Topics in Convexity: Theory of Valuations.
Here is a brief overview of the course "Topics in Convexity: theory of valuations.".
The central notion discussed in the course is the notion of valuaiton
on convex sets. A valuation is an additive functional on the family
of convex compact subsets of a finite dimensional real vector space. The basic examples of valuations
are Lebesgue measure and the Euler characteristic. More examples include
mixed volumes (which will be discussed in the course in some detail).
Syllabus of the course:
Prerequisites: introduction to functional analysis on the level of
the Hahn-Banach theorem, the Banach inverse theorem, basics of the theory of generalized
functions (distributions). Some familiarity with manifolds, differential
forms, and vector bundles would be useful, but not strictly necessary.