integrable holomorphic functions on Ω is a Hilbert space with the standard inner product. This space is denoted by L2h(Ω) and is known as the Bergman space of Ω. It can be shown that the evaluation functionals Ez:L2h(Ω)C given by Ez(f)=f(z) are continuous on L2h(Ω), and hence via the Riesz representation theorem there exists a K(,z)L2h(Ω) that reproduces square-integrable holomorphic functions on Ω. This function (on Ω×Ω) is known as the Bergman kernel of Ω, and has had a profound impact on the theory of holomorphic functions of several complex variables. This theory can also be generalized to weighted L2-spaces, given that the weight function is sufficiently "nice". This will be a mostly expository talk on Bergman kernel, with an emphasis on weighted Bergman kernels. Time allowing I will sketch some work I have done regarding the zeroes of weighted Bergman kernels.