In this thesis, we will be studying Lie groups and their connection to certain orthogonal polynomials. We will look into the classical Krawtchouk, Meixner and Laguerre polynomials, and the multivariate Krawtchouk and Meixner polynomials as defined by Iliev. Using representations of the Lie groups SU(2) and SU(1,1), it will be shown that the three classical polynomials can be described as matrix coefficients of the representations. Using this connection of the polynomials to Lie groups, we derive various properties of the polynomials from the unitarity of the representation and the associated Lie algebra representation. Next, the representations are generalised to higher dimensional spaces. Doing so, a new connection is shown between the Lie group SU(d+1) and the multivariate Krawtchouk polynomials, extending the known theory for the univariate polynomials. Another new result that will be established is the connection between the multivariate Meixner polynomials and Lie theory. This will be done by defining a representation of SU(1,d) in the Bergman space of the d-dimensional unit ball. Similar as for the univariate polynomials, we will derive the orthogonality, recurrence relations and difference equations from the associated Lie theory.

Stationary sets are important tools in proofs of properties in sets of uncountable cardinality. In this thesis we look at mapping properties between stationary sets. First, the theory necessary for the construction and evaluation of stationary sets is made. That is the theory of ordinal, cardinal and regular cardinal numbers is build up from the level of knowledge of a mathematics student. Two important theorems for stationary sets, Fodor's theorem and the theorem of Ulam and Solovay, are proven. Next mapping properties of stationary subsets of a regular cardinal $\kappa$ under measurable functions is looked into. With these properties we construct a necessary condition for the existence of homeomorphisms between stationary sets; they may only differ on a non-stationary set. Lastly the amount of stationary subsets of a regular cardinal k without a homeomorphism between them is estimated as the cardinality of the power set of k. We find that there are 2 to the power k topologically incomparable subsets of k.