In this talk, we shall consider equivariant subproduct system of Hilbert spaces and their Toeplitz and Cuntz–Pimsner algebras. We will provide results about their topological invariants through K(K)-theory. More specifically, we will show that the Toeplitz algebra of the subproduct system of an SU(2)-representation is equivariantly KKequivalent to the algebra of complex numbers so that the (K)K- theory groups of the Cuntz–Pimsner algebra can be effectively computed using a Gysin exact sequence involving an analogue of the Euler class of a sphere bundle. Finally, we will discuss why and how C*-algebras in this class satisfy KK-theoretic Poincaré duality.[-]

Two important examples of the determinantal point processes associated with the Hilbert spaces of holomorphic functions are the Ginibre point process and the set of zeros of the Gaussian Analytic Functions on the unit disk. In this talk, I will talk such class of determinantal point processes in greater generality. The main topics concerned are the equivalence of the reduced Palm measures and the quasi-invariance of these point processes under certain natural group action of the group of compactly supported diffeomorphisms of the phase space. This talk is based partly on the joint works with Alexander I. Bufetov and partly on a more recent joint work with Alexander I. Bufetov and Shilei Fan.[-]

Generalizing results of Rossi and Vergne for the holomorphic discrete series on symmetric domains, on the one hand, and of Chailuek and Hall for Toeplitz operators on the ball, on the other hand, we establish existence of analytic continuation of weighted Bergman spaces, in the weight (Wallach) parameter, as well as of the associated Toeplitz operators (with sufficiently nice symbols), on any smoothly bounded strictly pseudoconvex domain. Still further extension to Sobolev spaces of holomorphic functions is likewise treated.[-]