Description
The theory of valuations (finitely additive measures) on the finite unions of convex bodies goes back to the classical results about equi-dissectablity of polygons of equal area due to Farkas Bolyai, and the non-equi-dissectability of cubes and regular simplices of the same volume due to Dehn solving Hilbert's third problem.
The talk discusses tensor valued finitely additive measures that are polynomial with respect to translations, and equivariant with respect to a subgroup of $GL(n,\mathbb R)$. The case of $SL(m,\mathbb C)$ equvariant valuations when $n=2m$ and $\mathbb R^n=\mathbb C^m$ is discussed in more detail. Representations of classical grpous on Frechet spaces and on symmetric tensors have an important role in the arguments.