Hodge-Riemann relations for valuations and geometric inequalities

J. Kotrbatý has recently conjectured and proved in several special cases 
an analogue for valuations of the Hodge-Riemann relations. As first 
observed by S. Alesker, these results directly imply both classical and 
new geometric inequalities between the mixed volumes of convex bodies. 
In this talk we report on these developments. In particular, we present 
results from a joint paper with J. Kotrbatý on mixed Hodge-Riemann 
relations and the geometric inequalities they imply.