**PhD**

My thesis was on the first order theory of the structure M=(R, +,.,<, (Algebraic real numbers), (the integer powers of two)). This structure is interesting because it has a discrete subgroup as well as a dense codense subset of real numbers in it. The two structures (R,+,.,<, (the integer powers of two)) and (R, +,.,<,(real Algebraic numbers)) had been previously studied by van den Dries and my work merges these two structures and establishes the properties of the obtained structure. We have proved the completeness of the theory T which axiomatises the structure M and have proved quantifier elimination for T in the presence of predicates for certain existential formulas. We described the definable sets and types in models of T and showed that the definable sets in M follow a similar pattern. One particular and important result is that this structure is not subject to the Gödel phenomenon (the structure (Z,+,.,<) is not interpretable in it). In the final chapter we also proved that T is a dependent theory (or it has Not the Independence Property).

I did my viva on 23 Jul 2013 and my examiners were Prof. Boris Zilber (Oxford) and Dr. Gareth Jones (Manchester).

The first order theory of a dense pair and a discrete group.

**MSc**

I was working on Domain theory. Domain theory is the link between topology and theoretical computer science. The aim of my thesis was to provide a domain theoretic and computability framework for the cone-metric spaces. The title of the thesis was ‘cone and quasi-metric spaces via domain theory’.

fazAhAye metrike maxruti va shebhe metrik az didgAhe nazariyeye dAmane.