Exploring complex energy landscape is a challenging
issue in many applications. Besides locating equilibrium
states, it is often also important to identify the
transition states given by saddle points. In this talk,
we will discuss the mathematics and algorithms, in
particular, the shrinking dimer dynamics, developed to
compute transition states. Some applications will be
considered, including the study of critical nuclei
morphology in solid state transformations, optimal
photonic crystal design and the generalized Thomson problem.