An algebraic approach to inequalities in convex geometry

The classical isoperimetric inequality and more generally the Alexandrov-Fenchel inequality are among the most fundamental results in convex geometry; in fact, their significance extends far beyond convexity. In our talk, a conjectural broad generalization of these inequalities will be formulated in the language of Alesker's algebraic theory of valuations on convex bodies. We will also report on recent progress on this conjecture, obtained jointly with Thomas Wannerer and using the representation of valuations by differential forms as well as methods of harmonic analysis.