Rohini Ramadas: The field of complex dynamics began in the early 1900s with the study of iterating polynomials with complex coefficients. It gained momentum in the 1980s with important results on the structure of the Mandelbrot set by Douady-Hubbard and others, and connections established by Thurston, Sullivan and others with surface topology and hyperbolic 3-manifolds. The last decade has seen many breakthroughs achieved via new tools from number theory, measure theory and algebraic geometry. I will discuss some of my results, proved via degeneration techniques from algebraic geometry. The talk will highlight the rich interplay between topology on one hand and algebraic geometry/number theory on the other hand.