Last week, equivariant cohomology for $G$-spaces was defined. In this talk, we will define equivariant cohomology in the setting of smooth manifolds under Lie group action. We will first construct an algebraic model for the universal $G$-bundle (called the Weil algebra). Using this we shall define an equivariant version of de Rham complex. Lastly, we will work out the Weil algebra for circle actions.

Meeting ID: 997 4840 9440 Passcode: 911104

TBA

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.