The tasks a Mathematician is expected to perform include:

Solving mathematical problems by using known principles or by evolving new principles
Evaluating new techniques and applying mathematical principles
Developing new methods of analysis for solving various economic and scientific problems
Developing and improving the existing principles
Formulating own mathematical tools and ways for applying to abstract theoretical problems
Working on problems of mathematical nature in other branches of science such as mechanics, hydrodynamics, communication, power transmission, thermodynamics, etc.