Abstract: Building on her seminal work regarding moduli space volumes for Riemann surfaces, Mirzakhani also calculated expected values for various geometric functions on moduli space. Notably, she examined the expected Cheeger constant, the injectivity radius at a random point, and the statistical distribution of different types of curves on surfaces of large genus. We will review several of Mirzakhani’s key results, which collectively offer insights into the geometry of random surfaces in high genus. Following this, we will explore some extensions of her findings in the context of translation surfaces.

Abstract:

(Based on work by recent MSc Thesis student, George Shillcock)

Understanding the capacity of pathogens to cause severe disease is of fundamental importance to human health and preserving biodiversity. Many of those pathogens are not only transmitted horizontally between unrelated hosts but also vertically between parents and their progeny. It is widely accepted that vertical transmission leads to the evolution of less virulent pathogens, but this idea stems from research that neglects the evolutionary response of hosts. Here, we use a game-theory model of coevolution between pathogen and host to show that vertical transmission does not always lead to more benign pathogens. We highlight scenarios in which vertical transmission results in pathogens exhibiting more virulence. However, we also predict that more benign outcomes are still possible (a) when generating new horizontal infections inflicts too much damage on hosts, (b) when clearing an infection is too costly for the host, and (c) when vertical transmission is promoted by a greater growth rate of the host population. Though our work offers a new perspective on the role of vertical transmission in pathogen–host systems, it does agree with previous experimental work.