**Thesis Type:** Doctorate

**Institution Of The Thesis:** Orta Doğu Teknik Üniversitesi, Faculty of Arts and Sciences, Department of Mathematics, Turkey

**Approval Date:** 2013

**Student:** NESLİHAN SİPAHİ

**Supervisor: **MOHAN LAL BHUPAL

In this thesis, our main objects of interest are piecewise polynomial functions (splines). For a polyhedral complex $\Delta$ in $\mathbb{\R}^n$, $C^{r}(\Delta)$ denotes the set of piecewise polynomial functions defined on $\Delta$. Determining the dimension of the space of splines with polynomials having degree at most $k$, denoted by $C^r_k(\Delta)$, is an important problem, which has many applications. In this thesis, we first give an exposition on splines and introduce different algebraic geometric methods used to compute the dimension of splines both on polyhedral and simplicial complexes. Then we generalize the important result of Mcdonald and Schenck \cite{McdSch} on planar splines on a polyhedral complex. Also, by using the method in \cite{GeraSch}, we make generalizations on the dimension of the spaces of splines on simplicial complexes in dimension three. This generalizaton includes simplicial complexes having no interior points, and octahedrons with one interior point. In the latter case, we make some generalizations by considering the number of linearly independent interior planes.