By Zannier, Umberto (ed.)

Contributions by way of numerous authors reminiscent of Michael G. Cowling.- Joseph A. Wolf.- Gisbert Wustholz and David Mumford

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Angew. Math. 354 (1984), 164–174. ¨ , Zum Periodenproblem, Invent. Math. 78 (1984), [10] G. W USTHOLZ 381–391. [11] Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen, Annals of Math. 129 (1989), 501–517. ¨ , On Leibniz’ conjecture, periods and motives, in [12] G. W USTHOLZ preparation. The geometry and curvature of shape spaces David Mumford The idea that the set of all smooth submanifolds of a ﬁxed ambient ﬁnite dimensional differentiable manifold forms a manifold in its own right, albeit one of inﬁnite dimension, goes back to Riemann.

This uses the topological group structure and the rotation–invariant measure on S. One has a similar situation when the compact group S is replaced by a ﬁnite dimensional real vector space V . Let V ∗ denote its linear dual space. If f ∈ L 1 (V ) ∩ L 2 (V ) the Fourier inversion formula is f (x) = 1 m/2 2π V∗ f (ξ )ei x·ξ dξ where the Fourier transform is f (ξ ) = 1 m/2 2π f (x)e−i x·ξ dx = f, χξ V L 2 (V ) . 21 Classical analysis and nilpotent Lie groups Again, f is a linear combination (this time it is a continuous linear combination) of the unitary characters χξ (x) = ei x·ξ on V , and the coefﬁcients of the linear combination are given by the Fourier transform f .

The Integral Conjecture gives a precise geometric description of this set. We denote by D the reduced polar divisor of ξ and by U the Zariski open set X \ D. Let ι : H → H1 (U, Z) be a mixed Hodge substructure of the mixed Hodge structure H1 (U, Z) and ι∨ dual to ι. Then H ⊥ = ker ι∨ is a mixed Hodge substructure of H 1 (U, Z) = H1 (U, Z)∨ . The mixed Hodge structure H 1 (U, C) contains H 0 (U, U1 ) by Hodge theory. We 38 Gisbert Wüstholz deﬁne HC = H ⊗Z C and HC ⊥ = H ⊥ ⊗Z C and introduce the spaces H H = PU (Q) × F−1 /F0 HC and V H = H 0 (U, U1 ) × H 1 (U,C) HC ⊥ .