Given a group G and a manifold M, can one describe all the actions of G on M? This is a basic, natural question in geometric topology, but also a very difficult one -- even in the case where M is 1-dimensional, and G is a familiar, finitely generated group.  
This talk will introduce the theory of groups acting on 1-manifolds, through the study of orderable groups.  I will describe some connections between this theory and themes in topology and dynamics (like rigidity and foliation theory ), some current open problems, and indicate new approaches coming from recent joint work with C. Rivas.

This talk introduces one of surprising empirical regularities observed in economics: Pareto distributions are everywhere. Wealth and income, the size of cities and firms, stock market returns, to list but a few, are all known to follow a Pareto distribution. I first highlight key empirical facts and describe some economic theories that have been proposed to explain the regularity. To be more specific, I will put emphasis on inequality in income distributions. A simple mechanism as well as more complex random growth models that give rise to Pareto distributions will be discussed to explore dynamics of income inequality.

Gross and Zagier made a conjecture on the algebraicity of values of certain `higher' Greens functions at special points. Mellit proved a few cases by linking it to the existence of certain higher Chow cycles. Viazovska proved a few cases by linking it to Borcherds lifts of modular forms. We formulate a conjecture linking modular forms and higher Chow cycles which relates the two approaches and also describe a construction of higher Chow cycles which allows us to prove special cases of the Gross-Zagier conjecture as well as provide evidence for our conjecture. 

Recently, there has been considerable interest in both inference and predictions for compartmental epidemic models on multiple physical scales. For instance, one could be interested in analyzing response of immune system to infection within a single host or in  describing infectuous interactions in a population of hosts. Both viral invasions and global pandemics are often described by similar mathematical constructs known as SIR models. In this talk I will review some basic concepts related to such models across scales and present a simple unifying framework that allows to conceptually connect both deterministic (e.g., population level) and stochastic (e.g., molecular level) SIR models with the help of tools of statistical theory of survival analysis.