In the first part of this thesis, we prove two density theorems for quadrature domains in Cn ,n≥2. It is shown that quadrature domains are dense in the class of all product domains of the form D×Ωwhere D⊂Cn−1 is a smoothly bounded pseudoconvex domain satisfying Bell’s Condition R and Ω⊂Cis a smoothly bounded domain. It is also shown that quadrature domains are dense in the class of all smoothly bounded complete Hartogs domains in C2. In the second part of this thesis, we study the behaviour of the critical points of the Green’s function when a sequence of domains Dk⊂Rn con-verges to a limiting domain Din the C∞-topology. It is shown that the limit-ing set of the critical points of the Green’s functions Gkfor domains Dk⊂Care the zeroes of the Bergman kernel of D. This generalizes a result of Solynin and Gustafsson, Sebbar.

<p dir="ltr">This thesis presents applications of one-point quadrature domains to encryption and decryption as well as a method for estimating average temperature. In addition, it investigates methods for finding explicit formulas for certain functions and introduces results regarding quadrature domains for harmonic functions and for double quadrature domains. We use properties of quadrature domains to encrypt and decrypt locations in two dimensions. Results by Bell, Gustafsson, and Sylvan are used to encrypt a planar location as a point in a quadrature domain. A decryption method using properties of quadrature domains is then presented to uncover the location. We further demonstrate how to use data from the decryption algorithm to find an explicit formula for the Schwarz function for a one-point area quadrature domain. Given a double quadrature domain, we show that the fixed points within the area and arc length quadrature identities must be the same, but that the orders at each point may differ between these identities. In the realm of harmonic functions, we demonstrate how to uncover a one-point quadrature identity for harmonic functions from the quadrature identity for a simply-connected one-point quadrature domain for holomorphic functions. We use this result to state theorems for the density of one-point quadrature domains for harmonic functions in the realm of smooth domains with $C^{\infty}$-smooth boundary. These density theorems then lead us to discuss applications of quadrature domains for harmonic functions to estimating average temperature. We end by illustrating examples of the encryption process and discussing the building blocks for future work.</p>