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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 $\mathbb{C}^n$"
Room: MC 108

Abstract: Given a domain $\Omega\subseteq\mathbb{C}^n$, the space of square-integrable holomorphic functions on $\Omega$ is a Hilbert space with the standard inner product. This space is denoted by $L^2_h(\Omega)$ and is known as the Bergman space of $\Omega$. It can be shown that the evaluation functionals $E_z:L^2_h(\Omega)\to\mathbb{C}$ given by $E_z(f)=f(z)$ are continuous on $L^2_h(\Omega)$, and hence via the Riesz representation theorem there exists a $K(\,\cdot\,,z)\in L^2_h(\Omega)$ that reproduces square-integrable holomorphic functions on $\Omega$. This function (on $\Omega\times\Omega$) is known as the Bergman kernel of $\Omega$, and has had a profound impact on the theory of holomorphic functions of several complex variables. This theory can also be generalized to weighted $L^2$-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.