Georgia Institute of Technology College of Sciences

Aim of the talk is to make a survey on some recent results concerning analysis over spaces with Ricci curvature bounded from below. I will show that the heat flow in such setting can be equivalently built either as gradient flow of the natural Dirichlet energy in L^2 or as gradient flow if the relative entropy in the Wasserstein space. I will also show how such identification can lead to interesting analytic and geometric insights on the structures of the spaces themselves. From a collaboration with L.Ambrosio and G.Savare

Micro-organisms are known to respond to certain dissolved chemicalsubstances in their environment by moving preferentially away or towardtheir source in a process called chemotaxis. We study such chemotacticresponses at the population level when the micro-swimmers arehydrodynamically coupled to each-other as well as the chemicalconcentration. We include a chemotactic bias based on the known bacteriarun-and-tumble phenomenon in a kinetic model of motile suspension dynamicsdeveloped recently to study hydrodynamic interactions. The chemicalsubstance can be produced or consumed by the swimmers themselves, as wellas be advected by the fluid flows created by their movement. The linearstability analysis of the system will be discussed, as well as the entropyanalysis. Nonlinear dynamics are investigated using numerical simulationin two dimensions of the full system of equations. We show examples ofaggregation in suspensions of pullers (front-actuated swimmers) anddiscuss how chemotaxis affects the mixing flows in suspensions of pushers(rear-actuated swimmers). Last, I will discuss recent work on numericalsimulations of discrete particle/swimmer suspensions that have achemotactic bias.

"Tail risk" refers to an 'unholy trinity' Fat Tails, Micro Correlations, and Tail Dependence, that confound traditional risk analysis and are very much under-appreciated. The talk illustrates this with some punchy data. Of great interest is the question: when does aggregation amplify tail dependence? I'll show some data and new results. Tail obesity is not well defined mathematically, we have at least three definitions, leptokurtic, regularly varying and subexponential. A measure of tail obesity for finite data sets is proposed, and some theoretical properties explored.

I'll present a simple model of market where the use of (commodity) money naturally arisefrom the agents interaction. I'll introduce the relevant notion of (Nash) equilibrium and discuss itsexistence and properties.