Variants of the method of Chabauty and Coleman to compute rational points on certain curves

The method of Chabauty and Coleman is one of the most powerful methods to determine rational points on curves. In this talk, we will introduce this method and present some examples. However, this method does not apply to all curves, only to those satisfying a certain rank condition. For curves that satisfy a stronger condition, we can apply a variant of this method called symmetric Chabauty to determine points on curves defined over number fields and its applications. We will discuss the need to extend the method to the curves that do not satisfy the rank condition and explain how to extend the method to this case.