All continuous linearly intertwining symmetric Blaschke valuations on convex bodies are completely classified. It is shown that there is a unique non-trivial such valuation. On symmetric bodies, this valuation is the curvature image operator.

The proof by Hironaka [5] of resolution of singularities of algebraic varieties over fields of characteristic zero raised the problem of classifying singularities by looking at the resolution process. Thus, equisingularity of plane curve singularities was introduced and developed by Zariski in [15] by showing that the combinatorics of the resolution processes is equivalent data to Puiseux invar...

In sequential logic there is an order in which the atomic propositions in an expression are evaluated. This order allows the same atomic proposition to have different values depending on which atomic propositions have already been evaluated. In the sequential propositional logic introduced by Bergstra and Ponse in [5], such valuations are called “reactive” valuations, in contrast to “static” va...

It is known that the basic tensor valuations which, by a result of S. Alesker, span the vector space of tensor-valued, continuous, isometry covariant valuations on convex bodies, are not linearly independent. P. McMullen has discovered linear dependences between these basic valuations and has implicitly raised the question as to whether these are essentially the only ones. The present paper pro...

We consider allocation rules that choose both an outcome and transfers, based on the agents’ reported valuations of the outcomes. Under a given allocation rule, a bribing situation exists when agent j could pay agent i to misreport his valuations, resulting in a net gain to both agents. A rule is bribe-proof if such opportunities never arise. The central result is that when a bribe-proof rule i...

We continue studying the properties of the multiplicative structure on valuations. We prove a new version of the hard Lefschetz theorem for even translation invariant continuous valuations, and discuss problems of integral geometry staying behind these properties. Then we formulate a conjectural analogue of this result for odd valuations.

We develop a valuation theory for generalized polygons similar to the existing theory for dense near polygons. This valuation theory has applications for the study and classification of generalized polygons that have full subpolygons as subgeometries.

Centroid and difference bodies define SL(n) equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of SL(n) equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of SL(n) contravariant Minkowski valuations and of Lp-Minkowski valuations. 2000 AMS subject c...

The topic of this paper is the extension of the basic facts of valuation theory to noncommutative systems. The purpose of this generalization is twofold. First, the theory of valuations with commutative groups of values is placed in the framework of the theory of /-groups, and secondly the general theory leads to the construction of a new class of infinite division algebras. These division alge...

Part I of the course focused on special cases where we can achieve all three properties — gross substitutes (GS) valuations and special cases thereof. In Part II of the course, we focused on more general valuation classes in which the exact version of the second property is already incompatible with the third (assuming P 6= NP ). We focused further on special cases where, ignoring incentive con...