We'll survey some of the ways that hyperbolic groups have been studied
using analysis on their boundaries at infinity.
I will give a brief introduction to the Steenrod squares and move on to show some applications of them in Topology and Geometry.
After a quick-and-dirty introduction to nonstandard analysis, we will
define the asymptotic cones of a metric space and we will play around
with nonstandard tools to show some results about them.
For example, we will hopefully prove that any separable asymptotic cone
is proper and we will classify real trees appearing as asymptotic cones
of groups.
We will state a theorem of Shouhei Ma (2008) relating the Cusps of the Kaehler moduli space to the set of Fourier--Mukai partners of a K3 surface. Then we explain the relationship to the Bridgeland stability manifold and comment on our work relating stability conditions "near" to a cusp to the associated Fourier--Mukai partner.
A brief survey of the above.