Effective methods for Diophantine problems

This talk will be a survey of recent effective methods for solving a variety of Diophantine problems, both classical and modern. In particular, we will discuss current progress in solving Diophantine equations corresponding to curves of positive genus. We will emphasize results based upon techniques from Diophantine approximation such as the hypergeometric method of Thue and Siegel and lower bounds for linear forms in logarithms of algebraic numbers. If time permits, we will mention approaches for curves of genus > = 2, via the method of Coleman-Chabauty, and also results for Fermat-like Diophantine equations, `a la Wiles.