Diophantine Approximations, Diophantine Equations, Transcendence and Applications

This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c > 0 depending only on α such that | α− p q |> c qn for all rational numbers p q with q > 0. This inequality is from diophantine approximations. Any non-trivial improvement of this inequality shows that certain class of diophantine equations, known as Thue equations, has only finitely many integral solutions. Also, the above inequality can be applied to establish the transcendence of