A boy standing on a rotating table with heavy spheres in his hands, suddenly brings his hands close to his body. The angular velocity of the table

Since, there is no external force, angular momentum of this system is conserved and when angular momentum is constant, angular velocity is inversely proportional to moment of inertia. 

When the boy brings his hands close to his body, he is reducing his moment of inertia. By reducing his moment of inertia,  the angular velocity of the table will increase.