International Congress of Mathematicians

ICAl 2002 • Vol. II • 757-764Algebraic Structures on Valuations,Their Properties and ApplicationsSemyon Alesker*AbstractWe describe various structures of algebraic nature on the space of continuousvaluations on convex sets, their properties (like versions of Poincaréduality and hard Lefschetz theorem), and their relations and applications tointegral geometry.2000 Mathematics Subject Classification: 46, 47.Keywords and Phrases: Valuations, Convex sets, Kinematic formulas, ReductiveLie group.0. IntroductionThe theory of continuous valuations on convex sets generalizes, in a sense, boththe measure theory and the theory of the Euler characteristic. Roughly speaking oneshould think of a continuous valuation on a real linear space V as a finite additivemeasure on a class of compact nice subsets of V (say piecewise smooth submanifoldswith corners) which satisfy the following additional property (instead of the usualsigma-additivity): the restriction of to the subclass of convex compact domainswith smooth boundary extends by continuity to the class /C(V) of all convex compactsubsets of V. Here the continuity is understood in the sense of the Hausdorff metricon /C(V). Remind that the Hausdorff metric dn on /C(V) depends on the choiceof the Euclidean metric on V and it is defined as follows: dn(A,B) := infje >0|A c (B) e and B C (A) e }, where (U) e denotes the e-neighborhood of a set U.This condition of continuity turns out to be very strong restriction and has a lot ofconsequences on purely algebraic level. These properties will be discussed in thispaper. The simplest examples of such valuations are any smooth measure on V andthe Euler characteristic. Also it turns out that one of the main tools used recentlyin investigations of valuations is the representation theory of real reductive groupsand the Beilinson-Bernstein theory of D-modules.Now let us give the formal definition of valuation.* Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel. E-mail:semyon@post.tau.ac.il