This course will introduce the Einstein equations and describe some mathematical problems which arise in their study. We will focus on areas in which recent progress has been made.

Topics will include the Cauchy problem and the constraint equations. We will describe questions concerning the asymptotic behavior of solutions of the constraint equations and density theorems. These will elucidate the behavior of the energy and linear and angular momentum for asymptotically flat spacetimes. We will describe recent progress on mass/angular momentum inequalities.

We will also discuss questions relating to gravitational energy and positive energy theorems as well as notions of quasilocal mass. Some of the tools here include minimal hypersurface theory, inverse mean curvature flow, and the Dirac operator.

This course will introduce the Einstein equations and describe some mathematical problems which arise in their study. We will focus on areas in which recent progress has been made.

Topics will include the Cauchy problem and the constraint equations. We will describe questions concerning the asymptotic behavior of solutions of the constraint equations and density theorems. These will elucidate the behavior of the energy and linear and angular momentum for asymptotically flat spacetimes. We will describe recent progress on mass/angular momentum inequalities.

We will also discuss questions relating to gravitational energy and positive energy theorems as well as notions of quasilocal mass. Some of the tools here include minimal hypersurface theory, inverse mean curvature flow, and the Dirac operator.

A piecewise expanding unimodal map on the unit interval admits a unique absolutely continuous invariant measure. By Birkhoff's Ergodic Theorem Lebesgue almost every point in the unit interval is typical for this measure, i.e., for each continuous function its time average along the forward orbit of a typical point is equal to its space average over the absolutely continuous invariant measure. In this talk we look at one-parameter families of piecewise expanding unimodal maps and we show that in the generic case there exists a set of full Lebesgue measure in the parameter space such that for each map corresponding to a parameter in this set the turning point is typical for the absolutely continuous invariant measure. This almost sure typicality result in the parameter space also applies to points different from the turning point.