By Jean-Luc Brylinski (auth.), Israel M. Gelfand, Lawrence Corwin, James Lepowsky (eds.)

This Seminar begun in Moscow in November 1943 and has endured with no interruption as much as the current. we're chuffed that with this vol­ ume, Birkhiiuser has began to submit papers of talks from the Seminar. It was once, regrettably, tough to arrange their e-book prior to 1990. on account that 1990, lots of the talks have taken position at Rutgers college in New Brunswick, New Jersey. Parallel seminars have been additionally held in Moscow, and through July, 1992, at IRES in Bures-sur-Yvette, France. audio system have been invited to post papers of their personal variety, and to difficult on what they mentioned within the Seminar. we are hoping that readers will locate the variety of types beautiful, and realize that to some degree this displays the range of types in a mathematical society. The central objective was once to have fascinating talks, whether the subject used to be now not specially renowned on the time. The papers indexed within the desk of Contents mirror a few of the wealthy number of principles awarded within the Seminar. now not all of the audio system publish­ ted papers. one of the fascinating talks that encouraged the seminar in a huge manner, allow us to point out, for instance, that of R. Langlands on consistent with­ colation conception and people of J. Conway and J. McKay on sporadic teams. moreover, there have been many extemporaneous talks in addition to brief discus­ sions.

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1984. [8] Gelfand, I. , Graev M. 1 and Vilenkin N. , Generalized Jilunctions, vol. Y. 1966. , Graev M. , Differential forms and integral geometry, Funct. Analysis and its Applications, 3, (1969) 101-114. [10] Gelfand, I. , Graev M. , Integral geometry on k-dimensional planes, Funct. Analysis and its Applications, 1, (1967) 14-27. , Radon Transform (to appear). , Hilbert integrals, singular integrals and Radon transform, Acta Math. , Some problems in integral geometry and some related problems of microlocal analysis, Amer.

3p+1, Xn = al n Xl + ... 3n . )(n-p) C H;t,p. We want to note that all (n - k)-planes will be oriented if we fix the orientation of the space of variables x' = (Xl, ... , xp). Gelfand and Smimov 44 We shall rewrite the previous equation in the form x" = Ax' +B, where A is a p x (n - p) matrix of aij, B is the column of {3i and x' is the column of (Xl, . Xp)T. Here X = (X', x"). Proposition 3. Let (I(Xj vI, ... , Vk) be an odd Crofton density defined in R n. l( ()d( be its dual measure. l(A, B) the dual function.

So we have a fiber bundle Hn,n-k --+ Gn,n-k where Gn,n-k is the Grassman manifold of (n - k)-dimensional subspaces in Rn. The fiber over the point l E Gn,n-k is canonically isomorphic to the k-dimensional space R n Il. Definition 1. Let Mk be a smooth k-dimensional manifold in Rn. To every manifold we co7Tespond a function on Hn,n-k which we call a characteristic Crofton function and which we denote by CrofM " ((), where (E Hn,n-k. This function is defined by the equality CrofM ,,(() = number of the intersection points of the plane (with Mk.

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