Trait evolution is frequently understood as a gradual process, in which diverging species accumulate differences over the course of millions of years. However, the fossil record contains evidence of short periods of rapid evolution, followed by very long periods of stasis. This mode of evolution is referred to as punctuated evolution. We present a method which uses trait measurements from present-day species and the topology of their evolutionary tree to localise the positions of these shifts, their typical size, and the typical strength of the associated bounded evolution. We then present evidence of pulsed evolution in the birdsong of East African sunirds, and describe its effects on their speciation. Time permitting, we then switch the discussion from inference over long time scales to modelling over small length scales and present a stochastic model for a molecular motor stepping along a dynamic system of microtubules. This form of active transport is responsible for the delivery of various cell components to different locations and is critical in making a cell function correctly. Using these models, we investigate how the preferred stepping direction of the motor quantitatively and qualitatively affects the cargo delivery. Based on our findings, we present a new characterization of the different motor families, and report new experimental findings which were inspired by this classification.

Gleb completed his PhD in math at the University of Arizona. He then spent two years as a postdoc in the School of Maths at the University of Edinburgh. He's interested in applications of probability and statistics in areas such as evolution, intracellular transport, and particle systems.